Example by Nicky Vazou, unfinished

Require Import Arith.
Require Import Coq.Classes.DecidableClass.
Require Import Coq.Program.Wf.
Require Import List Lia.
Require Import PeanoNat.
Require Import Program.
Require Import Wf.
Require Import Equations.

Import ListNotations.

Set Keyed Unification.

Class Associative {T: Type} (op: T T T) :=
  {
    associativity: x y z, op x (op y z) = op (op x y) z;
  }.

Class Monoid (T: Type) :=
  MkMonoid {
    unit: T;
    op: T T T;
    monoid_associative: Associative op;
    monoid_left_identity: x, op unit x = x;
    monoid_right_identity: x, op x unit = x;
  }.

Instance app_Associative: T, Associative (@app T).
Proof.
  intro T.
  constructor.
  induction x.
  { reflexivity. }
  { simpl. congruence. }
Defined.

Instance list_Monoid: T, Monoid (list T).
Proof.
  intro T.
  apply MkMonoid with (unit := []) (op := @app T).
  { auto with typeclass_instances. }
  { reflexivity. }
  { induction x.
    { reflexivity. }
    { simpl. congruence. }
  }
Defined.

Notation "a ** b" := (op a b) (at level 50).

Class MonoidMorphism
      {Tn Tm: Type}
      `{Mn: Monoid Tn} `{Mm: Monoid Tm}
      (f: Tn Tm)
  :=
  {
    morphism_unit: f unit = unit;
    morphism_op: x y, f (x ** y) = f x ** f y;
  }.

Class ChunkableMonoid (T: Type) `{Monoid T} :=
  MkChunkableMonoid {
    length: T nat;
    drop: nat T T;
    take: nat T T;
    drop_spec:
       i x,
        length (drop i x) = length x - i;
    take_spec:
       i x,
        i length x
        length (take i x) = i;
    take_drop_spec:
       i x, x = take i x ** drop i x;
  }.

Fixpoint list_take {T: Type} i (l: list T) :=
  match i, l with
  | 0, _[]
  | _, [][]
  | S i, h::th :: list_take i t
  end.

Fixpoint list_drop {T: Type} i (l: list T) :=
  match i, l with
  | 0, _l
  | _, [][]
  | S i, h::tlist_drop i t
  end.

Instance list_ChunkableMonoid: T, ChunkableMonoid (list T).
Proof.
  intro T.
  apply MkChunkableMonoid
  with (length := @List.length T) (drop := list_drop) (take := list_take);
    intros.
  { generalize dependent x.
    induction i, x; intros; intuition.
  }
  { generalize dependent x.
    induction i, x; intros; intuition.
    simpl in ×.
    rewrite IHi; intuition.
  }
  { generalize dependent x.
    induction i, x; intros; intuition.
    simpl in ×.
    rewrite <- IHi.
    reflexivity.
  }
Defined.

Section Chunk.
  Context{T : Type} `{M : ChunkableMonoid T}.
  Set Program Mode.
  Equations? chunk (i: { i : nat | i > 0 }) (x : T) : list T by wf (length x) lt :=
  chunk i x with dec (length x <=? i) :=
    { | left _[x] ;
      | right ptake i x :: chunk i (drop i x) }.
  Proof. apply leb_complete_conv in p. rewrite drop_spec. lia. Qed.
End Chunk.

Theorem if_flip_helper {B: Type} {b: bool}
        (C E: true = b B) (D F: false = b B):
  ( (r: true = b), C r = E r)
  ( (r: false = b), D r = F r)
  (if b as an return an = b B then C else D) eq_refl =
  (if b as an return an = b B then E else F) eq_refl.
Proof.
  intros.
  destruct b.
  apply H.
  apply H0.
Qed.

(* Transparent chunk.
Eval compute in (chunk (exist _ 3 _) 0; 1; 2; 3; 4; 5; 6; 7; 8; 9). *)

(*
  = [0; 1; 2]; [3; 4; 5]; [6; 7; 8]; [9]
  : list (list nat)
 *)


Section mconcat.
  Context {M : Type} `{Monoid M}.

