Library UniMath.Combinatorics.FiniteSequences

Finite sequences

Vectors and matrices defined in March 2018 by Langston Barrett (@siddharthist).


  • Vectors
  • Matrices
  • Sequences
    • Definitions
    • Lemmas

Require Export UniMath.Combinatorics.FiniteSets.
Require Export UniMath.Combinatorics.Lists.

Require Import UniMath.MoreFoundations.PartA.
Require Import UniMath.MoreFoundations.Tactics.

Local Open Scope transport.


A Vector of length n with values in X is an ordered n-tuple of elements of X, encoded here as a function ⟦n⟧ → X.
Definition Vector (X : UU) (n : nat) : UU := stn n X.

hlevel of vectors
Lemma vector_hlevel (X : UU) (n : nat) {m : nat} (ism : isofhlevel m X) :
  isofhlevel m (Vector X n).
  apply impred; auto.

Constant vector
Definition const_vec {X : UU} {n : nat} (x : X) : Vector X n := λ _, x.

The unique empty vector
Definition iscontr_vector_0 X : iscontr (Vector X 0).
  intros. apply (@iscontrweqb _ (empty X)).
  - apply invweq. apply weqbfun. apply weqstn0toempty.
  - apply iscontrfunfromempty.

Definition empty_vec {X : UU} : Vector X 0 := iscontrpr1 (iscontr_vector_0 X).

Every type is equivalent to vectors of length 1 on that type.
Lemma weq_vector_1 {X : UU} : X Vector X 1.
  intermediate_weq (unit X).
  - apply invweq, weqfunfromunit.
  - apply weqbfun.
    exact weqstn1tounit.

Section Append.

  Context {X : UU} {n : nat} (vec : Vector X n) (x : X).

  Definition append_vec : Vector X (S n).
    intros i.
    induction (natlehchoice4 (pr1 i) n (pr2 i)) as [c|d].
    - exact (vec (pr1 i,,c)).
    - exact x.

  Definition append_vec_compute_1 i : append_vec (dni lastelement i) = vec i.
    induction i as [i b]; simpl.
    rewrite replace_dni_last.
    unfold append_vec; simpl.
    induction (natlehchoice4 i n (natlthtolths i n b)) as [p|p].
    - simpl. apply maponpaths. apply isinjstntonat; simpl. reflexivity.
    - simpl. destruct p. induction (isirreflnatlth i b).

  Definition append_vec_compute_2 : append_vec lastelement = x.
    intros; unfold append_vec; simpl.
    induction (natlehchoice4 n n (natgthsnn n)) as [a|a]; simpl.
    - contradicts a (isirreflnatlth n).
    - reflexivity.

End Append.

Lemma drop_and_append_vec {X n} (x : Vector X (S n)) :
  append_vec (x dni_lastelement) (x lastelement) = x.
  apply funextfun; intros [i b].
  induction (natlehchoice4 i n b) as [p|p].
  - unfold funcomp; simpl.
    unfold append_vec. simpl.
    induction (natlehchoice4 i n b) as [q|q].
    + simpl. apply maponpaths. apply isinjstntonat; simpl. reflexivity.
    + induction q. contradicts p (isirreflnatlth i).
  - induction p.
    unfold append_vec; simpl.
    induction (natlehchoice4 i i b) as [r|r].
    × simpl. unfold funcomp; simpl. apply maponpaths.
      apply isinjstntonat; simpl. reflexivity.
    × simpl. apply maponpaths. apply isinjstntonat; simpl. reflexivity.

An induction principle for vectors: If a statement is true for the empty vector, and if it is true for vectors of length n it is also true for those of length S n, then it is true for all vectors.
Definition Vector_rect {X : UU} {P : n, Vector X n UU}
           (p0 : P 0 empty_vec)
           (ind : (n : nat) (vec : Vector X n) (x : X),
                  P n vec P (S n) (append_vec vec x))
           {n : nat} (vec : Vector X n) : P n vec.
  induction n as [|n IH].
  - refine (transportf (P 0) _ p0).
    apply proofirrelevance, isapropifcontr, iscontr_vector_0.
  - exact (transportf (P _) (drop_and_append_vec vec)
                      (ind _ (vec dni_lastelement)
                             (vec lastelement)
                             (IH (vec dni_lastelement)))).

Section Lemmas.

  Context {X : UU} {n : nat}.

  Definition vectorEquality {m : nat} (f : Vector X n) (g : Vector X m) (p : n = m) :
    ( i, f i = g (transportf stn p i))
     transportf (Vector X) p f = g.
    intros ?.
    induction p.
    apply funextfun.

