# Library Coq.Arith.PeanoNat

Implementation of NAxiomsSig by nat

Module Nat
<: NAxiomsSig
<: UsualDecidableTypeFull
<: OrderedTypeFull
<: TotalOrder.

Operations over nat are defined in a separate module

Include Coq.Init.Nat.

When including property functors, inline t eq zero one two lt le succ

All operations are well-defined (trivial here since eq is Leibniz)

Definition eq_equiv : Equivalence (@eq nat) := eq_equivalence.
Local Obligation Tactic := simpl_relation.
#[global] Program Instance succ_wd : Proper (eq==>eq) S.
#[global] Program Instance pred_wd : Proper (eq==>eq) pred.
#[global] Program Instance add_wd : Proper (eq==>eq==>eq) plus.
#[global] Program Instance sub_wd : Proper (eq==>eq==>eq) minus.
#[global] Program Instance mul_wd : Proper (eq==>eq==>eq) mult.
#[global] Program Instance pow_wd : Proper (eq==>eq==>eq) pow.
#[global] Program Instance div_wd : Proper (eq==>eq==>eq) div.
#[global] Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
#[global] Program Instance lt_wd : Proper (eq==>eq==>iff) lt.
#[global] Program Instance testbit_wd : Proper (eq==>eq==>eq) testbit.

Bi-directional induction.

Theorem bi_induction :
forall A : nat -> Prop, Proper (eq==>iff) A ->
A 0 -> (forall n : nat, A n <-> A (S n)) -> forall n : nat, A n.

Recursion function

Definition recursion {A} : A -> (nat -> A -> A) -> nat -> A :=
nat_rect (fun _ => A).

#[global] Instance recursion_wd {A} (Aeq : relation A) :
Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) recursion.

Theorem recursion_0 :
forall {A} (a : A) (f : nat -> A -> A), recursion a f 0 = a.

Theorem recursion_succ :
forall {A} (Aeq : relation A) (a : A) (f : nat -> A -> A),
Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
forall n : nat, Aeq (recursion a f (S n)) (f n (recursion a f n)).

## Remaining constants not defined in Coq.Init.Nat

NB: Aliasing le is mandatory, since only a Definition can implement an interface Parameter...

Definition eq := @Logic.eq nat.
Definition le := Peano.le.
Definition lt := Peano.lt.

## Basic specifications : pred add sub mul

Lemma pred_succ n : pred (S n) = n.

Lemma pred_0 : pred 0 = 0.

Lemma one_succ : 1 = S 0.

Lemma two_succ : 2 = S 1.

Lemma add_0_l n : 0 + n = n.

Lemma add_succ_l n m : (S n) + m = S (n + m).

Lemma sub_0_r n : n - 0 = n.

Lemma sub_succ_r n m : n - (S m) = pred (n - m).

Lemma mul_0_l n : 0 * n = 0.

Lemma mul_succ_l n m : S n * m = n * m + m.

Lemma lt_succ_r n m : n < S m <-> n <= m.

## Boolean comparisons

Lemma eqb_eq n m : eqb n m = true <-> n = m.

#[global]
Instance Decidable_eq_nat : forall (x y : nat), Decidable (eq x y) := {
Decidable_spec := Nat.eqb_eq x y
}.

Lemma leb_le n m : (n <=? m) = true <-> n <= m.

#[global]
Instance Decidable_le_nat : forall (x y : nat), Decidable (x <= y) := {
Decidable_spec := Nat.leb_le x y
}.

Lemma ltb_lt n m : (n <? m) = true <-> n < m.

## Decidability of equality over nat.

Lemma eq_dec : forall n m : nat, {n = m} + {n <> m}.

## Ternary comparison

With nat, it would be easier to prove first compare_spec, then the properties below. But then we wouldn't be able to benefit from functor BoolOrderFacts

## Minimum, maximum

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

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

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

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

Some more advanced properties of comparison and orders, including compare_spec and lt_irrefl and lt_eq_cases.

Include BoolOrderFacts.

We can now derive all properties of basic functions and orders, and use these properties for proving the specs of more advanced functions.

## Power

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

Lemma pow_0_r a : a^0 = 1.

Lemma pow_succ_r a b : 0<=b -> a^(S b) = a * a^b.

## Square

Lemma square_spec n : square n = n * n.

## Parity

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

Module Private_Parity.

Lemma Even_0 : Even 0.

Lemma Even_1 : ~ Even 1.

Lemma Even_2 n : Even n <-> Even (S (S n)).

Lemma Odd_0 : ~ Odd 0.

Lemma Odd_1 : Odd 1.

Lemma Odd_2 n : Odd n <-> Odd (S (S n)).

End Private_Parity.
Import Private_Parity.

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

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

## Division

Lemma divmod_spec : forall x y q u, u <= y ->
let (q',u') := divmod x y q u in
x + (S y)*q + (y-u) = (S y)*q' + (y-u') /\ u' <= y.

