(** *****************************************************************************

Internal lattice objects in a category

Contents:

- Lattice objects ([latticeob])
- Bounded lattice objects ([bounded_latticeob])
- Proof that a subobject of a (bounded) lattice object is a lattice object
  ([sublatticeob], [sub_bounded_latticeob])

Based on "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk (Section IV.8, page 198)

Written by: Anders Mörtberg, 2017

*********************************************************************************)

Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.Tactics.

Require Import UniMath.OrderTheory.Lattice.Lattice.
Require Import UniMath.OrderTheory.Lattice.Bounded.

Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Monics.
Require Import UniMath.CategoryTheory.Limits.BinProducts.
Require Import UniMath.CategoryTheory.Limits.Terminal.

Local Open Scope cat.

(** * Definition of lattice objects and bounded lattice objects *)
Section LatticeObject_def.

Context {C : category} {BPC : BinProducts C}.

Local Notation "c ⊗ d" := (BinProductObject C (BPC c d)) : cat.
Local Notation "f '××' g" := (BinProductOfArrows _ _ _ f g) (at level 80) : cat.
Local Notation "1" := (identity _) : cat.

Let π1 {x y} : C⟦x ⊗ y,x⟧ := BinProductPr1 _ (BPC x y).
Let π2 {x y} : C⟦x ⊗ y,y⟧ := BinProductPr2 _ (BPC x y).

Definition binprod_assoc (x y z : C) : C⟦(x ⊗ y) ⊗ z,x ⊗ (y ⊗ z)⟧ :=
  BinProductArrow _ _ (π1 · π1) (BinProductArrow _ _ (π1 · π2) π2).
Let α {x y z} : C⟦(x ⊗ y) ⊗ z,x ⊗ (y ⊗ z)⟧ := binprod_assoc x y z.

Definition binprod_delta (x : C) : C⟦x,x ⊗ x⟧ :=
  BinProductArrow _ _ (identity x) (identity x).
Let δ {x} : C⟦x,x ⊗ x⟧ := binprod_delta x.

Definition binprod_swap (x y : C) : C⟦x ⊗ y,y ⊗ x⟧ :=
  BinProductArrow _ _ (BinProductPr2 _ _) (BinProductPr1 _ _).
Let τ {x y} : C⟦x ⊗ y,y ⊗ x⟧ := binprod_swap x y.


(** Equation witnessing that a morphism representing a binary operation is
    associative as illustrated by the diagram:
<<
               f×1
 (L ⊗ L) ⊗ L -------> L ⊗ L
     |                  |
   α |                  |
     V                  |
 L ⊗ (L ⊗ L)            | f
     |                  |
 1×f |                  |
     V                  V
   L ⊗ L ----------->   L
              f
>>
*)
Definition isassoc_cat {L} (f : C⟦L ⊗ L,L⟧) : UU := (f ×× 1) · f = α · (1 ×× f) · f.


(** Equation witnessing that a morphism representing a binary operation is
    commutative as illustrated by the diagram:
<<
   L ⊗ L
     |   \
     |     \
   τ |       \ f
     |         \
     |          V
   L ⊗ L -----> L
           f
>>
*)
Definition iscomm_cat {L} (f : C⟦L ⊗ L,L⟧) : UU := f = τ · f.


(** Equation witnessing the absorbtion law as illustrated by the diagram:
<<
           δ×1                   α
   L ⊗ L ------> (L ⊗ L) ⊗ L -------> L ⊗ (L ⊗ L)
     |                                    |
  π1 |                                    | 1×g
     V                                    V
     L <------------------------------- L ⊗ L
                       f
>>

If f is ∧ and g is ∨ this expresses: x ∧ (x ∨ y) = x

*)
Definition isabsorb_cat {L} (f g : C⟦L ⊗ L,L⟧) : UU :=
  (δ ×× 1) · α · (1 ×× g) · f = π1.

Definition latticeop_cat {L} (meet_mor join_mor : C⟦L ⊗ L,L⟧) :=
    (isassoc_cat meet_mor × iscomm_cat meet_mor)
  × (isassoc_cat join_mor × iscomm_cat join_mor)
  × (isabsorb_cat meet_mor join_mor × isabsorb_cat join_mor meet_mor).

(** A lattice object L has operation meet and join satisfying the above laws *)
Definition latticeob (L : C) : UU :=
  ∑ (meet_mor join_mor : C⟦L ⊗ L,L⟧), latticeop_cat meet_mor join_mor.

