Require Import UniMath.MoreFoundations.DecidablePropositions.
Require Import UniMath.Combinatorics.StandardFiniteSets.
Require Import UniMath.Algebra.Monoids_and_Groups.
Require Import UniMath.CategoryTheory.total2_paths.
Require Import UniMath.Ktheory.Utilities.
Require UniMath.Ktheory.Magma UniMath.Ktheory.QuotientSet.
Local Notation Hom := monoidfun (only parsing).
Local Notation "x * y" := ( op x y ).
Local Notation "g ∘ f" := (monoidfuncomp f g) (only parsing).
Definition funEquality G H (p q : Hom G H) : pr1 p = pr1 q → p = q.
  revert p q; intros [p i] [q j] v. simpl in v. destruct v.
  destruct (pr1 (isapropismonoidfun p i j)). reflexivity. Qed.
Definition unitproperty {G H} (p:Hom G H) : p (unel G) = unel H
  := pr2 (pr2 p).
Definition multproperty {G H} (p:Hom G H) (g g':G) : p(g × g') = p g × p g'
  := pr1 (pr2 p) g g'.
Definition zero : monoid.
  ∃ Magma.zero. split. intros x y z. reflexivity.
  ∃ tt. split. intros []. reflexivity. intros []. reflexivity.
Defined.
Definition zero_map (A B:monoid) : Hom A B.
  intros. ∃ (λ _, unel B).
  { split. - intros x y. apply pathsinv0. apply lunax.
           - reflexivity. } Defined.
Lemma zero_is_initial (A:monoid) : iscontr (Hom zero A).
Proof. intros. ∃ (zero_map zero A).
       intros [f e]. apply funEquality; simpl.
       apply funextsec. intro t. induction t. exact (pr2 e). Defined.
Lemma zero_is_final (A:monoid) : iscontr (Hom A zero).
Proof. intros. ∃ (zero_map A zero).
       intros [f e]. apply funEquality; simpl.
       apply funextsec; intro a. induction (f a). reflexivity. Defined.

Require UniMath.Combinatorics.FiniteSequences.

Module Presentation'.

  Import UniMath.Combinatorics.FiniteSequences.

  Definition word X := Sequence X.
  Definition word_length {X} : word X → nat := length.
  Definition word_unit {X} : word X := nil.
  Definition word_gen {X} (x:X) : word X.
  Proof. intros. exact (1,,λ i, x). Defined.

  Definition word_op {X}: word X → word X → word X := concatenate.

  Lemma word_left_unit {X} (w:word X) : word_op word_unit w = w.
  Abort.

  Lemma word_right_unit {X} (w:word X) : word_op w word_unit = w.
  Abort.

  Lemma word_assoc {X} (u v w:word X) : (word_op (word_op u v) w) = (word_op u (word_op v w)).
  Abort.

  Definition reln X := word X × word X.

  Definition lhs {X} : reln X → word X := pr1.

  Definition rhs {X} : reln X → word X := pr2.

  Definition make_reln {X} : word X → word X → reln X := λ v w, v,,w.

  Definition isAdequateRelation {X I} (R:I→reln X) (r : eqrel (word X)) :=
         (∏ i, r (lhs (R i)) (rhs (R i))) ×
         (∏ u v w, r v w → r (word_op u v) (word_op u w)) ×
         (∏ u v w, r u v → r (word_op u w) (word_op v w)).
  Definition base {X I} {R:I→reln X} {r} (ad : isAdequateRelation R r) := pr1 ad.
  Definition left_compat {X I} {R:I→reln X} {r} (ad : isAdequateRelation R r) := pr1(pr2 ad).
  Definition right_compat {X I} {R:I→reln X} {r} (ad : isAdequateRelation R r) := pr2(pr2 ad).

  Definition AdequateRelation {X I} (R:I→reln X) := ∑ r, isAdequateRelation R r.
  Definition toeqrel {X I} (R:I→reln X) : AdequateRelation R → eqrel (word X) := pr1.
  Coercion toeqrel : AdequateRelation >-> eqrel.

