Can we ban arguments that don't meet conditions? - coq

Question

I want to partially derive functions whose input is a dependent list.
deriveP has an error because EucAppend fk (pk ::: lk) of length is not always n, but I always expects a list whose length is P.
This is due to the definition of lastk and firstk.
To solve this problem, lastk and firstk must return only Euc k, not Euc n.
I want to ban arguments that n and k don't meet k <= n in lastk and firstk.
I don't know how to do it. Please tell 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 : forall n, Euc n -> Euc (Nat.min k n) :=
match k with
| 0%nat => fun _ _ => []
| S k' => fun n =>
match n return Euc n -> Euc (Nat.min (S k') n) with
| 0%nat => fun _ => []
| S n' => fun v =>
snoc (lastk k' _ (but_last v)) (last v)
end
end.
(* extract first k elements *)
Fixpoint firstk k :forall n, Euc n -> Euc (Nat.min k n) :=
match k with
| 0%nat => fun _ _ => []
| S k' => fun n =>
match n return Euc n -> Euc (Nat.min (S k') n) with
| 0%nat => fun _ => []
| S n' => fun v => (head v) ::: firstk k' _ (tail v)
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' n' (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.
deriveP partially derive fnctions. I (EucAppend fk (pk ::: lk)) is where the error is.
Definition deriveP {n A} (k:nat) (I:Euc n -> Euc A) (p :Euc n) :=
let fk := firstk k P p in
let lk := lastk (P-(k+1)) P p in
(Derive (fun pk => I (EucAppend fk (pk ::: lk)) )) (EucNth k (P-1) p).

Answer 1

You can work with an order relation as you mentioned. My recommendation is to avoid proofs inside your definition (which makes proving after more complex), example :
Fixpoint lastk k n : Euc n -> k < n -> Euc k :=
match n with
|0 => fun _ (H : k < 0) => False_rect _ (Lt.lt_n_O _ H)
|S n => match k with
|S m => fun v H => snoc (lastk (but_last v) (le_S_n _ _ H)) (last v)
|0 => fun _ H => []
end
end.
Fixpoint firstk k n : Euc n -> k < n -> Euc k :=
match n with
|0 => fun _ (H : k < 0) => False_rect _ (Lt.lt_n_O _ H)
|S n => match k with
|S m => fun v H => (head v) ::: firstk (tail v) (le_S_n _ _ H)
|0 => fun _ H => []
end
end.
This definition is transparent which makes it easy to prove after using k n as inductions points.
The vectodef library works with Fin types(finite sequences). You can do a workaround to make it comfortable to extract the definition :
Fixpoint of_nat {n} (x : t n) : nat :=
match x with
|#F1 _ => 0
|#FS _ y => S (of_nat y)
end.
Fixpoint lastk n (H : t n) (v : Euc n) : Euc (of_nat H) :=
match H as t in (t n0) return (Euc n0 -> Euc (of_nat t)) with
| #F1 n0 => fun=> [ ]
| #FS n0 H1 =>
fun H2 : Euc (S n0) => snoc (lastk H1 (but_last H2)) (last H2)
end v.
Theorem of_nat_eq : forall y k (H : k < y), of_nat (of_nat_lt H) = k.
intros y k.
elim/#nat_double_ind : y/k.
intros;inversion H.
intros; auto.
intros; simply.
by rewrite -> (H (Lt.lt_S_n _ _ H0)).
Qed.
Definition last_leb n k (v : Euc n) : k < n -> Euc k.
intros.
rewrite <- (of_nat_eq H).
exact (#lastk _ (of_nat_lt H) v).
Show Proof.
Defined.
But..., as I mentioned this has proofs in the terms.
I think you probably will need another proof for deriveP, but I don't know the definition of Derive, please consider to specify at least the type definition.