  Equations mconcat (l: list M): M :=
  mconcat [] := unit;
  mconcat (cons x xs) := x ** mconcat xs.
End mconcat.
Transparent mconcat.

Theorem morphism_distribution:
   {M N: Type}
    `{MM: Monoid M} `{MN: Monoid N}
    `{CMM: @ChunkableMonoid N MN}
    (f: N M)
    `{MMf: @MonoidMorphism _ _ MN MM f},
     i x,
      f x = mconcat (map f (chunk i x)).
Proof.
  intros.
  funelim (chunk i x).
  { simpl. simp mconcat. now rewrite monoid_right_identity. }
  simpl. simp mconcat.
  rewrite <- H; auto.
  rewrite <- morphism_op.
  now rewrite <- take_drop_spec.
Qed.

Lemma length_list_drop: {T: Type} i (x: list T),
  i < Datatypes.length x
  Datatypes.length (list_drop i x) = Datatypes.length x - i.
Proof.
  intros.
  generalize dependent i.
  induction x; destruct i; simpl; intros.
  { reflexivity. }
  { reflexivity. }
  { reflexivity. }
  { apply IHx. intuition. }
Qed.

Lemma length_chunk_base:
   {T: Type} I (x: list T),
    let i := proj1_sig I in
    i > 1
    Datatypes.length x i
    Datatypes.length (chunk I x) = 1.
Proof.
  intros; subst i.
  funelim (chunk I x). reflexivity.
  simpl.
  apply leb_correct in H1. rewrite e0 in H1. discriminate.
Qed.
Ltac feed H :=
  match type of H with
  | ?foo _
    let FOO := fresh in
    assert foo as FOO; [|specialize (H FOO); clear FOO]
  end.

Lemma length_chunk_lt:
   {T: Type} I (x: list T),
    let i := proj1_sig I in
    i > 1
    Datatypes.length x > i
    Datatypes.length (chunk I x) < Datatypes.length x.
Proof.
  intros; subst i.
  funelim (chunk I x).
  simpl. lia.
  simpl.
  specialize (H H0).
  revert H. unfold drop. simpl.
  pose proof (drop_spec (` i) x). simpl in H.
  rewrite H by lia. clear H.
  simp chunk. clear Heq. destruct dec. simp chunk; simpl; intros; try lia. intros.
  feed H.
  clear H. apply leb_complete_conv in e.
  pose proof (drop_spec (` i) x). rewrite H in e; try lia;
                                    unfold length in *; simpl in *; lia.
  lia.
Qed.

Section pmconcat.
  Context {M : Type} `{ChunkableMonoid M}.

  Equations? pmconcat (I : { i : nat | i > 0 }) (x : list M) : M by wf (length x) lt :=
  pmconcat i x with dec ((` i <=? 1) || (length x <=? ` i))%bool ⇒ {
    | left Hmconcat x ;
    | right Hdpmconcat i (map mconcat (chunk i x)) }.
  Proof. clear pmconcat.
    rewrite map_length.
    rewrite Bool.orb_false_iff in Hd.
    destruct Hd. apply leb_complete_conv in H2. apply leb_complete_conv in H3.
    apply length_chunk_lt; simpl; auto.
  Qed. (* 0.264s from 1.571s *)
End pmconcat.

Instance mconcat_mon T : MonoidMorphism (@mconcat (list T) _).

Next Obligation.
Proof.
  funelim (mconcat x). reflexivity.
  simpl. rewrite H. now rewrite <- app_assoc.
Qed.

Theorem concatEquivalence: {T: Type} i (x: list (list T)),
    pmconcat i x = mconcat x.
Proof.
  intros.
  funelim (pmconcat i x).
  reflexivity.
  rewrite H. now rewrite <- (morphism_distribution mconcat).
Qed.

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