# Natural numbers

Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.PartA.

Local Open Scope nat.

Definition := nat.

Module Uniqueness.

Lemma helper_A (P:natType) (p0:P 0) (IH: n, P nP(S n))
(f: n, P n) :
weq ( n, f n = nat_rect P p0 IH n)
(f 0=p0 × n, f(S n)=IH n (f n)).
Proof.
intros. simple refine (_,,isweq_iso _ _ _ _).
{ intros h. split.
{ exact (h 0). } { intros. exact (h (S n) @ maponpaths (IH n) (! h n)). } }
{ intros [h0 h'] ?. induction n as [|n' IHn'].
{ exact h0. } { exact (h' n' @ maponpaths (IH n') IHn'). } }
{ simpl. intros h. apply funextsec; intros n; simpl. induction n as [|n IHn].
{ simpl. reflexivity. }
{ simpl. rewrite <- path_assoc. simple refine (_ @ pathscomp0rid _).
rewrite <- maponpathscomp0. rewrite IHn. rewrite pathsinv0l.
simpl. reflexivity. } }
{ intros [h0 h']. apply maponpaths. apply funextsec; intro n; simpl.
rewrite <- path_assoc. rewrite <- maponpathscomp0. rewrite pathsinv0r.
apply pathscomp0rid. }
Defined.

Lemma helper_B (P:natType) (p0:P 0) (IH: n, P nP(S n))
(f: n, P n) :
weq (f = nat_rect P p0 IH)
((f 0=p0) × ( n, f(S n)=IH n (f n))).
Proof.
intros.
exact (weqcomp (weqtoforallpaths _ _ _) (helper_A _ _ _ _)).
Defined.

Lemma helper_C (P:natType) (p0:P 0) (IH: n, P nP(S n)) :
( f: n, P n, f = nat_rect P p0 IH)

( f: n, P n, f 0=p0 × n, f(S n)=IH n (f n)).
Proof.
intros. apply weqfibtototal. intros f. apply helper_B.
Defined.

Lemma hNatRecursionUniq (P:natType) (p0:P 0) (IH: n, P nP(S n)) :
∃! (f: n, P n), f 0=p0 × n, f(S n) = IH n (f n).
Proof.
intros. exact (iscontrweqf (helper_C _ _ _) (iscontrcoconustot _ _)).
Defined.

Lemma helper_D (P:natType) (p0:P 0) (IH: n, P nP(S n)) :
( f: n, P n, (f 0=p0) × ( n, f(S n)=IH n (f n)))

(@hfiber
( (f: n, P n), n, f(S n)=IH n (f n))
(P 0)
(λ fh, pr1 fh 0)
p0).
Proof.
intros. simple refine (make_weq _ (isweq_iso _ _ _ _)).
{ intros [f [h0 h']]. exact ((f,,h'),,h0). }
{ intros [[f h'] h0]. exact (f,,(h0,,h')). }
{ intros [f [h0 h']]. reflexivity. }
{ intros [[f h'] h0]. reflexivity. }
Defined.

Lemma hNatRecursion_weq (P:natType) (IH: n, P nP(S n)) :
weq (total2 (fun f: n, P n n, f(S n)=IH n (f n))) (P 0).
Proof.
intros. (λ f, pr1 f 0). intro p0.
apply (iscontrweqf (helper_D _ _ _)). apply hNatRecursionUniq.
Defined.

End Uniqueness.

Fixpoint nat_dist (m n:nat) : nat :=
match m , n with
| S m, S nnat_dist m n
| 0, nn
| m, 0 ⇒ m end.

Module Discern.

Fixpoint nat_discern (m n:nat) : UU :=
match m , n with
| S m, S nnat_discern m n
| 0, S nempty
| S m, 0 ⇒ empty
| 0, 0 ⇒ unit end.

Goal m n, nat_discern m n nat_discern (S m) (S n).
Proof.
intros ? ? e. exact e.
Defined.

Lemma nat_discern_inj m n : nat_discern (S m) (S n) nat_discern m n.
Proof.
intros e. induction m.
{ induction n. { exact tt. } { simpl in e. exact (fromempty e). } }
{ induction n. { simpl in e. exact (fromempty e). } { simpl in e. exact e. } }
Defined.

Lemma nat_discern_isaprop m n : isaprop (nat_discern m n).
Proof.
revert n; induction m as [|m IHm].
{ intros n. induction n as [|n IHn].
{ apply isapropifcontr. apply iscontrunit. }
{ simpl. apply isapropempty. } }
{ intros n. induction n as [|n IHn].
{ simpl. apply isapropempty. }
{ simpl. apply IHm. } }
Defined.