Lemma div_mod_eq x y : x = y*(x/y) + x mod y.

The y <> 0 hypothesis is needed to fit in NAxiomsSig.
Lemma div_mod x y : y <> 0 -> x = y*(x/y) + x mod y.

Lemma mod_bound_pos x y : 0<=x -> 0<y -> 0 <= x mod y < y.

## Square root

Lemma sqrt_iter_spec : forall k p q r,
q = p+p -> r<=q ->
let s := sqrt_iter k p q r in
s*s <= k + p*p + (q - r) < (S s)*(S s).

Lemma sqrt_specif n : (sqrt n)*(sqrt n) <= n < S (sqrt n) * S (sqrt n).

Definition sqrt_spec a (Ha:0<=a) := sqrt_specif a.

Lemma sqrt_neg a : a<0 -> sqrt a = 0.

## Logarithm

Lemma log2_iter_spec : forall k p q r,
2^(S p) = q + S r -> r < 2^p ->
let s := log2_iter k p q r in
2^s <= k + q < 2^(S s).

Lemma log2_spec n : 0<n ->
2^(log2 n) <= n < 2^(S (log2 n)).

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

## Properties of iter

Lemma iter_swap_gen A B (f:A -> B) (g:A -> A) (h:B -> B) :
(forall a, f (g a) = h (f a)) -> forall n a,
f (iter n g a) = iter n h (f a).

Lemma iter_swap :
forall n (A:Type) (f:A -> A) (x:A),
iter n f (f x) = f (iter n f x).

Lemma iter_succ :
forall n (A:Type) (f:A -> A) (x:A),
iter (S n) f x = f (iter n f x).

Lemma iter_succ_r :
forall n (A:Type) (f:A -> A) (x:A),
iter (S n) f x = iter n f (f x).

forall p q (A:Type) (f:A -> A) (x:A),
iter (p+q) f x = iter p f (iter q f x).

Lemma iter_ind (A:Type) (f:A -> A) (a:A) (P:nat -> A -> Prop) :
P 0 a ->
(forall n a', P n a' -> P (S n) (f a')) ->
forall n, P n (iter n f a).

Lemma iter_rect (A:Type) (f:A -> A) (a:A) (P:nat -> A -> Type) :
P 0 a ->
(forall n a', P n a' -> P (S n) (f a')) ->
forall n, P n (iter n f a).

Lemma iter_invariant :
forall (n:nat) (A:Type) (f:A -> A) (Inv:A -> Prop),
(forall x:A, Inv x -> Inv (f x)) ->
forall x:A, Inv x -> Inv (iter n f x).

## Gcd

Definition divide x y := exists z, y=z*x.
Notation "( x | y )" := (divide x y) (at level 0) : nat_scope.

Lemma gcd_divide : forall a b, (gcd a b | a) /\ (gcd a b | b).

Lemma gcd_divide_l : forall a b, (gcd a b | a).

Lemma gcd_divide_r : forall a b, (gcd a b | b).

Lemma gcd_greatest : forall a b c, (c|a) -> (c|b) -> (c|gcd a b).

Lemma gcd_nonneg a b : 0<=gcd a b.

## Bitwise operations

Lemma div2_double n : div2 (2*n) = n.

Lemma div2_succ_double n : div2 (S (2*n)) = n.

Lemma le_div2 n : div2 (S n) <= n.

Lemma lt_div2 n : 0 < n -> div2 n < n.

Lemma div2_decr a n : a <= S n -> div2 a <= n.

Lemma double_twice : forall n, double n = 2*n.

Definition double_S : forall n, double (S n) = S (S (double n))
:= fun n => add_succ_r (S n) n.

Definition double_add : forall n m, double (n + m) = double n + double m
:= fun n m => add_shuffle1 n m n m.

Lemma testbit_0_l : forall n, testbit 0 n = false.

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

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

Lemma testbit_odd_succ' a n : testbit (2*a+1) (S n) = testbit a n.

Lemma testbit_even_succ' a n : testbit (2*a) (S n) = testbit a n.

Lemma shiftr_specif : forall a n m,
testbit (shiftr a n) m = testbit a (m+n).

Lemma shiftl_specif_high : forall a n 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 div2_bitwise : forall op n a b,
div2 (bitwise op (S n) a b) = bitwise op n (div2 a) (div2 b).

Lemma odd_bitwise : forall op n a b,
odd (bitwise op (S n) a b) = op (odd a) (odd b).

Lemma testbit_bitwise_1 : forall op, (forall b, op false b = false) ->
forall n m a b, a<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m).

Lemma testbit_bitwise_2 : forall op, op false false = false ->
forall n m a b, a<=n -> b<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m).

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

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

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

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

Lemma div2_spec a : div2 a = shiftr a 1.