  Definition tail (vecsn : Vector X (S n)) : Vector X n :=
    vecsn dni (0,, natgthsn0 n).

It doesn't matter what the proofs are in the stn inputs.
  Definition vector_stn_proofirrelevance {vec : Vector X n}
            {i j : stn n} : (stntonat _ i = stntonat _ j) vec i = vec j.
    intros ?.
    apply maponpaths, isinjstntonat; assumption.
End Lemmas.


Local Open Scope stn.

An m × n matrix is an m-length vector of n-length vectors (rows). <--- n ---> | [ * * * * ] m [ * * * * ] | [ * * * * ]
Since Vectors are encoded as functions ⟦n⟧ → X, a matrix is a function (of two arguments). Thus, the (i, j)-entry of a matrix Mat is simply Mat i j.
Definition Matrix (X : UU) (m n : nat) : UU := Vector (Vector X n) m.

The transpose is obtained by flipping the arguments.
Definition transpose {X : UU} {n m : nat} (mat : Matrix X m n) : Matrix X n m :=
  flip mat.

Definition row {X : UU} {m n : nat} (mat : Matrix X m n) : m Vector X n := mat.

Definition col {X : UU} {m n : nat} (mat : Matrix X m n) : n Vector X m := transpose mat.

Definition row_vec {X : UU} {n : nat} (vec : Vector X n) : Matrix X 1 n :=
  λ i j, vec j.

Definition col_vec {X : UU} {n : nat} (vec : Vector X n) : Matrix X n 1 :=
  λ i j, vec i.

hlevel of matrices
Lemma matrix_hlevel (X : UU) (n m : nat) {o : nat} (ism : isofhlevel o X) :
  isofhlevel o (Matrix X n m).
  do 2 apply vector_hlevel; assumption.

Constant matrix
Definition const_matrix {X : UU} {n m : nat} (x : X) : Matrix X n m :=
  const_vec (const_vec x).

Every type is equivalent to 1 × 1 matrices on that type.
Lemma weq_matrix_1_1 {X : UU} : X Matrix X 1 1.
  intermediate_weq (Vector X 1); apply weq_vector_1.



A Sequence is a Vector of any length.


Definition composeSequence {X Y} (f:XY) : Sequence X Sequence Y := λ x, functionToSequence (f x).

Definition composeSequence' {X m n} (f:stn n X) (g:stn m stn n) : Sequence X
  := functionToSequence (f g).

Definition composeUnorderedSequence {X Y} (f:XY) : UnorderedSequence X UnorderedSequence Y
  := λ x, functionToUnorderedSequence(f x).

Definition weqListSequence {X} : list X Sequence X.
  apply weqfibtototal; intro n.
  apply weqlistfun.

Definition transport_stn m n i (b:i<m) (p:m=n) :
  transportf stn p (i,,b) = (i,,transportf (λ m,i<m) p b).
Proof. intros. induction p. reflexivity. Defined.

Definition sequenceEquality2 {X} (f g:Sequence X) (p:length f=length g) :
  ( i, f i = g (transportf stn p i)) f = g.
  intros e. induction f as [m f]. induction g as [n g]. simpl in p.
  apply (total2_paths2_f p). now apply vectorEquality.

The following two lemmas are the key lemmas that allow to prove (transportational) equality of sequences whose lengths are not definitionally equal. In particular, these lemmas can be used in the proofs of such results as associativity of concatenation of sequences and the right unity axiom for the empty sequence.

Definition seq_key_eq_lemma {X :UU}( g g' : Sequence X)(e_len : length g = length g')
           (e_el : ( i : nat )(ltg : i < length g )(ltg' : i < length g' ),
               g (i ,, ltg) = g' (i ,, ltg')) : g=g'.
  induction g as [m g]; induction g' as [m' g']. simpl in e_len, e_el.
  intermediate_path (m' ,, transportf (λ i, stn i X) e_len g).
  - apply transportf_eq.
  - apply maponpaths.
    intermediate_path (g transportb stn e_len).
    + apply transportf_fun.
    + apply funextfun. intro x. induction x as [ i b ].
      simple refine (_ @ e_el _ _ _).
      × unfold funcomp.
        apply maponpaths.
        apply transport_stn.

The following lemma requires in the assumption e_el only one comparison i < length g and one comparison i < length g' for each i instead of all such comparisons as in the original version seq_key_eq_lemma .