  Definition smallestAdequateRelation {X I} (R:I→reln X) : AdequateRelation R.
  Proof.
    intros.
    simple refine (_,,_).
    { simple refine (_,,_).
      { intros v w.
        ∃ (∏ r, isAdequateRelation R r → r v w).
        apply impred; intros r; apply impred_prop. }
      { simple refine (_,,_).
        { simple refine (_,,_).
          { intros u v w uv vw r ad; simpl in uv, vw.
            apply (eqreltrans r u v w).
            { apply uv. exact ad. }
            { apply vw. exact ad. }}
          { intros w r ad. apply eqrelrefl. } }
        { intros v w vw r ad. apply eqrelsymm. apply vw. exact ad. }}}
    { simple refine (_,,_); simpl.
      { intros i r ad. apply (base ad i). }
      { simple refine (_,,_).
        { intros u v w min r ad. apply (left_compat ad). apply min. exact ad. }
        { intros u v w min r ad. apply (right_compat ad). apply min. exact ad. }}}
  Defined.

  Definition univ_set {X I} (R:I→reln X) : hSet :=
    setquotinset (smallestAdequateRelation R).

  Definition univ_unit {X I} (R:I→reln X) := setquotpr _ word_unit : univ_set R.

  Definition univ_binop {X I} (R:I→reln X) : binop (univ_set R).
    intros. simple refine (setquotfun2 _ _ word_op _).
    induction (smallestAdequateRelation R) as [r ad]; simpl.
    intros v v' w w' vv' ww'.
    apply (eqreltrans _ _ (word_op v w') _ (left_compat ad _ _ _ ww') (right_compat ad _ _ _ vv')).
  Defined.

  Definition univ_setwithbinop {X I} (R:I→reln X) := setwithbinoppair (univ_set R) (univ_binop R).

  Lemma eqreleq {X} (r : eqrel X) (x y:X) : x=y → r x y.
  Proof.
    intros e.
    induction e.
    apply eqrelrefl.
  Defined.


  Definition univ_binop_islunit {X I} (R:I→reln X) : islunit (univ_binop R) (univ_unit R).
  Abort.

  Definition univ_binop_isrunit {X I} (R:I→reln X) : isrunit (univ_binop R) (univ_unit R).
  Abort.



End Presentation'.

Module Presentation.

  Inductive word X : Type :=
    | word_unit : word X
    | word_gen : X → word X
    | word_op : word X → word X → word X.
  Arguments word_unit {X}.
  Arguments word_gen {X} x.
  Arguments word_op {X} v w.
  Record reln X := make_reln { lhs : word X; rhs : word X }.
  Arguments lhs {X} r.
  Arguments rhs {X} r.
  Arguments make_reln {X} _ _.
  Record MarkedPreMonoid X :=
    make_preMonoid {
        elem :> Type;
        op0 : elem;
        op1 : X → elem;
        op2 : elem → elem → elem }.
  Arguments elem {X} M : rename.
  Arguments op0 {X M} : rename.
  Arguments op1 {X M} x : rename.
  Arguments op2 {X M} v w : rename.
  Definition wordop X := make_preMonoid X (word X) word_unit word_gen word_op.
  Fixpoint evalword {X} (Y:MarkedPreMonoid X) (w:word X) : elem Y.
    revert w; intros [|x|v w]. { exact op0. } { exact (op1 x). }
    { exact (op2 (evalword X Y v) (evalword X Y w)). } Defined.
  Definition MarkedPreMonoid_to_hrel {X}
             (M:MarkedPreMonoid X) (is:isaset (elem M)) :
      hrel (word X) :=
    λ v w, (evalword M v = evalword M w) ,, is _ _.

  Fixpoint reassemble {X I} (R:I→reln X) (v:wordop X) : evalword (wordop X) v = v.
  Proof. revert v; intros [|x|v w]. { reflexivity. } { reflexivity. }
         { exact (aptwice word_op (reassemble _ _ R v) (reassemble _ _ R w)). } Qed.

  Record AdequateRelation {X I} (R:I→reln X) (r : hrel (word X)) :=
    make_AdequateRelation {
        base: ∏ i, r (lhs (R i)) (rhs (R i));
        reflex : ∏ w, r w w;
        symm : ∏ v w, r v w → r w v;
        trans : ∏ u v w, r u v → r v w → r u w;
        left_compat : ∏ u v w, r v w → r (word_op u v) (word_op u w);
        right_compat: ∏ u v w, r u v → r (word_op u w) (word_op v w);
        left_unit : ∏ w, r (word_op word_unit w) w;
        right_unit : ∏ w, r (word_op w word_unit) w;
        assoc : ∏ u v w, r (word_op (word_op u v) w) (word_op u (word_op v w))
      }.
  Arguments make_AdequateRelation {X I} R r _ _ _ _ _ _ _ _ _.
  Arguments base {X I R r} _ _.
  Definition adequacy_to_eqrel {X I} (R:I→reln X) (r : hrel (word X)) :
    AdequateRelation R r → eqrel (word X).
  Proof. intros ra. ∃ r.
         abstract ( split; [ split; [ exact (trans R r ra) | exact (reflex R r ra) ] |
                             exact (symm R r ra)]). Defined.