Aliases with extra dummy hypothesis, to fulfil the interface

Definition testbit_odd_succ a n (_:0<=n) := testbit_odd_succ' a n.
Definition testbit_even_succ a n (_:0<=n) := testbit_even_succ' a n.
Lemma testbit_neg_r a n (H:n<0) : testbit a n = false.

Definition shiftl_spec_high a n m (_:0<=m) := shiftl_specif_high a n m.
Definition shiftr_spec a n m (_:0<=m) := shiftr_specif a n m.

Lemma div_0_r a : a / 0 = 0.

Lemma mod_0_r a : a mod 0 = a.

Properties of advanced functions (pow, sqrt, log2, ...)
Properties of tail-recursive addition and multiplication

Definition Even_Odd_dec n : {Even n} + {Odd n}.

Even (n + m) -> Even n /\ Even m \/ Odd n /\ Odd m.

Odd (n + m) -> Odd n /\ Even m \/ Even n /\ Odd m.

Definition Even_Even_add n m: Even n -> Even m -> Even (n + m).

Definition Odd_add_l n m : Odd n -> Even m -> Odd (n + m).

Definition Odd_add_r n m : Even n -> Odd m -> Odd (n + m).

Definition Odd_Odd_add n m : Odd n -> Odd m -> Even (n + m).

(Odd (n + m) <-> Odd n /\ Even m \/ Even n /\ Odd m) /\
(Even (n + m) <-> Even n /\ Even m \/ Odd n /\ Odd m).

Definition Even_add_Even_inv_r n m : Even (n + m) -> Even n -> Even m.

Definition Even_add_Even_inv_l n m : Even (n + m) -> Even m -> Even n.

Definition Even_add_Odd_inv_r n m : Even (n + m) -> Odd n -> Odd m.

Definition Even_add_Odd_inv_l n m : Even (n + m) -> Odd m -> Odd n.

Definition Odd_add_Even_inv_l n m : Odd (n + m) -> Odd m -> Even n.

Definition Odd_add_Even_inv_r n m : Odd (n + m) -> Odd n -> Even m.

Definition Odd_add_Odd_inv_l n m : Odd (n + m) -> Even m -> Odd n.

Definition Odd_add_Odd_inv_r n m : Odd (n + m) -> Even n -> Odd m.

Definition Even_mul_aux n m :
(Odd (n * m) <-> Odd n /\ Odd m) /\ (Even (n * m) <-> Even n \/ Even m).

Definition Even_mul_l n m : Even n -> Even (n * m).

Definition Even_mul_r n m : Even m -> Even (n * m).

Definition Even_mul_inv_r n m : Even (n * m) -> Odd n -> Even m.

Definition Even_mul_inv_l n m : Even (n * m) -> Odd m -> Even n.

Definition Odd_mul n m : Odd n -> Odd m -> Odd (n * m).

Definition Odd_mul_inv_l n m : Odd (n * m) -> Odd n.

Definition Odd_mul_inv_r n m : Odd (n * m) -> Odd m.

Definition Even_div2 n : Even n -> div2 n = div2 (S n).

Definition Odd_div2 n : Odd n -> S (div2 n) = div2 (S n).

Definition div2_Even n : div2 n = div2 (S n) -> Even n.

Definition div2_Odd n : S (div2 n) = div2 (S n) -> Odd n.

Definition Even_Odd_div2 n :
(Even n <-> div2 n = div2 (S n)) /\ (Odd n <-> S (div2 n) = div2 (S n)).

Definition Even_Odd_double n :
(Even n <-> n = double (div2 n)) /\ (Odd n <-> n = S (double (div2 n))).

Definition Even_double n : Even n -> n = double (div2 n).
Proof proj1 (proj1 (Even_Odd_double n)).

Definition double_Even n : n = double (div2 n) -> Even n.
Proof proj2 (proj1 (Even_Odd_double n)).

Definition Odd_double n : Odd n -> n = S (double (div2 n)).
Proof proj1 (proj2 (Even_Odd_double n)).

Definition double_Odd n : n = S (double (div2 n)) -> Odd n.
Proof proj2 (proj2 (Even_Odd_double n)).

Inductive definition of even and odd
Inductive Even_alt : nat -> Prop :=
| Even_alt_O : Even_alt 0
| Even_alt_S : forall n, Odd_alt n -> Even_alt (S n)
with Odd_alt : nat -> Prop :=
| Odd_alt_S : forall n, Even_alt n -> Odd_alt (S n).

Definition Even_alt_Even : forall n, Even_alt n <-> Even n.

Definition Odd_alt_Odd : forall n, Odd_alt n <-> Odd n.

Scheme Odd_alt_Even_alt_ind := Minimality for Odd_alt Sort Prop
with Even_alt_Odd_alt_ind := Minimality for Even_alt Sort Prop.