Definition seq_key_eq_lemma' {X :UU} (g g' : Sequence X) :
  length g = length g'
  ( i, ltg : i < length g, ltg' : i < length g',
                                        g (i ,, ltg) = g' (i ,, ltg'))
  intros k r.
  apply seq_key_eq_lemma.
  × assumption.
  × intros.
    induction (r i) as [ p [ q e ]].
    simple refine (_ @ e @ _).
    - now apply maponpaths, isinjstntonat.
    - now apply maponpaths, isinjstntonat.

Notation fromstn0 := empty_vec.

Definition nil {X} : Sequence X.
Proof. intros. exact (0,, empty_vec). Defined.

Definition append {X} : Sequence X X Sequence X.
Proof. intros x y. exact (S (length x),, append_vec (pr2 x) y).

Definition drop_and_append {X n} (x : stn (S n) X) :
  append (n,,x dni_lastelement) (x lastelement) = (S n,, x).
  intros. apply pair_path_in2. apply drop_and_append_vec.

Local Notation "s □ x" := (append s x) (at level 64, left associativity).

Definition nil_unique {X} (x : stn 0 X) : nil = (0,,x).
  intros. unfold nil. apply maponpaths. apply isapropifcontr. apply iscontr_vector_0.

Definition isaset_transportf {X : hSet} (P : X UU) {x : X} (e : x = x) (p : P x) :
  transportf P e p = p.
Proof. induction (pr1 ((setproperty _) _ _ (idpath _) e)).

Definition iscontr_rect' X (i : iscontr X) (x0 : X) (P : X UU) (p0 : P x0) : x:X, P x.
Proof. intros. induction (pr1 (isapropifcontr i x0 x)). exact p0. Defined.

Definition iscontr_rect_compute' X (i : iscontr X) (x : X) (P : X UU) (p : P x) :
  iscontr_rect' X i x P p x = p.
  unfold iscontr_rect'.
  induction (pr1 (isasetifcontr i x x (idpath _) (pr1 (isapropifcontr i x x)))).

Definition iscontr_rect'' X (i : iscontr X) (P : X UU) (p0 : P (pr1 i)) : x:X, P x.
Proof. intros. exact (invmap (weqsecovercontr P i) p0 x). Defined.

Definition iscontr_rect_compute'' X (i : iscontr X) (P : X UU) (p : P(pr1 i)) :
  iscontr_rect'' X i P p (pr1 i) = p.
Proof. try reflexivity. intros. exact (homotweqinvweq (weqsecovercontr P i) p).

Definition iscontr_adjointness X (is:iscontr X) (x:X) : pr1 (isapropifcontr is x x) = idpath x.
Proof. intros. now apply isasetifcontr. Defined.

Definition iscontr_rect X (is : iscontr X) (x0 : X) (P : X UU) (p0 : P x0) : x:X, P x.
Proof. intros. exact (transportf P (pr1 (isapropifcontr is x0 x)) p0). Defined.

Definition iscontr_rect_compute X (is : iscontr X) (x : X) (P : X UU) (p : P x) :
  iscontr_rect X is x P p x = p.
Proof. intros. unfold iscontr_rect. now rewrite iscontr_adjointness. Defined.

Corollary weqsecovercontr':
   (X:UU) (P:XUU) (is:iscontr X), ( x:X, P x) P (pr1 is).
  set (x0 := pr1 is).
  set (secs := x : X, P x).
  set (fib := P x0).
  set (destr := (λ f, f x0) : secsfib).
  set (constr:= iscontr_rect X is x0 P : fibsecs).
  apply (isweq_iso destr constr).
  - intros f. apply funextsec; intros x.
    unfold destr, constr.
    apply transport_section.
  - apply iscontr_rect_compute.

Definition nil_length {X} (x : Sequence X) : length x = 0 x = nil.
  intros. split.
  - intro e. induction x as [n x]. simpl in e.
    induction (!e). apply pathsinv0. apply nil_unique.
  - intro h. induction (!h). reflexivity.

Definition drop {X} (x:Sequence X) : length x != 0 Sequence X.
  revert x. intros [n x] h.
  induction n as [|n].
  - simpl in h. contradicts h (idpath 0).
  - exact (n,,x dni_lastelement).

Definition drop' {X} (x:Sequence X) : x != nil Sequence X.
Proof. intros h. exact (drop x (pr2 (logeqnegs (nil_length x)) h)). Defined.

