# Library Coq.Numbers.Natural.SpecViaZ.NSigNAxioms

Require Import ZArith OrdersFacts Nnat NAxioms NSig.

# The interface NSig.NType implies the interface NAxiomsSig

Module NTypeIsNAxioms (Import NN : NType').

Hint Rewrite
spec_0 spec_1 spec_2 spec_succ spec_add spec_mul spec_pred spec_sub
spec_div spec_modulo spec_gcd spec_compare spec_eqb spec_ltb spec_leb
spec_square spec_sqrt spec_log2 spec_max spec_min spec_pow_pos spec_pow_N
spec_pow spec_even spec_odd spec_testbit spec_shiftl spec_shiftr
spec_land spec_lor spec_ldiff spec_lxor spec_div2 spec_of_N
: nsimpl.
Ltac nsimpl := autorewrite with nsimpl.
Ltac ncongruence := unfold eq, to_N; repeat red; intros; nsimpl; congruence.
Ltac zify := unfold eq, lt, le, to_N in *; nsimpl.
Ltac omega_pos n := generalize (spec_pos n); omega with *.

Instance eq_equiv : Equivalence eq.

Program Instance succ_wd : Proper (eq==>eq) succ.
Program Instance pred_wd : Proper (eq==>eq) pred.
Program Instance sub_wd : Proper (eq==>eq==>eq) sub.
Program Instance mul_wd : Proper (eq==>eq==>eq) mul.

Theorem pred_succ : forall n, pred (succ n) == n.

Theorem one_succ : 1 == succ 0.

Theorem two_succ : 2 == succ 1.

Definition N_of_Z z := of_N (Z.to_N z).

Lemma spec_N_of_Z z : (0<=z)%Z -> [N_of_Z z] = z.

Section Induction.

Variable A : NN.t -> Prop.
Hypothesis A_wd : Proper (eq==>iff) A.
Hypothesis A0 : A 0.
Hypothesis AS : forall n, A n <-> A (succ n).

Let B (z : Z) := A (N_of_Z z).

Lemma B0 : B 0.

Lemma BS : forall z : Z, (0 <= z)%Z -> B z -> B (z + 1).

Lemma B_holds : forall z : Z, (0 <= z)%Z -> B z.

Theorem bi_induction : forall n, A n.

End Induction.

Theorem add_0_l : forall n, 0 + n == n.

Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m).

Theorem sub_0_r : forall n, n - 0 == n.

Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m).

Theorem mul_0_l : forall n, 0 * n == 0.

Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m.

Order

Lemma eqb_eq x y : eqb x y = true <-> x == y.

Lemma leb_le x y : leb x y = true <-> x <= y.

Lemma ltb_lt x y : ltb x y = true <-> x < y.

Lemma compare_eq_iff n m : compare n m = Eq <-> n == m.

Lemma compare_lt_iff n m : compare n m = Lt <-> n < m.

Lemma compare_le_iff n m : compare n m <> Gt <-> n <= m.

Lemma compare_antisym n m : compare m n = CompOpp (compare n m).

Include BoolOrderFacts NN NN NN [no inline].

Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare.

Instance eqb_wd : Proper (eq ==> eq ==> Logic.eq) eqb.

Instance ltb_wd : Proper (eq ==> eq ==> Logic.eq) ltb.

Instance leb_wd : Proper (eq ==> eq ==> Logic.eq) leb.

Instance lt_wd : Proper (eq ==> eq ==> iff) lt.

Theorem lt_succ_r : forall n m, n < succ m <-> n <= m.

Theorem min_l : forall n m, n <= m -> min n m == n.

Theorem min_r : forall n m, m <= n -> min n m == m.

Theorem max_l : forall n m, m <= n -> max n m == n.

Theorem max_r : forall n m, n <= m -> max n m == m.

Properties specific to natural numbers, not integers.

Theorem pred_0 : pred 0 == 0.