Lemma nat_discern_unit m : nat_discern m m = unit.
Proof.
induction m as [|m IHm]. { reflexivity. } { simpl. apply IHm. }
Defined.

Lemma nat_discern_iscontr m : iscontr (nat_discern m m).
Proof.
apply iscontraprop1.
{ apply nat_discern_isaprop. }
{ induction m as [|m IHm]. { exact tt. } { simpl. exact IHm. } }
Defined.

Fixpoint helper_A m n : nat_dist m n = 0 nat_discern m n.
Proof.
destruct m as [|m'].
{ destruct n as [|n'].
{ intros _. exact tt. } { simpl. exact (negpathssx0 n'). } }
{ destruct n as [|n'].
{ simpl. exact (negpathssx0 m'). } { simpl. exact (helper_A m' n'). } }
Defined.

Fixpoint helper_B m n : nat_discern m n m = n.
Proof.
destruct m as [|m'].
{ destruct n as [|n'].
{ intros _. reflexivity. } { simpl. exact fromempty. } }
{ destruct n as [|n'].
{ simpl. exact fromempty. }
{ simpl. intro i. assert(b := helper_B _ _ i); clear i.
destruct b. reflexivity. } }
Defined.

Goal m n (e:nat_discern m n), maponpaths S (helper_B m n e) = helper_B (S m) (S n) e.
Proof.
reflexivity.
Defined.

Fixpoint helper_C m n : m = n nat_discern m n.
Proof.
intros e. destruct e.
exact (cast (! nat_discern_unit m) tt).
Defined.

Lemma apSC m n (e:m=n) : helper_C m n e = helper_C (S m) (S n) (maponpaths S e).
Proof.
intros. apply proofirrelevance. apply nat_discern_isaprop.
Defined.

Definition helper_D m n : isweq (helper_B m n).
Proof.
intros. simple refine (isweq_iso _ (helper_C _ _) _ _).
{ intro e. assert(p := ! helper_B _ _ e). destruct p.
apply proofirrelevancecontr. apply nat_discern_iscontr. }
{ intro e. destruct e. induction m as [|m IHm].
{ reflexivity. }
{ exact ( maponpaths (helper_B (S m) (S m)) (! apSC _ _ (idpath m))
@ maponpaths (maponpaths S) IHm). } }
Defined.

Definition E m n : (nat_discern m n) (m = n).
Proof.
intros. exact (make_weq (helper_B _ _) (helper_D _ _)).
Defined.

Definition nat_dist_anti m n : nat_dist m n = 0 m = n.
Proof.
intros i. exact (helper_B _ _ (helper_A _ _ i)).
Defined.

End Discern.

Fixpoint nat_dist_symm m n : nat_dist m n = nat_dist n m.
Proof.
destruct m as [|m'].
{ destruct n as [|n']. { reflexivity. } { simpl. reflexivity. } }
{ destruct n as [|n'].
{ simpl. reflexivity. }
{ simpl. apply nat_dist_symm. } }
Defined.

Fixpoint nat_dist_ge m n : m n nat_dist m n = m-n.
Proof.
induction m as [|m'].
{ induction n as [|n']. { reflexivity. } { intro f. now induction (!natleh0tois0 f). } }
{ induction n as [|n']. { reflexivity. } { exact (nat_dist_ge m' n'). } }
Defined.

Definition nat_dist_0m m : nat_dist 0 m = m.
Proof.
reflexivity.
Defined.

Definition nat_dist_m0 m : nat_dist m 0 = m.
Proof.
destruct m. { reflexivity. } { reflexivity. }
Defined.

Fixpoint nat_dist_plus m n : nat_dist (m + n) m = n.
Proof.
revert m n; intros [|m'] ?.
{ simpl. apply nat_dist_m0. }
{ simpl. apply nat_dist_plus. }
Defined.

Fixpoint nat_dist_le m n : m n nat_dist m n = n-m.
Proof.
destruct m as [|m'].
{ destruct n as [|n']. { reflexivity. } { simpl. intros _. reflexivity. } }
{ destruct n as [|n'].
{ intro f. now induction (!natleh0tois0 f). }
{ exact (nat_dist_le m' n'). } }
Defined.

Definition nat_dist_minus m n : m n nat_dist (n - m) n = m.
Proof.
intros e. set (k := n-m). assert(b := ! minusplusnmm n m e).
rewrite (idpath _ : n-m = k) in b. rewrite b.
rewrite nat_dist_symm. apply nat_dist_plus.
Qed.