Definition make_latticeob {L : C} {meet_mor join_mor : C⟦L ⊗ L,L⟧} :
  latticeop_cat meet_mor join_mor → latticeob L :=
    λ (isL : latticeop_cat meet_mor join_mor), meet_mor,, join_mor ,, isL.

Definition meet_mor {L : C} (isL : latticeob L) : C⟦L ⊗ L,L⟧ := pr1 isL.
Definition join_mor {L : C} (isL : latticeob L) : C⟦L ⊗ L,L⟧ := pr1 (pr2 isL).

(** Bounded lattice objects *)

Context {TC : Terminal C}.

Let ι {x : C} : C⟦x,TC ⊗ x⟧ :=
  BinProductArrow _ _ (TerminalArrow _ _) (identity x).

(** Given u : C⟦TC,L⟧ the equation witnessing the left unit law is given
    by the diagram:
<<
          ι
     L ------> 1 ⊗ L
     |           |
   1 |           | u×1
     V           V
     L <------ L ⊗ L
          f
>>
*)
Definition islunit_cat {L} (f : C⟦L ⊗ L,L⟧) (u : C⟦TC,L⟧) : UU :=
  ι · (u ×× 1) · f = 1.

Definition bounded_latticeop_cat {L} (l : latticeob L) (bot top : C⟦TC,L⟧) :=
  (islunit_cat (join_mor l) bot) × (islunit_cat (meet_mor l) top).

Definition bounded_latticeob (L : C) : UU :=
  ∑ (l : latticeob L) (bot top : C⟦TC,L⟧), bounded_latticeop_cat l bot top.

Definition make_bounded_latticeob {L} {l : latticeob L} {bot top : C⟦TC,L⟧} :
  bounded_latticeop_cat l bot top → bounded_latticeob L := λ bl, l,, bot,, top,, bl.

Definition bounded_latticeob_to_latticeob X : bounded_latticeob X → latticeob X := pr1.
Coercion bounded_latticeob_to_latticeob : bounded_latticeob >-> latticeob.

Definition bot_mor {L} (isL : bounded_latticeob L) : C⟦TC,L⟧ := pr1 (pr2 isL).
Definition top_mor {L} (isL : bounded_latticeob L) : C⟦TC,L⟧ := pr1 (pr2 (pr2 isL)).

End LatticeObject_def.

Arguments latticeob {_} _ _.
Arguments bounded_latticeob {_} _ _ _.

Section LatticeObject_accessors.

Context {C : category} (BPC : BinProducts C) {L : C} (isL : latticeob BPC L).

Definition isassoc_meet_mor : isassoc_cat (meet_mor isL) :=
  pr1 (pr1 (pr2 (pr2 isL))).
Definition iscomm_meet_mor : iscomm_cat (meet_mor isL) :=
  pr2 (pr1 (pr2 (pr2 isL))).
Definition isassoc_join_mor : isassoc_cat (join_mor isL) :=
  pr1 (pr1 (pr2 (pr2 (pr2 isL)))).
Definition iscomm_join_mor : iscomm_cat (join_mor isL) :=
  pr2 (pr1 (pr2 (pr2 (pr2 isL)))).
Definition meet_mor_absorb_join_mor : isabsorb_cat (meet_mor isL) (join_mor isL) :=
  pr1 (pr2 (pr2 (pr2 (pr2 isL)))).
Definition join_mor_absorb_meet_mor : isabsorb_cat (join_mor isL) (meet_mor isL) :=
  pr2 (pr2 (pr2 (pr2 (pr2 isL)))).

End LatticeObject_accessors.

Section BoundedLatticeObject_accessors.

Context {C : category} (BPC : BinProducts C) (TC : Terminal C).
Context {L : C} (l : bounded_latticeob BPC TC L).

Definition islunit_join_mor_bot_mor : islunit_cat (join_mor l) (bot_mor l) :=
  pr1 (pr2 (pr2 (pr2 l))).

Definition islunit_meet_mor_top_mor : islunit_cat (meet_mor l) (top_mor l) :=
  pr2 (pr2 (pr2 (pr2 l))).

End BoundedLatticeObject_accessors.


(** * Definition of sublattice objects  *)
Section SublatticeObject.