Lemma append_and_drop_fun {X n} (x : stn n X) y :
  append_vec x y dni lastelement = x.
  apply funextsec; intros i.
  unfold funcomp.
  unfold append_vec.
  induction (natlehchoice4 (pr1 (dni lastelement i)) n (pr2 (dni lastelement i))) as [I|J].
  - simpl. apply maponpaths. apply subtypePath_prop. simpl. apply di_eq1. exact (stnlt i).
  - apply fromempty. simpl in J.
    assert (P : di n i = i).
    { apply di_eq1. exact (stnlt i). }
    induction (!P); clear P.
    induction i as [i r]. simpl in J. induction J.
    exact (isirreflnatlth _ r).

Definition drop_and_append' {X n} (x : stn (S n) X) :
  append (drop (S n,,x) (negpathssx0 _)) (x lastelement) = (S n,, x).
  intros. simpl. apply pair_path_in2. apply drop_and_append_vec.

Definition disassembleSequence {X} : Sequence X coprod unit (X × Sequence X).
  intros x.
  induction x as [n x].
  induction n as [|n].
  - exact (ii1 tt).
  - exact (ii2(x lastelement,,(n,,x dni_lastelement))).

Definition assembleSequence {X} : coprod unit (X × Sequence X) Sequence X.
  intros co.
  induction co as [t|p].
  - exact nil.
  - exact (append (pr2 p) (pr1 p)).

Lemma assembleSequence_ii2 {X} (p : X × Sequence X) :
  assembleSequence (ii2 p) = append (pr2 p) (pr1 p).
Proof. reflexivity. Defined.

Theorem SequenceAssembly {X} : Sequence X unit ⨿ (X × Sequence X).
  intros. disassembleSequence. apply (isweq_iso _ assembleSequence).
  { intros. induction x as [n x]. induction n as [|n].
    { apply nil_unique. }
    apply drop_and_append'. }
  intros co. induction co as [t|p].
  { unfold disassembleSequence; simpl. apply maponpaths.
    apply proofirrelevancecontr. apply iscontrunit. }
  induction p as [x y]. induction y as [n y].
  apply (maponpaths (@inr unit (X × Sequence X))).
  unfold append_vec, lastelement, funcomp; simpl.
  unfold append_vec. simpl.
  induction (natlehchoice4 n n (natgthsnn n)) as [e|e].
  { contradicts e (isirreflnatlth n). }
  simpl. apply maponpaths, maponpaths.
  apply funextfun; intro i. clear e. induction i as [i b].
  unfold funcomp, dni_lastelement; simpl.
  induction (natlehchoice4 i n (natlthtolths i n b)) as [d|d].
  { simpl. apply maponpaths. now apply isinjstntonat. }
  simpl. induction d; contradicts b (isirreflnatlth i).

Definition Sequence_rect {X} {P : Sequence X UU}
           (p0 : P nil)
           (ind : (x : Sequence X) (y : X), P x P (append x y))
           (x : Sequence X) : P x.
Proof. intros. induction x as [n x]. induction n as [|n IH].
  - exact (transportf P (nil_unique x) p0).
  - exact (transportf P (drop_and_append x)
                      (ind (n,,x dni_lastelement)
                           (x lastelement)
                           (IH (x dni_lastelement)))).

Lemma Sequence_rect_compute_nil {X} {P : Sequence X UU} (p0 : P nil)
      (ind : (s : Sequence X) (x : X), P s P (append s x)) :
  Sequence_rect p0 ind nil = p0.
  try reflexivity.
  unfold Sequence_rect; simpl.
  change p0 with (transportf P (idpath nil) p0) at 2.
  apply (maponpaths (λ e, transportf P e p0)).
  exact (maponpaths (maponpaths functionToSequence) (iscontr_adjointness _ _ _)).

Lemma Sequence_rect_compute_cons
      {X} {P : Sequence X UU} (p0 : P nil)
      (ind : (s : Sequence X) (x : X), P s P (append s x))
      (p := Sequence_rect p0 ind) (x:X) (l:Sequence X) :
  p (append l x) = ind l x (p l).

Lemma append_length {X} (x:Sequence X) (y:X) :
  length (append x y) = S (length x).
Proof. intros. reflexivity. Defined.

Definition concatenate {X : UU} : binop (Sequence X)
  := λ x y, functionToSequence (concatenate' x y).

Definition concatenate_length {X} (x y:Sequence X) :
  length (concatenate x y) = length x + length y.
Proof. intros. reflexivity. Defined.

