Library UniMath.MoreFoundations.Nat
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.PartA.
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) @ 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. }
{ 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. }
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.
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.
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.
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 (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. }
{ 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.
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. } }
{ 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. } }
{ 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. } }
{ 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'). } }
{ 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. } }
{ 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), maponpaths 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.
Lemma apSC m n (e:m=n) : helper_C m n e = helper_C (S m) (S n) (maponpaths S e).
Show proof.
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 ( maponpaths (helper_B (S m) (S m)) (! apSC _ _ (idpath m))
@ maponpaths (maponpaths S) IHm). } }
{ 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). } }
Definition E m n : (nat_discern m n) ≃ (m = n).
Show proof.
Definition nat_dist_anti m n : nat_dist m n = 0 -> m = n.
Show proof.
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. } }
{ 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'). } }
{ 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.
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'). } }
{ 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.
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 (maponpaths S). apply nat_dist_m0. }
{ simpl. apply nat_dist_gt. exact i. } }
{ 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. } }
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.
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. }
{ 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)).
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. }
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. }
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. } } } }
{ 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.
Lemma plusmn0m0 m n : m + n = 0 -> m = 0.
Show proof.
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).
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.
rewrite natpluscomm in b. rewrite (minusplusnmm _ _ i) in b.
exact b.
Lemma natplusminus m n k : k=m+n -> k-n=m.
Show proof.
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. }
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).
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. } }
{ 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.
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.
Theorem nat_le_diff
{n m : ℕ}
(p : n ≤ m)
: ∑ (k : ℕ), n + k = m.
Show proof.