Context {C : category} (BPC : BinProducts C) {M L : C}.
Context {i : C⟦M,L⟧} (Hi : isMonic i) (l : latticeob BPC L).

Local Notation "c ⊗ d" := (BinProductObject C (BPC c d)) : cat.
Local Notation "f '××' g" := (BinProductOfArrows _ _ _ f g) (at level 90) : cat.

(** This asserts that i is a lattice homomorphism internally *)
Context {meet_mor_M : C⟦M ⊗ M,M⟧} (Hmeet : meet_mor_M · i = (i ×× i) · meet_mor l).
Context {join_mor_M : C⟦M ⊗ M,M⟧} (Hjoin : join_mor_M · i = (i ×× i) · join_mor l).

Local Lemma identity_comm : identity M · i = i · identity L.
Proof.
  rewrite id_left, id_right. reflexivity.
Qed.

Local Lemma binprod_assoc_comm :
  ((i ×× i) ×× i) · @binprod_assoc _ BPC L L L =
  @binprod_assoc _ BPC M M M · (i ×× (i ×× i)).
Proof.
unfold binprod_assoc; rewrite postcompWithBinProductArrow.
apply BinProductArrowUnique.
- rewrite <-assoc, BinProductPr1Commutes.
  rewrite assoc, BinProductOfArrowsPr1, <- assoc, BinProductOfArrowsPr1, assoc.
  reflexivity.
- rewrite postcompWithBinProductArrow.
  apply BinProductArrowUnique.
  + etrans; [ apply cancel_postcomposition; rewrite <-assoc;
              apply maponpaths, BinProductPr2Commutes |].
    rewrite <- assoc, BinProductPr1Commutes.
    rewrite assoc, BinProductOfArrowsPr1, <- assoc, BinProductOfArrowsPr2, assoc.
    reflexivity.
  + etrans; [ apply cancel_postcomposition; rewrite <-assoc;
              apply maponpaths, BinProductPr2Commutes |].
    rewrite <- assoc, BinProductPr2Commutes, BinProductOfArrowsPr2.
    reflexivity.
Qed.

Local Lemma binprod_delta_comm :
  i · @binprod_delta _ BPC L = @binprod_delta _ BPC M · (i ×× i).
Proof.
unfold binprod_delta; rewrite postcompWithBinProductArrow.
apply BinProductArrowUnique.
- rewrite <-assoc, BinProductPr1Commutes, identity_comm. reflexivity.
- rewrite <-assoc, BinProductPr2Commutes, identity_comm. reflexivity.
Qed.

Local Lemma isassoc_cat_comm {f : C⟦M ⊗ M,M⟧} {g : C⟦L ⊗ L,L⟧} (Hfg : f · i = (i ×× i) · g) :
  isassoc_cat g → isassoc_cat f.
Proof.
unfold isassoc_cat; intros H; apply Hi.
rewrite <-!assoc, !Hfg, !assoc, BinProductOfArrows_comp, Hfg, <- !assoc, identity_comm.
rewrite <- BinProductOfArrows_comp, <- assoc, H, !assoc.
apply cancel_postcomposition.
rewrite <-!assoc, BinProductOfArrows_comp, Hfg, identity_comm.
rewrite <- BinProductOfArrows_comp, !assoc, binprod_assoc_comm.
reflexivity.
Qed.

Local Lemma iscomm_cat_comm {f : C⟦M ⊗ M,M⟧} {g : C⟦L ⊗ L,L⟧} (Hfg : f · i = (i ×× i) · g) :
  iscomm_cat g → iscomm_cat f.
Proof.
unfold iscomm_cat; intros H; apply Hi.
rewrite <- !assoc, !Hfg.
etrans; [eapply maponpaths, H|].
rewrite !assoc; apply cancel_postcomposition.
unfold binprod_swap; rewrite postcompWithBinProductArrow.
apply BinProductArrowUnique; rewrite <- assoc.
* now rewrite BinProductPr1Commutes, BinProductOfArrowsPr2.
* now rewrite BinProductPr2Commutes, BinProductOfArrowsPr1.
Qed.