  Definition smallestAdequateRelation0 {X I} (R:I→reln X) : hrel (word X).
    intros v w.
    ∃ (∏ r: hrel (word X), AdequateRelation R r → r v w).
    abstract (apply impred; intro r; apply impred_prop).
  Defined.
  Lemma adequacy {X I} (R:I→reln X) :
    AdequateRelation R (smallestAdequateRelation0 R).
  Proof. intros. simple refine (make_AdequateRelation R _ _ _ _ _ _ _ _ _ _).
         { intros ? r ra. apply base. exact ra. }
         { intros ? r ra. apply (reflex R). exact ra. }
         { intros ? ? p r ra. apply (symm R). exact ra. exact (p r ra). }
         { exact (λ u v w p q r ra, trans R r ra u v w (p r ra) (q r ra)). }
         { intros ? ? ? p r ra. apply (left_compat R). exact ra. exact (p r ra). }
         { intros ? ? ? p r ra. apply (right_compat R). exact ra. exact (p r ra). }
         { intros ? r ra. apply (left_unit R). exact ra. }
         { intros ? r ra. apply (right_unit R). exact ra. }
         { exact (λ u v w r ra, assoc R r ra u v w). }
  Qed.
  Definition smallestAdequateRelation {X I} (R:I→reln X) : eqrel (word X).
    intros. exact (adequacy_to_eqrel R _ (adequacy R)). Defined.

  Definition universalMarkedPreMonoid0 {X I} (R:I→reln X) : hSet :=
    setquotinset (smallestAdequateRelation R).
  Lemma op2_compatibility {X I} (R:I→reln X) :
    QuotientSet.iscomprelrelfun2
      (smallestAdequateRelation R) (smallestAdequateRelation R) (smallestAdequateRelation R)
      word_op.
  Proof. intros. split.
    { intros x x' y p r ra. exact (right_compat R r ra x x' y (p r ra)). }
    { intros x y y' p r ra. exact ( left_compat R r ra x y y' (p r ra)). } Qed.

  Definition univ_binop {X I} (R:I→reln X) : binop (universalMarkedPreMonoid0 R).
    intros. simple refine (QuotientSet.setquotfun2 word_op _). apply op2_compatibility. Defined.
  Definition univ_setwithbinop {X I} (R:I→reln X) : setwithbinop
             := setwithbinoppair (universalMarkedPreMonoid0 R) (univ_binop R).

  Definition universalMarkedPreMonoid {X I} (R:I→reln X) : MarkedPreMonoid X.
    intros. simple refine (make_preMonoid X (universalMarkedPreMonoid0 R) _ _ _).
    { exact (setquotpr _ word_unit). }
    { exact (λ x, setquotpr _ (word_gen x)). }
    { exact (univ_binop _). } Defined.