Fixpoint nat_dist_gt m n : m > n S (nat_dist m (S n)) = nat_dist m n.
Proof.
destruct m as [|m'].
{ unfold natgth; simpl. intro x.
apply fromempty. apply nopathsfalsetotrue. exact x. }
{ intro i. simpl.
destruct n as [|n'].
{ apply (maponpaths S). apply nat_dist_m0. }
{ simpl. apply nat_dist_gt. exact i. } }
Defined.

Definition nat_dist_S m n : nat_dist (S m) (S n) = nat_dist m n.
Proof.
reflexivity.
Defined.

Definition natminuseqlr m n x : mn n-m = x n = x+m.
Proof.
intros i j.
rewrite <- (minusplusnmm _ _ i). rewrite j. reflexivity.
Defined.

Definition nat_dist_between_le m n a b : m n nat_dist m n = a + b
x, nat_dist x m = a × nat_dist x n = b.
Proof.
intros i j. (m+a). split.
{ apply nat_dist_plus. }
{ rewrite (nat_dist_le m n i) in j.
assert (k := natminuseqlr _ _ _ i j); clear j.
assert (l := nat_dist_plus (m+a) b).
rewrite nat_dist_symm. rewrite (natpluscomm (a+b) m) in k.
rewrite (natplusassoc m a b) in l. rewrite <- k in l. exact l. }
Defined.

Definition nat_dist_between_ge m n a b :
n m nat_dist m n = a + b x:nat, nat_dist x m = a × nat_dist x n = b.
Proof.
intros i j.
rewrite nat_dist_symm in j.
rewrite natpluscomm in j.
(pr1 (nat_dist_between_le n m b a i j)).
apply (weqdirprodcomm _ _).
exact (pr2 (nat_dist_between_le n m b a i j)).
Defined.

Definition nat_dist_between m n a b :
nat_dist m n = a + b x:nat, nat_dist x m = a × nat_dist x n = b.
Proof.
intros j.
induction (natgthorleh m n) as [r|s].
{ apply nat_dist_between_ge. apply natlthtoleh. exact r. exact j. }
{ apply nat_dist_between_le. exact s. exact j. }
Defined.

Definition natleorle m n : (mn) ⨿ (nm).
Proof.
intros.
induction (natgthorleh m n) as [r|s].
{ apply ii2. apply natlthtoleh. exact r. }
{ apply ii1. exact s. }
Defined.

Definition nat_dist_trans x y z : nat_dist x z nat_dist x y + nat_dist y z.
Proof.
intros. induction (natleorle x y) as [r|s].
{ rewrite (nat_dist_le _ _ r).
induction (natleorle y z) as [t|u].
{ assert (u := istransnatgeh _ _ _ t r). rewrite (nat_dist_le _ _ t).
rewrite (nat_dist_le _ _ u). apply (natlehandplusrinv _ _ x).
rewrite (minusplusnmm _ _ u). rewrite (natpluscomm _ x).
rewrite <- natplusassoc. rewrite (natpluscomm x).
rewrite (minusplusnmm _ _ r). rewrite (natpluscomm y).
rewrite (minusplusnmm _ _ t). apply isreflnatleh. }
{ rewrite (nat_dist_ge _ _ u).
induction (natleorle x z) as [p|q].
{ rewrite (nat_dist_le _ _ p). apply (natlehandplusrinv _ _ x).
rewrite (minusplusnmm _ _ p). rewrite natpluscomm.
rewrite <- natplusassoc. rewrite (natpluscomm x).
rewrite (minusplusnmm _ _ r). apply (natlehandplusrinv _ _ z).
rewrite natplusassoc. rewrite (minusplusnmm _ _ u).
apply (istransnatleh (m := y+z)).
{ apply natlehandplusr. exact u. }
{ apply natlehandplusl. exact u. } }
{ rewrite (nat_dist_ge _ _ q). apply (natlehandplusrinv _ _ z).
rewrite (minusplusnmm _ _ q). rewrite natplusassoc.
rewrite (minusplusnmm _ _ u). rewrite natpluscomm.
apply (natlehandplusrinv _ _ x). rewrite natplusassoc.
rewrite (minusplusnmm _ _ r). apply (istransnatleh (m := x+y)).
{ apply natlehandplusl. assumption. }
{ apply natlehandplusr. assumption. } } } }
{ rewrite (nat_dist_ge _ _ s).
induction (natleorle z y) as [u|t].
{ assert (w := istransnatleh u s). rewrite (nat_dist_ge _ _ w).
rewrite (nat_dist_ge _ _ u). apply (natlehandplusrinv _ _ z).
rewrite (minusplusnmm _ _ w). rewrite natplusassoc.
rewrite (minusplusnmm _ _ u). rewrite (minusplusnmm _ _ s).
apply isreflnatleh. }
{ rewrite (nat_dist_le _ _ t).
induction (natleorle x z) as [p|q].
{ rewrite (nat_dist_le _ _ p). apply (natlehandplusrinv _ _ x).
rewrite (minusplusnmm _ _ p). apply (natlehandpluslinv _ _ y).
rewrite (natplusassoc (x-y)). rewrite <- (natplusassoc y).
rewrite (natpluscomm y (x-y)). rewrite (minusplusnmm _ _ s).
apply (natlehandplusrinv _ _ y). rewrite (natplusassoc x).
rewrite (natplusassoc _ x y). rewrite (natpluscomm x y).
rewrite <- (natplusassoc _ y x). rewrite (minusplusnmm _ _ t).
rewrite (natpluscomm z x). rewrite <- (natplusassoc x).
rewrite (natplusassoc y). rewrite (natpluscomm z y).
rewrite <- (natplusassoc y). apply (natlehandplusr _ _ z).
apply (istransnatleh (m := x+y)).
{ apply natlehandplusr. assumption. }
{ apply natlehandplusl. assumption. } }
{ rewrite (nat_dist_ge _ _ q). apply (natlehandplusrinv _ _ z).
rewrite (minusplusnmm _ _ q). apply (natlehandpluslinv _ _ y).
rewrite (natplusassoc (x-y)). rewrite <- (natplusassoc y).
rewrite (natpluscomm y (x-y)). rewrite (minusplusnmm _ _ s).
apply (natlehandplusrinv _ _ y). rewrite (natplusassoc x).
rewrite (natplusassoc _ z y). rewrite (natpluscomm z y).
rewrite <- (natplusassoc _ y z). rewrite (minusplusnmm _ _ t).
rewrite (natpluscomm y x). rewrite (natplusassoc x).
apply natlehandplusl. apply (istransnatleh (m := z+y)).
{ apply natlehandplusr. assumption. }
{ apply natlehandplusl. assumption. } } } }
Defined.

