Library UniMath.Ktheory.Nat


Natural numbers


Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.Algebra.Monoids.
Require Import UniMath.Foundations.NaturalNumbers
               UniMath.MoreFoundations.NegativePropositions
               UniMath.Foundations.UnivalenceAxiom
               UniMath.CategoryTheory.total2_paths
               UniMath.Ktheory.Utilities.

Local Open Scope nat.

Definition ℕ := nat.

Module Uniqueness.

  Lemma helper_A (P:nat->Type) (p0:P 0) (IH:∏ n, P n->P(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)).
  Show proof.
intros. simple refine (_,,isweq_iso _ _ _ _).
         { intros h. split.
           { exact (h 0). } { intros. exact (h (S n) @ ap (IH n) (! h n)). } }
         { intros [h0 h'] ?. induction n as [|n' IHn'].
           { exact h0. } { exact (h' n' @ ap (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. }

  Lemma helper_B (P:nat->Type) (p0:P 0) (IH:∏ n, P n->P(S n))
        (f:∏ n, P n) :
    weq (f = nat_rect P p0 IH)
        ((f 0=p0) × (∏ n, f(S n)=IH n (f n))).
  Show proof.
intros.
         exact (weqcomp (weqtoforallpaths _ _ _) (helper_A _ _ _ _)).

  Lemma helper_C (P:nat->Type) (p0:P 0) (IH:∏ n, P n->P(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)).
  Show proof.
intros. apply weqfibtototal. intros f. apply helper_B.

  Lemma hNatRecursionUniq (P:nat->Type) (p0:P 0) (IH:∏ n, P n->P(S n)) :
    ∃! (f:∏ n, P n), f 0=p0 × ∏ n, f(S n) = IH n (f n).
  Show proof.
intros. exact (iscontrweqf (helper_C _ _ _) (iscontrcoconustot _ _)).

  Lemma helper_D (P:nat->Type) (p0:P 0) (IH:∏ n, P n->P(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).
  Show proof.
intros. simple refine (weqpair _ (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. }

  Lemma hNatRecursion_weq (P:nat->Type) (IH:∏ n, P n->P(S n)) :
    weq (total2 (fun f:∏ n, P n => ∏ n, f(S n)=IH n (f n))) (P 0).
  Show proof.
intros. exists (λ f, pr1 f 0). intro p0.
         apply (iscontrweqf (helper_D _ _ _)). apply hNatRecursionUniq.

End Uniqueness.

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

Module Discern.

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

  Goal ∏ m n, nat_discern m n -> nat_discern (S m) (S n).
  Show proof.
intros ? ? e. exact e.

  Lemma nat_discern_inj m n : nat_discern (S m) (S n) -> nat_discern m n.
  Show 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. } }

  Lemma nat_discern_isaprop m n : isaprop (nat_discern m n).
  Show 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. } }

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

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

  Fixpoint helper_A m n : nat_dist m n = 0 -> nat_discern m n.
  Show 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'). } }

  Fixpoint helper_B m n : nat_discern m n -> m = n.
  Show 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. } }

  Goal ∏ m n (e:nat_discern m n), ap S (helper_B m n e) = helper_B (S m) (S n) e.
  Show proof.
reflexivity.

  Fixpoint helper_C m n : m = n -> nat_discern m n.
  Show proof.
intros e. destruct e.
         exact (cast (! nat_discern_unit m) tt).

  Lemma apSC m n (e:m=n) : helper_C m n e = helper_C (S m) (S n) (ap S e).
  Show proof.
intros. apply proofirrelevance. apply nat_discern_isaprop.

  Definition helper_D m n : isweq (helper_B m n).
  Show 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 ( ap (helper_B (S m) (S m)) (! apSC _ _ (idpath m))
                    @ ap (ap S) IHm). } }

  Definition E m n : (nat_discern m n) ≃ (m = n).
  Show proof.
intros. exact (weqpair (helper_B _ _) (helper_D _ _)).

  Definition nat_dist_anti m n : nat_dist m n = 0 -> m = n.
  Show proof.
intros i. exact (helper_B _ _ (helper_A _ _ i)).

End Discern.

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

Fixpoint nat_dist_ge m n : m ≥ n -> nat_dist m n = m-n.
Show 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'). } }

Definition nat_dist_0m m : nat_dist 0 m = m.
Show proof.
reflexivity.

Definition nat_dist_m0 m : nat_dist m 0 = m.
Show proof.
destruct m. { reflexivity. } { reflexivity. }

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

Fixpoint nat_dist_le m n : m ≤ n -> nat_dist m n = n-m.
Show 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'). } }

Definition nat_dist_minus m n : m ≤ n -> nat_dist (n - m) n = m.
Show 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.

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

Definition nat_dist_S m n : nat_dist (S m) (S n) = nat_dist m n.
Show proof.
reflexivity.

Definition natminuseqlr m n x : m≤n -> n-m = x -> n = x+m.
Show proof.
intros i j.
       rewrite <- (minusplusnmm _ _ i). rewrite j. reflexivity.

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.
Show proof.
intros i j. exists (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. }

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.
Show proof.
intros i j.
       rewrite nat_dist_symm in j.
       rewrite natpluscomm in j.
       exists (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)).

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.
Show 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. }

Definition natleorle m n : (m≤n) ⨿ (n≤m).
Show proof.
intros.
       induction (natgthorleh m n) as [r|s].
       { apply ii2. apply natlthtoleh. exact r. }
       { apply ii1. exact s. }

Definition nat_dist_trans x y z : nat_dist x z ≤ nat_dist x y + nat_dist y z.
Show 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. } } } }

Lemma plusmn0n0 m n : m + n = 0 -> n = 0.
Show proof.
intros i. assert (a := natlehmplusnm m n). rewrite i in a.
       apply natleh0tois0. assumption.

Lemma plusmn0m0 m n : m + n = 0 -> m = 0.
Show proof.
intros i. assert (a := natlehnplusnm m n). rewrite i in a.
       apply natleh0tois0. assumption.

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

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

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

Lemma natminusminus n m : m ≤ n -> n - (n - m) = m.
Show proof.
intros i. assert (b := plusminusnmm m (n-m)).
       rewrite natpluscomm in b. rewrite (minusplusnmm _ _ i) in b.
       exact b.

Lemma natplusminus m n k : k=m+n -> k-n=m.
Show proof.
intros i. rewrite i. apply plusminusnmm.

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

Lemma natltminus1 m n : m < n -> m ≤ n - 1.
Show 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).

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

Definition natminusplusltcomm m n k : k ≤ n -> m ≤ n - k -> k ≤ n - m.
Show 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.