  Lemma lift {X I} (R:I→reln X) : issurjective (setquotpr (smallestAdequateRelation R)).
  Proof. intros. exact (issurjsetquotpr (smallestAdequateRelation R)). Qed.
  Lemma is_left_unit_univ_binop {X I} (R:I→reln X) (w:universalMarkedPreMonoid0 R) :
    ((univ_binop _) (setquotpr _ word_unit) w) = w.
  Proof. isaprop_goal ig. { apply setproperty. }
    apply (squash_to_prop (lift R w) ig); intros [w' []].
    exact (iscompsetquotpr (smallestAdequateRelation R) _ _
                           (λ r ra, left_unit R r ra w')). Qed.
  Lemma is_right_unit_univ_binop {X I} (R:I→reln X) (w:universalMarkedPreMonoid0 R) :
    ((univ_binop _) w (setquotpr _ word_unit)) = w.
  Proof. isaprop_goal ig. { apply setproperty. }
    apply (squash_to_prop (lift R w) ig); intros [w' []].
    exact (iscompsetquotpr (smallestAdequateRelation R) _ _
                           (λ r ra, right_unit R r ra w')). Qed.
  Lemma isassoc_univ_binop {X I} (R:I→reln X) : isassoc(univ_binop R).
  Proof. intros. set (e := smallestAdequateRelation R). intros u' v' w'.
         isaprop_goal ig. { apply setproperty. }
         apply (squash_to_prop (lift R u') ig); intros [u i]; destruct i.
         apply (squash_to_prop (lift R v') ig); intros [v j]; destruct j.
         apply (squash_to_prop (lift R w') ig); intros [w []].
         exact (iscompsetquotpr e _ _ (λ r ra, assoc R r ra u v w)). Qed.
  Fixpoint reassemble_pr {X I} (R:I→reln X) (v:word X) :
    evalword (universalMarkedPreMonoid R) v = setquotpr _ v.
  Proof. revert v; intros [|x|v w]. { reflexivity. } { reflexivity. }
         { simpl. assert (p := ! reassemble_pr _ _ R v). destruct p.
                  assert (q := ! reassemble_pr _ _ R w). destruct q.
                  reflexivity. } Qed.
  Lemma pr_eval_compat {X I} (R:I→reln X) (w:word X) :
    setquotpr (smallestAdequateRelation R) (evalword (wordop X) w)
    = evalword (universalMarkedPreMonoid R) w.
  Proof. intros. destruct w as [|x|v w]. { reflexivity. } { reflexivity. }
    { assert (p := !reassemble R (word_op v w)). destruct p.
      exact (!reassemble_pr R (word_op v w)). } Qed.

  Definition toMarkedPreMonoid {X I} (R:I→reln X) (M:monoid) (el:X→M) :
      MarkedPreMonoid X.
    intros. exact {| elem := M; op0 := unel _; op1 := el; op2 := op |}.
  Defined.
  Record MarkedMonoid {X I} (R:I→reln X) :=
    make_MarkedMonoid {
        m_base :> monoid;
        m_mark : X → m_base;
        m_reln : ∏ i, evalword (toMarkedPreMonoid R m_base m_mark) (lhs (R i)) =
                           evalword (toMarkedPreMonoid R m_base m_mark) (rhs (R i)) }.
  Arguments make_MarkedMonoid {X I} R _ _ _.
  Arguments m_base {X I R} _.
  Arguments m_mark {X I R} _ x.
  Definition toMarkedPreMonoid' {X I} {R:I→reln X} (M:MarkedMonoid R) : MarkedPreMonoid X :=
    toMarkedPreMonoid R (m_base M) (m_mark M).
  Definition evalwordMM {X I} {R:I→reln X} (M:MarkedMonoid R) : word X → M :=
    evalword (toMarkedPreMonoid' M).
  Definition MarkedMonoid_to_hrel {X I} {R:I→reln X} (M:MarkedMonoid R) : hrel (word X) :=
    λ v w , eqset (evalwordMM M v) (evalwordMM M w).
  Lemma abelian_group_adequacy {X I} (R:I→reln X) (M:MarkedMonoid R) :
    AdequateRelation R (MarkedMonoid_to_hrel M).
  Proof. intros. simple refine (make_AdequateRelation R _ _ _ _ _ _ _ _ _ _).
         { exact (λ i, m_reln R M i). } { reflexivity. }
         { intros ? ?. exact pathsinv0. } { intros ? ? ?. exact pathscomp0. }
         { intros ? ? ? p. simpl in p; simpl.
           unfold evalwordMM,evalword in ×. destruct p. reflexivity. }
         { intros ? ? ? p. simpl in p; simpl.
           unfold evalwordMM,evalword in ×. destruct p. reflexivity. }
         { intros. apply lunax. } { intros. apply runax. } { intros. apply assocax. }
  Qed.
  Record MarkedMonoidMap {X I} {R:I→reln X} (M N:MarkedMonoid R) :=
    make_MarkedMonoidMap {
        map_base :> Hom M N;
        map_mark : ∏ x, map_base (m_mark M x) = m_mark N x }.
  Arguments map_base {X I R M N} m.
  Arguments map_mark {X I R M N} m x.
  Lemma MarkedMonoidMapEquality {X I} {R:I→reln X} {M N:MarkedMonoid R}
        (f g:MarkedMonoidMap M N) : map_base f = map_base g → f = g.
  Proof. intros j.
         destruct f as [f ft], g as [g gt]; simpl in j. destruct j.
         assert(k : ft = gt). { apply funextsec; intro x. apply setproperty. } destruct k.
         reflexivity. Qed.
  Fixpoint MarkedMonoidMap_compat {X I} {R:I→reln X}
           {M N:MarkedMonoid R} (f:MarkedMonoidMap M N) (w:word X) :
    map_base f (evalwordMM M w) = evalwordMM N w.
  Proof. intros. destruct w as [|x|v w].
         { exact (unitproperty f). }
         { exact (map_mark f x). }
         { exact (multproperty f (evalwordMM M v) (evalwordMM M w)
                  @ aptwice (λ r s, r × s)
                            (MarkedMonoidMap_compat _ _ _ _ _ f v)
                            (MarkedMonoidMap_compat _ _ _ _ _ f w)). } Qed.
  Lemma MarkedMonoidMap_compat2 {X I} {R:I→reln X}
           {M N:MarkedMonoid R} (f g:MarkedMonoidMap M N) (w:word X) :
    map_base f (evalwordMM M w) = map_base g (evalwordMM M w).
  Proof. intros.
         exact (MarkedMonoidMap_compat f w @ !MarkedMonoidMap_compat g w). Qed.