Local Lemma isabsorb_cat_comm {f1 f2 : C⟦M ⊗ M,M⟧} {g1 g2 : C⟦L ⊗ L,L⟧}
  (Hfg1 : f1 · i = (i ×× i) · g1) (Hfg2 : f2 · i = (i ×× i) · g2) :
  isabsorb_cat g1 g2  → isabsorb_cat f1 f2.
Proof.
unfold isabsorb_cat; intros H; apply Hi.
assert (HH : BinProductPr1 C (BPC M M) · i = (i ×× i) · BinProductPr1 C (BPC L L)).
{ now rewrite BinProductOfArrowsPr1. }
rewrite HH, <- H, <-!assoc, Hfg1, !assoc.
apply cancel_postcomposition.
rewrite <-!assoc, BinProductOfArrows_comp, Hfg2, identity_comm, !assoc.
rewrite BinProductOfArrows_comp, <-identity_comm, binprod_delta_comm.
etrans; [| eapply pathsinv0; do 2 apply cancel_postcomposition;
           now rewrite <-BinProductOfArrows_comp].
rewrite <-!assoc; apply maponpaths.
rewrite assoc, binprod_assoc_comm, <-assoc; apply maponpaths.
now rewrite identity_comm, BinProductOfArrows_comp.
Qed.

Definition sub_latticeob : latticeob BPC M.
Proof.
use make_latticeob.
- apply meet_mor_M.
- apply join_mor_M.
- repeat split.
  + now apply (isassoc_cat_comm Hmeet), (isassoc_meet_mor _ l).
  + now apply (iscomm_cat_comm Hmeet), (iscomm_meet_mor _ l).
  + now apply (isassoc_cat_comm Hjoin), (isassoc_join_mor _ l).
  + now apply (iscomm_cat_comm Hjoin), (iscomm_join_mor _ l).
  + now apply (isabsorb_cat_comm Hmeet Hjoin), (meet_mor_absorb_join_mor _ l).
  + now apply (isabsorb_cat_comm Hjoin Hmeet), (join_mor_absorb_meet_mor _ l).
Defined.

End SublatticeObject.

Section SubboundedlatticeObject.

Context {C : category} (BPC : BinProducts C) (TC : Terminal C).
Context {M L : C} {i : C⟦M,L⟧} (Hi : isMonic i) (l : bounded_latticeob BPC TC L).

Local Notation "c ⊗ d" := (BinProductObject C (BPC c d)) : cat.
Local Notation "f '××' g" := (BinProductOfArrows _ _ _ f g) (at level 90) : cat.

Context {meet_mor_M : C⟦M ⊗ M,M⟧} (Hmeet : meet_mor_M · i = (i ×× i) · meet_mor l).
Context {join_mor_M : C⟦M ⊗ M,M⟧} (Hjoin : join_mor_M · i = (i ×× i) · join_mor l).
Context {bot_mor_M : C⟦TC,M⟧} (Hbot : bot_mor_M · i = bot_mor l).
Context {top_mor_M : C⟦TC,M⟧} (Htop : top_mor_M · i = top_mor l).

Lemma islunit_cat_comm
  {fM : C⟦M ⊗ M,M⟧} {fL : C⟦L ⊗ L,L⟧} (Hf : fM · i = (i ×× i) · fL)
  {gM : C ⟦TC,M⟧} {gL : C⟦TC,L⟧} (Hg : gM · i = gL) :
  islunit_cat fL gL → islunit_cat fM gM.
Proof.
unfold islunit_cat; intros H; apply Hi.
rewrite <-!assoc.
etrans; [ do 2 apply maponpaths; apply Hf |].
rewrite identity_comm, <-H, postcompWithBinProductArrow, !assoc.
apply cancel_postcomposition.
rewrite !postcompWithBinProductArrow, <-assoc, Hg, !id_left.
apply pathsinv0, BinProductArrowUnique.
- rewrite <- assoc, BinProductPr1Commutes, assoc.
  now apply cancel_postcomposition, TerminalArrowUnique.
- now rewrite <- assoc, BinProductPr2Commutes, id_right.
Qed.

Definition sub_bounded_latticeob : bounded_latticeob BPC TC M.
Proof.
use make_bounded_latticeob.
- exact (sub_latticeob BPC Hi l Hmeet Hjoin).
- exact bot_mor_M.
- exact top_mor_M.
- split.
  + now apply (islunit_cat_comm Hjoin Hbot), (islunit_join_mor_bot_mor BPC TC l).
  + now apply (islunit_cat_comm Hmeet Htop), (islunit_meet_mor_top_mor BPC TC l).
Defined.

End SubboundedlatticeObject.
