Set Implicit Arguments.
Unset Strict Implicit.

Require Export pointwise.
Require Export modify_seq.
Require Export vecspaces_verybasic.

Definition mult_by_scalars (R : ring) (V : module R)
  (N : Nat) (a : Map (fin N) R) (v : Map (fin N) V) :
  MAP (fin N) V := pointwise (uncurry (MODULE_comp (Mod:=V))) a v.

Lemma mult_by_scalars_comp :
 ∀ (R : ring) (V : module R) (N : Nat) (a b : MAP (fin N) R)
   (v w : MAP (fin N) V),
 a =' b in _ →
 v =' w in _ → mult_by_scalars a v =' mult_by_scalars b w in _.
intros.
unfold mult_by_scalars in |- ×.
apply (pointwise_comp_general (A:=fin N) (B:=R) (C:=V) (D:=V));
 auto with algebra.
Qed.

Lemma mult_by_scalars_fun2_compatible :
 ∀ (R : ring) (V : module R) (N : Nat),
 fun2_compatible
   (mult_by_scalars (V:=V) (N:=N):MAP _ _ → MAP _ _ → MAP _ _).
exact mult_by_scalars_comp.
Qed.

Hint Resolve mult_by_scalars_fun2_compatible mult_by_scalars_comp: algebra.

Lemma mult_by_scalars_Seqtl :
 ∀ (R : ring) (V : module R) (N : Nat) (a : Map (fin N) R)
   (v : Map (fin N) V),
 mult_by_scalars (Seqtl a) (Seqtl v) =' Seqtl (mult_by_scalars a v) in _.
intros.
unfold mult_by_scalars in |- ×.
apply Sym; auto with algebra.
Qed.

Hint Resolve mult_by_scalars_Seqtl: algebra.

Lemma mult_by_scalars_concat :
 ∀ (R : ring) (V : module R) (N M : Nat) (a : Map (fin N) R)
   (b : Map (fin M) R) (v : Map (fin N) V) (w : Map (fin M) V),
 mult_by_scalars (a ++ b) (v ++ w) ='
 mult_by_scalars a v ++ mult_by_scalars b w in _.
intros.
unfold mult_by_scalars in |- ×.
auto with algebra.
Qed.

Hint Resolve mult_by_scalars_concat: algebra.

Lemma mult_by_scalars_modify_seq :
 ∀ (F : field) (V : vectorspace F) (n : Nat) (p : seq n F)
   (k : seq n V) (i : fin n) (q : F),
 mult_by_scalars (modify_seq p i q) k ='
 modify_seq (mult_by_scalars p k) i (q mX k i) in seq _ _.
simple induction n.
intros; inversion i; inversion in_range_prf.
intros.
case i.
intro; case index.
intros.
apply Trans with (mult_by_scalars (q;; Seqtl p) k); auto with algebra.
apply Trans with (mult_by_scalars (q;; Seqtl p) (head k;; Seqtl k)).
unfold mult_by_scalars in |- ×.
apply toMap; apply pointwise_comp; auto with algebra.
apply Trans with (q mX head k;; mult_by_scalars (Seqtl p) (Seqtl k)).
unfold mult_by_scalars in |- ×.
apply
 Trans
  with
    (uncurry (MODULE_comp (Mod:=V)) (couple q (head k));;
     pointwise (uncurry (MODULE_comp (R:=F) (Mod:=V))) (Seqtl p) (Seqtl k));
 auto with algebra.
apply Trans with (q mX head k;; Seqtl (mult_by_scalars p k)).
apply cons_comp; auto with algebra.
unfold mult_by_scalars in |- ×.
apply
 Trans
  with (q mX k (Build_finiteT in_range_prf);; Seqtl (mult_by_scalars p k));
 auto with algebra.
apply cons_comp; auto with algebra.

intros m Hm.
set (Hm' := lt_S_n _ _ Hm) in ×.
apply
 Trans with (hdtl (mult_by_scalars (modify_seq p (Build_finiteT Hm) q) k));
 auto with algebra.
apply
 Trans
  with
    (hdtl
       (modify_seq (mult_by_scalars p k) (Build_finiteT Hm)
          (q mX k (Build_finiteT Hm)))); auto with algebra.
unfold hdtl in |- ×.
apply
 Trans
  with
    (head (mult_by_scalars p k);;
     Seqtl
       (modify_seq (mult_by_scalars p k) (Build_finiteT Hm)
          (q mX k (Build_finiteT Hm)))); auto with algebra.
apply
 Trans
  with
    (head (mult_by_scalars (modify_seq p (Build_finiteT Hm) q) k);;
     mult_by_scalars (Seqtl (modify_seq p (Build_finiteT Hm) q)) (Seqtl k)).
auto with algebra.
apply
 Trans
  with
    (head (mult_by_scalars (modify_seq p (Build_finiteT Hm) q) k);;
     mult_by_scalars (modify_seq (Seqtl p) (Build_finiteT Hm') q) (Seqtl k)).
apply cons_comp; auto with algebra.
apply cons_comp; auto with algebra.
apply
 Trans
  with
    (mult_by_scalars (Seqtl (modify_seq p (Build_finiteT Hm) q)) (Seqtl k));
 auto with algebra.
apply
 Trans
  with
    (modify_seq (Seqtl (mult_by_scalars p k)) (Build_finiteT Hm')
       (q mX k (Build_finiteT Hm))); auto with algebra.
apply
 Trans
  with
    (modify_seq (mult_by_scalars (Seqtl p) (Seqtl k))
       (Build_finiteT Hm') (q mX k (Build_finiteT Hm)));
 auto with algebra.
apply
 Trans
  with
    (mult_by_scalars (modify_seq (Seqtl p) (Build_finiteT Hm') q) (Seqtl k)).
apply toMap; apply mult_by_scalars_comp; auto with algebra.
apply
 Trans
  with
    (modify_seq (mult_by_scalars (Seqtl p) (Seqtl k))
       (Build_finiteT Hm') (q mX Seqtl k (Build_finiteT Hm')));
 auto with algebra.
Qed.

Hint Resolve mult_by_scalars_modify_seq: algebra.