Lemma plusmn0n0 m n : m + n = 0 n = 0.
Proof.
intros i. assert (a := natlehmplusnm m n). rewrite i in a.
apply natleh0tois0. assumption.
Defined.

Lemma plusmn0m0 m n : m + n = 0 m = 0.
Proof.
intros i. assert (a := natlehnplusnm m n). rewrite i in a.
apply natleh0tois0. assumption.
Defined.

Lemma natminus0le {m n} : m-n = 0 n m.
Proof.
intros i. apply negnatgthtoleh. intro k.
assert (r := minusgth0 _ _ k); clear k.
induction (!i); clear i. exact (negnatgth0n 0 r).
Defined.

Lemma minusxx m : m - m = 0.
Proof.
induction m as [|m IHm]. reflexivity. simpl. assumption.
Defined.

Lemma minusSxx m : S m - m = 1.
Proof.
induction m as [|m IHm]. reflexivity. assumption.
Defined.

Lemma natminusminus n m : m n n - (n - m) = m.
Proof.
intros i. assert (b := plusminusnmm m (n-m)).
rewrite natpluscomm in b. rewrite (minusplusnmm _ _ i) in b.
exact b.
Defined.

Lemma natplusminus m n k : k=m+n k-n=m.
Proof.
intros i. rewrite i. apply plusminusnmm.
Defined.

Lemma natleplusminus k m n : k + m n k n - m.
Proof.
intros i.
apply (natlehandplusrinv _ _ m).
rewrite minusplusnmm.
{ exact i. }
{ change (m n).
simple refine (istransnatleh _ i); clear i.
apply natlehmplusnm. }
Defined.

Lemma natltminus1 m n : m < n m n - 1.
Proof.
intros i. assert (a := natlthp1toleh m (n - 1)).
assert (b := natleh0n m). assert (c := natlehlthtrans _ _ _ b i).
assert (d := natlthtolehsn _ _ c). assert (e := minusplusnmm _ _ d).
rewrite e in a. exact (a i).
Defined.

Fixpoint natminusminusassoc m n k : (m-n)-k = m-(n+k).
Proof.
intros. destruct m. { reflexivity. }
{ destruct n. { rewrite natminuseqn. reflexivity. }
{ simpl. apply natminusminusassoc. } }
Defined.

Definition natminusplusltcomm m n k : k n m n - k k n - m.
Proof.
intros i p.
assert (a := natlehandplusr m (n-k) k p); clear p.
assert (b := minusplusnmm n k i); clear i.
rewrite b in a; clear b. apply natleplusminus.
rewrite natpluscomm. exact a.
Qed.