Require Import UniMath.Algebra.GaussianElimination.Elimination.
Require Import UniMath.Algebra.GaussianElimination.RowOps.

Require Import UniMath.Algebra.Matrix.

Require Import UniMath.Combinatorics.Maybe.
Require Import UniMath.Combinatorics.StandardFiniteSets.
Require Import UniMath.Combinatorics.Vectors.

Require Import UniMath.Algebra.Archimedean.
Require Import UniMath.Algebra.BinaryOperations.
Require Import UniMath.Algebra.Domains_and_Fields.
Require Import UniMath.Algebra.IteratedBinaryOperations.
Require Import UniMath.Algebra.RigsAndRings.

(** Observing/testing the extent that it is
    possible to compute some of the Elimination material.

    Primary author: Daniel @Skantz (November 2022) *)

(** This section tests just the matrix/vector operations, avoiding use of the ring operations as far as possible. *)
Section Tests_1.

  Context (R : rig).
  Context (F : fld).

(* natcommrig is not available at this point (requires the later package NaturalNumbersAlgebra).
   Restating the below is sufficient for the tests, which could still be nice to have. *)
  Local Definition natcommrig' : commrig.
  Proof.
    split with (make_setwith2binop natset (make_dirprod (λ n m : nat, n + m) (λ n m : nat, n * m))).
    split.
    - split.
      + split with
        (make_dirprod
            (make_dirprod
               (make_dirprod natplusassoc (@make_isunital natset _ 0 (make_dirprod natplusl0 natplusr0)))
               natpluscomm)
            (make_dirprod natmultassoc (@make_isunital natset _ 1 (make_dirprod natmultl1 natmultr1)))).
        apply (make_dirprod natmult0n natmultn0).
      + apply (make_dirprod natldistr natrdistr).
    - unfold iscomm. apply natmultcomm.
  Defined.

  Local Definition nattorig' {X : rig} (n : nat) : X :=
      natmult (X := rigaddabmonoid X) n 1%rig.

  (* End duplicated definitions. *)

  Let v1 := (append_vec empty_vec 2).
  Let v2 := (append_vec empty_vec 3).

  Let eval1 := Eval compute in firstValue (@pointwise nat _ mul v1 v2).

  Local Lemma eq1 : eval1 = 6. Proof. apply idpath. Defined.

  Let v3 := (append_vec (append_vec empty_vec 2) 3).
  Let v4 := (append_vec (append_vec empty_vec 3) 4).

  Let eval2 := Eval compute in firstValue (@pointwise nat _ mul v1 v2).

  Local Lemma eq2 : eval2 = 6. Proof. apply idpath. Defined.

  Let eval3 := Eval compute in stnsum (@pointwise nat _ mul v3 v4).

  Local Lemma eq3 : eval3 = 18. Proof. apply idpath. Defined.

  Let e1 := (@nattorig natcommrig' 2).
  Let e2 := (@nattorig natcommrig' 3).

  Let v5 := (append_vec (append_vec empty_vec e1) e1).
  Let v6 := (append_vec (append_vec empty_vec e2) e2).

  Let m1 := append_vec (row_vec v5) v6.

  (* dot product of [2, 2], [3, 3] *)
  Let eval4 :=
      Eval compute
        in ((iterop_fun (@rigunel2 natcommrig') op1 (pointwise _ op2 v5 v6))).
  (* 12 *)

  Local Lemma eq4 : eval4 = 12. Proof. apply idpath. Defined.

  (* matrix product : [2, 2] [2, 2] = [10, 10]
                      [3, 3] [3, 3]   [15, 15] *)

  (* Computing entry (1, 1) : *)
  Let eval5 :=
      Eval compute in firstValue (firstValue (@matrix_mult _ _ _ m1 _ m1)).

  Local Lemma eq5 : eval5 = 10. Proof. apply idpath. Defined.

  Let m5 := (@identity_matrix natcommrig' 10).

  Let eval6 := Eval cbn in (@matrix_mult _ _ _ m5 _ m5).

  Local Lemma eq6 : firstValue (firstValue eval6) = (@nattorig natcommrig' 1).
  Proof. apply idpath. Defined.

  (* Switch : [1, 0] -> [0, 1]
              [0, 1]    [1, 0]*)

  Let eval7 := @switch_row _ 2 2 (0,, (natgthsnn 0)) (1,, (natgthsnn _)) (@identity_matrix R 2).

  Local Lemma eq7 : (firstValue (firstValue eval7)) = (@rigunel1 R).
  Proof. apply idpath. Defined.

  Local Lemma eq8 : (firstValue (lastValue eval7)) = (@rigunel2 R).
  Proof. apply idpath. Defined.

  Let m9 := (@identity_matrix natcommrig' 2).

  Let v9 := @iterop_fun (Matrix natcommrig' 2 2) m9 (λ X : Matrix natcommrig' 2 2, matrix_mult X).

  (* [ [I_2 * I_2 * I_2]_{1, 1}*)
  Let eval9 := Eval vm_compute in (firstValue (firstValue (v9 3 (λ X : stn 3, m9)))).

  Let eq9 : eval9 = (@rigunel2 natcommrig').
  Proof. apply idpath. Defined.

End Tests_1.

(** This section tests computation in the integers [hz] and rationals [hq].
    Requires importing NaturalNumberSystems.{RationalNumbers, Integers}.

Section Tests_2.

    (* Context {R: rig}.
    Context {F: fld}. *)

    (* Very slow *)
    (* Eval cbn in (1 + 1 + 1 + 1 + 1)%hz. *)

    Let v3 := (append_vec empty_vec (1 + 1)%hz).
    Let v4 := (append_vec empty_vec (1 + 1)%hq).

    (* Let eval6 := Eval cbn in (op2 (1 + 1) (1 + 1))%hz. *)
    (* rigtoringop2int natcommrig (make_dirprod 2 0) (make_dirprod 2 0)) *)

    (* < 1 second, not computing. *)
    (* Eval cbn in (firstValue (@pointwise hz _ op2 v3 v3)).
    (* rigtoringop2int _ (make_dirprod 2 0) (make_dirprod 2 0) *) *)

    (* < 1 second, not computing. *)
    (* Eval cbn in
        firstValue (firstValue (@matrix_mult hz _ _ (row_vec v3) _ (col_vec v3))). *)

    (* Let v9 := (append_vec (append_vec empty_vec (@rigunel1 hq)) (@rigunel2 hq)). *)

    (* Slow, not computing. Neither is the dual. *)
    (* Let eval9 := Eval native_compute in leading_entry_compute _ v9. *)

    (* A minute - and not computing *)
    (* Let eval6 := Eval cbn in
        ((@gaussian_elimination hq _ _ (row_vec v3))). *)

End Tests_2. *)