Lemma Odd_Even_ind (P Q : nat -> Prop) :
(forall n, Even n -> Q n -> P (S n)) ->
Q 0 -> (forall n, Odd n -> P n -> Q (S n)) -> forall n, Odd n -> P n.

Lemma Even_Odd_ind (P Q : nat -> Prop) :
(forall n, Even n -> Q n -> P (S n)) ->
Q 0 -> (forall n, Odd n -> P n -> Q (S n)) -> forall n, Even n -> Q n.

Scheme Odd_alt_Even_alt_sind := Minimality for Odd_alt Sort SProp
with Even_alt_Odd_alt_sind := Minimality for Even_alt Sort SProp.

Lemma Odd_Even_sind (P Q : nat -> SProp) :
(forall n, Even n -> Q n -> P (S n)) ->
Q 0 -> (forall n, Odd n -> P n -> Q (S n)) -> forall n, Odd n -> P n.

Lemma Even_Odd_sind (P Q : nat -> SProp) :
(forall n, Even n -> Q n -> P (S n)) ->
Q 0 -> (forall n, Odd n -> P n -> Q (S n)) -> forall n, Even n -> Q n.

additional versions of parity predicates in Type useful for eliminating into Type, but still with opaque proofs

Definition EvenT n := { m | n = 2 * m }.
Definition OddT n := { m | n = 2 * m + 1 }.

Lemma EvenT_0 : EvenT 0.

Lemma EvenT_2 n : EvenT n -> EvenT (S (S n)).

Lemma OddT_1 : OddT 1.

Lemma OddT_2 n : OddT n -> OddT (S (S n)).

Lemma EvenT_S_OddT n : EvenT (S n) -> OddT n.

Lemma OddT_S_EvenT n : OddT (S n) -> EvenT n.

Lemma even_EvenT : forall n, even n = true -> EvenT n.

Lemma odd_OddT : forall n, odd n = true -> OddT n.

Lemma EvenT_Even n : EvenT n -> Even n.

Lemma OddT_Odd n : OddT n -> Odd n.

Lemma Even_EvenT n : Even n -> EvenT n.

Lemma Odd_OddT n : Odd n -> OddT n.

Lemma EvenT_even n : EvenT n -> even n = true.

Lemma OddT_odd n : OddT n -> odd n = true.

Lemma EvenT_OddT_dec n : EvenT n + OddT n.

Lemma OddT_EvenT_rect (P Q : nat -> Type) :
(forall n, EvenT n -> Q n -> P (S n)) ->
Q 0 -> (forall n, OddT n -> P n -> Q (S n)) -> forall n, OddT n -> P n.

Lemma EvenT_OddT_rect (P Q : nat -> Type) :
(forall n, EvenT n -> Q n -> P (S n)) ->
Q 0 -> (forall n, OddT n -> P n -> Q (S n)) -> forall n, EvenT n -> Q n.

End Nat.

Re-export notations that should be available even when the Nat module is not imported.

Bind Scope nat_scope with Nat.t nat.

Infix "^" := Nat.pow : nat_scope.
Infix "=?" := Nat.eqb (at level 70) : nat_scope.
Infix "<=?" := Nat.leb (at level 70) : nat_scope.
Infix "<?" := Nat.ltb (at level 70) : nat_scope.
Infix "?=" := Nat.compare (at level 70) : nat_scope.
Infix "/" := Nat.div : nat_scope.
Infix "mod" := Nat.modulo (at level 40, no associativity) : nat_scope.

#[global] Hint Unfold Nat.le : core.
#[global] Hint Unfold Nat.lt : core.

#[local]
Definition lt_n_Sm_le := (fun n m => (proj1 (Nat.lt_succ_r n m))).
Register lt_n_Sm_le as num.nat.lt_n_Sm_le.
#[local]
Definition le_lt_n_Sm := (fun n m => (proj2 (Nat.lt_succ_r n m))).
Register le_lt_n_Sm as num.nat.le_lt_n_Sm.
#[local]
Definition lt_S_n := (fun n m => (proj2 (Nat.succ_lt_mono n m))).
Register lt_S_n as num.nat.lt_S_n.
Register Nat.le_lt_trans as num.nat.le_lt_trans.
#[local]
Definition pred_of_minus := (fun n => eq_sym (Nat.sub_1_r n)).
Register pred_of_minus as num.nat.pred_of_minus.
Register Nat.le_trans as num.nat.le_trans.
Register Nat.nlt_0_r as num.nat.nlt_0_r.

Nat contains an order tactic for natural numbers
Note that Nat.order is domain-agnostic: it will not prove 1<=2 or x<=x+x, but rather things like x<=y -> y<=x -> x=y.

Section TestOrder.
Let test : forall x y, x<=y -> y<=x -> x=y.
End TestOrder.