Definition concatenate_0 {X} (s t:Sequence X) : length t = 0 concatenate s t = s.
  induction s as [m s]. induction t as [n t].
  intro e; simpl in e. induction (!e).
  simple refine (sequenceEquality2 _ _ _ _).
  - simpl. apply natplusr0.
  - intro i; simpl in i. simpl.
    unfold concatenate'.
    rewrite weqfromcoprodofstn_invmap_r0.

Definition concatenateStep {X : UU} (x : Sequence X) {n : nat} (y : stn (S n) X) :
  concatenate x (S n,,y) = append (concatenate x (n,,y dni lastelement)) (y lastelement).
  revert x n y. induction x as [m l]. intros n y.
  use seq_key_eq_lemma.
  - cbn. apply natplusnsm.
  - intros i r s.
    unfold concatenate, concatenate', weqfromcoprodofstn_invmap; cbn.
    unfold append_vec, coprod_rect, funcomp; cbn.
    induction (natlthorgeh i m) as [H | H].
    + induction (natlehchoice4 i (m + n) s) as [H1 | H1].
      × reflexivity.
      × apply fromempty. induction (!H1); clear H1.
        set (tmp := natlehnplusnm m n).
        set (tmp2 := natlehlthtrans _ _ _ tmp H).
        exact (isirreflnatlth _ tmp2).
    + induction (natlehchoice4 i (m + n) s) as [I|J].
      × apply maponpaths, subtypePath_prop. rewrite replace_dni_last. reflexivity.
      × apply maponpaths, subtypePath_prop. simpl.
        induction (!J). rewrite natpluscomm. apply plusminusnmm.

Definition flatten {X : UU} : Sequence (Sequence X) Sequence X.
  intros x. (stnsum (length x)). exact (flatten' (sequenceToFunction x)).

Definition flattenUnorderedSequence {X : UU} : UnorderedSequence (UnorderedSequence X) UnorderedSequence X.
  intros x.
  use tpair.
  - exact (( i, pr1 (x i))%finset).
  - intros ij. exact (x (pr1 ij) (pr2 ij)). Defined.

Definition flattenStep' {X n}
           (m : stn (S n) nat)
           (x : i : stn (S n), stn (m i) X)
           (m' := m dni lastelement)
           (x' := x dni lastelement) :
  flatten' x = concatenate' (flatten' x') (x lastelement).
  apply funextfun; intro i.
  unfold flatten'.
  unfold funcomp.
  rewrite 2 weqstnsum1_eq'.
  unfold StandardFiniteSets.weqstnsum_invmap at 1.
  unfold concatenate'.
  unfold nat_rect, coprod_rect, funcomp.
  change (weqfromcoprodofstn_invmap (stnsum (λ r : stn n, m (dni lastelement r))))
  with (weqfromcoprodofstn_invmap (stnsum m')) at 1 2.
  induction (weqfromcoprodofstn_invmap (stnsum m')) as [B|C].
  - reflexivity.
  - now induction C. Defined.

Definition flattenStep {X} (x: NonemptySequence (Sequence X)) :
  flatten x = concatenate (flatten (composeSequence' x (dni lastelement))) (lastValue x).
  apply pair_path_in2.
  set (xlens := λ i, length(x i)).
  set (xvals := λ i, λ j:stn (xlens i), x i j).
  exact (flattenStep' xlens xvals).

Definition partition' {X n} (f:stn n nat) (x:stn (stnsum f) X) : stn n Sequence X.
Proof. intros i. (f i). intro j. exact (x(inverse_lexicalEnumeration f (i,,j))).

Definition partition {X n} (f:stn n nat) (x:stn (stnsum f) X) : Sequence (Sequence X).
Proof. intros. n. exact (partition' f x).

Definition flatten_partition {X n} (f:stn n nat) (x:stn (stnsum f) X) :
  flatten (partition f x) ¬ x.
  intros. intro i.
  change (x (weqstnsum1 f (pr1 (invmap (weqstnsum1 f) i),, pr2 (invmap (weqstnsum1 f) i))) = x i).
  apply maponpaths. apply subtypePath_prop. now rewrite homotweqinvweq.