  Definition universalMarkedMonoid0 {X I} (R:I→reln X) : monoid.
    intros.
    { ∃ (univ_setwithbinop R).
        { split.
          { exact (isassoc_univ_binop R). }
          { ∃ (setquotpr _ word_unit). split.
            { exact (is_left_unit_univ_binop R). }
            { exact (is_right_unit_univ_binop R). } } } }
Defined.
  Definition universalMarkedMonoid1 {X I} (R:I→reln X) : MarkedPreMonoid X :=
    (toMarkedPreMonoid R
                  (universalMarkedMonoid0 R)
                  (λ x : X, setquotpr (smallestAdequateRelation R) (word_gen x))).
  Lemma universalMarkedMonoid2 {X I} (R:I→reln X) (w:word X) :
    setquotpr (smallestAdequateRelation R) w = evalword (universalMarkedMonoid1 R) w.
  Proof. intros.
    exact (! (ap (setquotpr (smallestAdequateRelation R)) (reassemble R w))
           @ pr_eval_compat R w). Qed.
  Definition universalMarkedMonoid3 {X I} (R:I→reln X) (i:I) :
    evalword (universalMarkedMonoid1 R) (lhs (R i)) =
    evalword (universalMarkedMonoid1 R) (rhs (R i)).
  Proof. intros.
         exact (! universalMarkedMonoid2 R (lhs (R i))
                @ iscompsetquotpr (smallestAdequateRelation R) _ _ (λ r ra, base ra i)
                @ universalMarkedMonoid2 R (rhs (R i))). Qed.
  Definition universalMarkedMonoid {X I} (R:I→reln X) : MarkedMonoid R :=
    make_MarkedMonoid R (universalMarkedMonoid0 R)
                (λ x, setquotpr (smallestAdequateRelation R) (word_gen x))
                (universalMarkedMonoid3 R).
  Fixpoint agreement_on_gens0 {X I} {R:I→reln X} {M:monoid}
        (f g:Hom (universalMarkedMonoid R) M)
        (p:∏ i, f (setquotpr (smallestAdequateRelation R) (word_gen i)) =
                   g (setquotpr (smallestAdequateRelation R) (word_gen i)))
        (w:word X) :
          pr1 f (setquotpr (smallestAdequateRelation R) w) =
          pr1 g (setquotpr (smallestAdequateRelation R) w).
  Proof. intros. destruct w as [|x|v w].
         { intermediate_path (unel M). exact (unitproperty f). exact (!unitproperty g). }
         { apply p. }
         