Power

Program Instance pow_wd : Proper (eq==>eq==>eq) pow.

Lemma pow_0_r : forall a, a^0 == 1.

Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b.

Lemma pow_neg_r : forall a b, b<0 -> a^b == 0.

Lemma pow_pow_N : forall a b, a^b == pow_N a (to_N b).

Lemma pow_N_pow : forall a b, pow_N a b == a^(of_N b).

Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p).

Square

Lemma square_spec n : square n == n * n.

Sqrt

Lemma sqrt_spec : forall n, 0<=n ->
(sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)).

Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0.

Log2

Lemma log2_spec : forall n, 0<n ->
2^(log2 n) <= n /\ n < 2^(succ (log2 n)).

Lemma log2_nonpos : forall n, n<=0 -> log2 n == 0.

Even / Odd

Definition Even n := exists m, n == 2*m.
Definition Odd n := exists m, n == 2*m+1.

Lemma even_spec n : even n = true <-> Even n.

Lemma odd_spec n : odd n = true <-> Odd n.

Div / Mod

Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.

Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b).

Theorem mod_bound_pos : forall a b, 0<=a -> 0<b ->
0 <= modulo a b /\ modulo a b < b.

Gcd

Definition divide n m := exists p, m == p*n.

Lemma spec_divide : forall n m, (n|m) <-> Z.divide [n] [m].

Lemma gcd_divide_l : forall n m, (gcd n m | n).

Lemma gcd_divide_r : forall n m, (gcd n m | m).

Lemma gcd_greatest : forall n m p, (p|n) -> (p|m) -> (p|gcd n m).

Lemma gcd_nonneg : forall n m, 0 <= gcd n m.

Bitwise operations

Program Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit.

Lemma testbit_odd_0 : forall a, testbit (2*a+1) 0 = true.

Lemma testbit_even_0 : forall a, testbit (2*a) 0 = false.

Lemma testbit_odd_succ : forall a n, 0<=n ->
testbit (2*a+1) (succ n) = testbit a n.

Lemma testbit_even_succ : forall a n, 0<=n ->
testbit (2*a) (succ n) = testbit a n.

Lemma testbit_neg_r : forall a n, n<0 -> testbit a n = false.

Lemma shiftr_spec : forall a n m, 0<=m ->
testbit (shiftr a n) m = testbit a (m+n).

Lemma shiftl_spec_high : forall a n m, 0<=m -> n<=m ->
testbit (shiftl a n) m = testbit a (m-n).

Lemma shiftl_spec_low : forall a n m, m<n ->
testbit (shiftl a n) m = false.

Lemma land_spec : forall a b n,
testbit (land a b) n = testbit a n && testbit b n.

Lemma lor_spec : forall a b n,
testbit (lor a b) n = testbit a n || testbit b n.

Lemma ldiff_spec : forall a b n,
testbit (ldiff a b) n = testbit a n && negb (testbit b n).

Lemma lxor_spec : forall a b n,
testbit (lxor a b) n = xorb (testbit a n) (testbit b n).

Lemma div2_spec : forall a, div2 a == shiftr a 1.

Recursion

Definition recursion (A : Type) (a : A) (f : NN.t -> A -> A) (n : NN.t) :=
N.peano_rect (fun _ => A) a (fun n a => f (NN.of_N n) a) (NN.to_N n).

Instance recursion_wd (A : Type) (Aeq : relation A) :
Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) (@recursion A).

Theorem recursion_0 :
forall (A : Type) (a : A) (f : NN.t -> A -> A), recursion a f 0 = a.

Theorem recursion_succ :
forall (A : Type) (Aeq : relation A) (a : A) (f : NN.t -> A -> A),
Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
forall n, Aeq (recursion a f (succ n)) (f n (recursion a f n)).

End NTypeIsNAxioms.

Module NType_NAxioms (NN : NType)
<: NAxiomsSig <: OrderFunctions NN <: HasMinMax NN
:= NN <+ NTypeIsNAxioms.