Definition isassoc_concatenate {X : UU} (x y z : Sequence X) :
  concatenate (concatenate x y) z = concatenate x (concatenate y z).
  use seq_key_eq_lemma.
  - cbn. apply natplusassoc.
  - intros i ltg ltg'. cbn. unfold concatenate'. unfold weqfromcoprodofstn_invmap. unfold coprod_rect. cbn.
    induction (natlthorgeh i (length x + length y)) as [H | H].
    + induction (natlthorgeh (make_stn (length x + length y) i H) (length x)) as [H1 | H1].
      × induction (natlthorgeh i (length x)) as [H2 | H2].
        -- apply maponpaths. apply isinjstntonat. apply idpath.
        -- apply fromempty. exact (natlthtonegnatgeh i (length x) H1 H2).
      × induction (natchoice0 (length y)) as [H2 | H2].
        -- apply fromempty. induction H2. induction (! (natplusr0 (length x))).
           apply (natlthtonegnatgeh i (length x) H H1).
        -- induction (natlthorgeh i (length x)) as [H3 | H3].
           ++ apply fromempty. apply (natlthtonegnatgeh i (length x) H3 H1).
           ++ induction (natchoice0 (length y + length z)) as [H4 | H4].
              ** apply fromempty. induction (! H4).
                 use (isirrefl_natneq (length y)).
                 use natlthtoneq.
                 use (natlehlthtrans (length y) (length y + length z) (length y) _ H2).
                 apply natlehnplusnm.
              ** cbn. induction (natlthorgeh (i - length x) (length y)) as [H5 | H5].
                 --- apply maponpaths. apply isinjstntonat. apply idpath.
                 --- apply fromempty.
                     use (natlthtonegnatgeh (i - (length x)) (length y)).
                     +++ set (tmp := natlthandminusl i (length x + length y) (length x) H
                                                     (natlthandplusm (length x) _ H2)).
                         rewrite (natpluscomm (length x) (length y)) in tmp.
                         rewrite plusminusnmm in tmp. exact tmp.
                     +++ exact H5.
    + induction (natchoice0 (length z)) as [H1 | H1].
      × apply fromempty. cbn in ltg. induction H1. rewrite natplusr0 in ltg.
        exact (natlthtonegnatgeh i (length x + length y) ltg H).
      × induction (natlthorgeh i (length x)) as [H2 | H2].
        -- apply fromempty.
           use (natlthtonegnatgeh i (length x) H2).
           use (istransnatgeh i (length x + length y) (length x) H).
           apply natgehplusnmn.
        -- induction (natchoice0 (length y + length z)) as [H3 | H3].
           ++ apply fromempty. cbn in ltg'. induction H3. rewrite natplusr0 in ltg'.
              exact (natlthtonegnatgeh i (length x) ltg' H2).
           ++ cbn. induction (natlthorgeh (i - length x) (length y)) as [H4 | H4].
              ** apply fromempty.
                 use (natlthtonegnatgeh i (length x + length y) _ H).
                 apply (natlthandplusr _ _ (length x)) in H4.
                 rewrite minusplusnmm in H4.
                 --- rewrite natpluscomm in H4. exact H4.
                 --- exact H2.
              ** apply maponpaths. apply isinjstntonat. cbn. apply (! (natminusminus _ _ _)).


Definition reverse {X : UU} (x : Sequence X) : Sequence X :=
  functionToSequence (fun i : (stn (length x)) ⇒ x (dualelement i)).

Lemma reversereverse {X : UU} (x : Sequence X) : reverse (reverse x) = x.
  induction x as [n x].
  apply pair_path_in2.
  apply funextfun; intro i.
  unfold reverse, dualelement, coprod_rect. cbn.
  induction (natchoice0 n) as [H | H].
  + apply fromempty. rewrite <- H in i. now apply negstn0.
  + cbn. apply maponpaths. apply isinjstntonat. apply minusminusmmn. apply natgthtogehm1. apply stnlt.

Lemma reverse_index {X : UU} (x : Sequence X) (i : stn (length x)) :
  (reverse x) (dualelement i) = x i.
  cbn. unfold dualelement, coprod_rect.
  set (e := natgthtogehm1 (length x) i (stnlt i)).
  induction (natchoice0 (length x)) as [H' | H'].
  - apply maponpaths. apply isinjstntonat. cbn. apply (minusminusmmn _ _ e).
  - apply maponpaths. apply isinjstntonat. cbn. apply (minusminusmmn _ _ e).

Lemma reverse_index' {X : UU} (x : Sequence X) (i : stn (length x)) :
  (reverse x) i = x (dualelement i).
  cbn. unfold dualelement, coprod_rect.
  induction (natchoice0 (length x)) as [H' | H'].
  - apply maponpaths. apply isinjstntonat. cbn. apply idpath.
  - apply maponpaths. apply isinjstntonat. cbn. apply idpath.