         { simple refine (
               multproperty f (setquotpr (smallestAdequateRelation R) v)
                   (setquotpr (smallestAdequateRelation R) w)
             @ _ @ !
               multproperty g (setquotpr (smallestAdequateRelation R) v)
                   (setquotpr (smallestAdequateRelation R) w)).
           apply (aptwice (λ r s, r × s)).
           { apply agreement_on_gens0. assumption. }
           { apply agreement_on_gens0. assumption. } } Qed.
  Lemma agreement_on_gens {X I} {R:I→reln X} {M:monoid}
        (f g:Hom (universalMarkedMonoid R) M) :
        (∏ i, f (setquotpr (smallestAdequateRelation R) (word_gen i)) =
                   g (setquotpr (smallestAdequateRelation R) (word_gen i)))
          → f = g.
    intros p. apply funEquality.
    apply funextsec; intro t; simpl in t.
    apply (surjectionisepitosets _ _ _ (issurjsetquotpr _)).
    { apply setproperty. } { apply agreement_on_gens0. assumption. } Qed.
  Definition universality0 {X I} {R:I→reln X} (M:MarkedMonoid R) :
    universalMarkedMonoid0 R → M.
  Proof.
    apply (setquotuniv _ _ (evalwordMM M)).
    exact (λ _ _ r, r (MarkedMonoid_to_hrel M) (abelian_group_adequacy R M)).
  Defined.
  Definition universality1 {X I} (R:I→reln X)
                           (M:MarkedMonoid R) (v w:universalMarkedMonoid0 R) :
    universality0 M (v × w) = universality0 M v × universality0 M w.
  Proof. intros. isaprop_goal ig. { apply setproperty. }
    apply (squash_to_prop (lift R v) ig); intros [v' j]; destruct j.
    apply (squash_to_prop (lift R w) ig); intros [w' []].
    reflexivity. Qed.
  Definition universality2 {X I} {R:I→reln X} (M:MarkedMonoid R) :
    monoidfun (universalMarkedMonoid R) M.
  Proof. intros. ∃ (universality0 M).
      split. { intros v w. apply universality1. } { reflexivity. } Defined.

  Local Arguments pr1monoidfun {X Y} f x.
  Theorem iscontrMarkedMonoidMap {X I} {R:I→reln X} (M:MarkedMonoid R) :
        iscontr (MarkedMonoidMap (universalMarkedMonoid R) M).
  Proof. intros.
    assert (g := make_MarkedMonoidMap X I R
                           (universalMarkedMonoid R) M
                           (universality2 M) (λ x, idpath _)).
    ∃ g. intros f. apply MarkedMonoidMapEquality.
    apply funEquality. apply funextsec; intro v.
    isaprop_goal ig. { apply setproperty. }
    apply (squash_to_prop (lift R v) ig); intros [w []].
    exact ((ap f (universalMarkedMonoid2 R w))
         @ MarkedMonoidMap_compat2 f g w @ !(ap g (universalMarkedMonoid2 R w))).
  Defined.
End Presentation.
Module Free.
  Import Presentation.
  Definition make (X:Type) := @universalMarkedMonoid X empty fromempty.
End Free.
Definition NN := Free.make unit.
Module Product.
  Definition make {I} (X:I→monoid) : monoid.
    intros. ∃ (Magma.Product.make X). split.
    intros a b c. apply funextsec; intro. apply assocax.
    ∃ (λ i, unel (X i)). split.
    intros a. apply funextsec; intro. apply lunax.
    intros a. apply funextsec; intro. apply runax. Defined.
  Definition Proj {I} (X:I→monoid) (i:I) : Hom (make X) (X i).
    intros. ∃ (pr1 (Magma.Product.Proj X i)). split.
    exact (pr2 (Magma.Product.Proj X i)). simpl. reflexivity. Defined.
  Definition Fun {I} (X:I→monoid) (T:monoid) (g: ∏ i, Hom T (X i))
             : Hom T (make X).
    intros. ∃ (pr1 (Magma.Product.Fun X T g)).
    ∃ (pr2 (Magma.Product.Fun X T g)). apply funextsec; intro i.
    exact (pr2 (pr2 (g i))). Defined.
  Definition Eqn {I} (X:I→monoid) (T:monoid) (g: ∏ i, Hom T (X i))
             : ∏ i, Proj X i ∘ Fun X T g = g i.
    intros. apply funEquality. reflexivity. Qed.
  Lemma issurjective_projection {I} (X:I→monoid) (i:I) :
    isdeceq I → issurjective (Proj X i).

  Proof. intros decide_equality xi. apply hinhpr.
    simple refine (_,,_).
    { intro j. destruct (decide_equality i j) as [p|_].
      { exact (transportf X p xi). }
      { exact (unel (X j)). } }
    simpl. destruct (decide_equality i i) as [q|r]; simpl.
    { assert (e : idpath i = q).
      { apply isasetifdeceq. assumption. }
      destruct e. reflexivity. }
    { destruct (r (idpath i)). } Qed.
  Lemma issurjective_projection' {I} (X:I→monoid) (i:I) :
    LEM → isaset I → issurjective (Proj X i).
  Proof. intros lem is. apply issurjective_projection.
    apply LEM_for_sets. assumption. assumption. Qed.
End Product.