I want to partially derive functions whose input is a dependent list.
I tried to define deriveP with proving.
Derive is a function in Coquelicot.Derive.
Definition deriveP {P A B}(k:nat)(I:Euc (S P) -> Euc A -> Euc B)
(input:Euc A)(train:Euc B)(p :Euc (S P))
:(lt k (S P)) -> (lt ((S P)-(k+1)) (S P)) -> R.
intros.
pose fk := firstk k (S P) p H.
pose lk := lastk ((S P)-(k+1)) (S P) p H0.
pose pk := EucNth k p.
apply arith_basic in H.
exact ( Derive (fun PK => EucSum (QuadraticError (I (fk +++ (PK ::: lk)) input) train )) pk ).
I can not apply arith_basic poposed by Tiago because H is used in fk.
I can apply arith_basic to H before I make fk, but then I can not make fk because There is not k < P.+1.
I want to apply arith_basic to H while leaving k < P.+1.
Please help me.
(***********************************************************)
This is dependent list of R.
Require Import Coq.Reals.Reals.
Require Import Coquelicot.Coquelicot.
Inductive Euc:nat -> Type:=
|RO : Euc 0
|Rn : forall {n:nat}, R -> Euc n -> Euc (S n).
Notation "[ ]" := RO.
Notation "[ r1 , .. , r2 ]" := (Rn r1 .. ( Rn r2 RO ) .. ).
Infix ":::" := Rn (at level 60, right associativity).
Basic list operation.
Definition head {n} (v : Euc (S n)) : R :=
match v with
| x ::: _ => x
end.
Definition tail {n} (v : Euc (S n)) : Euc n :=
match v with
| _ ::: v => v
end.
(* extract the last element *)
Fixpoint last {n} : Euc (S n) -> R :=
match n with
| 0%nat => fun v => head v
| S n => fun v => last (tail v)
end.
(* eliminate last element from list *)
Fixpoint but_last {n} : Euc (S n) -> Euc n :=
match n with
| 0%nat => fun _ => []
| S n => fun v => head v ::: but_last (tail v)
end.
(* do the opposite of cons *)
Fixpoint snoc {n} (v : Euc n) (x : R) : Euc (S n) :=
match v with
| [] => [x]
| y ::: v => y ::: snoc v x
end.
(* extract last k elements *)
Fixpoint lastk k n : Euc n -> (lt k n) -> Euc k :=
match n with
|0%nat => fun _ (H : lt k 0) => False_rect _ (Lt.lt_n_O _ H)
|S n => match k with
|S m => fun v H => snoc (lastk m n (but_last v) (le_S_n _ _ H)) (last v)
|0%nat => fun _ H => []
end
end.
(* extract first k elements *)
Fixpoint firstk k n : Euc n -> (lt k n) -> Euc k :=
match n with
|0%nat => fun _ (H :lt k 0) => False_rect _ (Lt.lt_n_O _ H)
|S n => match k with
|S m => fun v H => (head v) ::: firstk m n (tail v) (le_S_n _ _ H)
|0%nat => fun _ _ => []
end
end.
(* extract nth element *)
(* 0 origine *)
Fixpoint EucNth (k:nat) :forall {n}, Euc (S n) -> R:=
match k with
| 0%nat => fun _ e => head e
| S k' => fun n =>
match n return Euc (S n) -> R with
| 0%nat => fun e => head e
| S n' => fun v => EucNth k' (tail v)
end
end.
Fixpoint EucAppend {n m} (e:Euc n) (f:Euc m) :Euc (n+m):=
match e with
|[] => f
|e' ::: es => e' ::: (EucAppend es f)
end.
Infix "+++" := EucAppend (at level 60, right associativity).
Fixpoint QuadraticError {n : nat} (b : Euc n) : Euc n -> Euc n.
refine (match b in Euc n return Euc n -> Euc n with
|@Rn m x xs => _
|@RO => fun H => []
end).
remember (S m).
intro H; destruct H as [| k y ys].
inversion Heqn0.
inversion Heqn0.
subst; exact ((x - y)^2 ::: QuadraticError _ xs ys).
Defined.
Fixpoint EucSum {A}(e:Euc A) :R:=
match e with
| [] => 0%R
| e' ::: es => e' + (EucSum es)
end.
I have solved on my own.
We can duplicate lemma in the sub-goal with generalize tactic.
Definition deriveP {P A B}(k:nat)(I:Euc (S P) -> Euc A -> Euc B)
(input:Euc A)(train:Euc B)(p :Euc (S P))
:(lt k (S P)) -> (lt ((S P)-(k+1)) (S P)) -> R.
intros.
generalize H.
intro H1.
apply arith_basic in H1.
pose lk := lastk ((S P)-(k+1)) (S P) p H0.
pose fk := firstk k (S P) p H.
pose pk := EucNth k p.
rewrite (_: (P.+1)%nat = (k + (P.+1 - (k + 1)%coq_nat)%coq_nat.+1)%coq_nat) in I.
exact ( Derive (fun PK => EucSum (QuadraticError (I (fk +++ (PK ::: lk)) input) train )) pk ).
apply H1.
Defined.