Fixpoint D (e : expr) n {struct e} : expr * domain :=
match e with
| Var v => (if ssrnat.eqn v n then Cst 1 else Cst 0, Always)
| Cst _ => (Cst 0, Always)
| AppExt k f le =>
let lnc := index_not_const le n in
let ld := map (fun e => D e n) le in
match lnc with
| nil => (Cst 0, Always)
| v :: nil =>
let '(d1,d2) := nth (Cst 0,Never) ld v in
(Binary Emult d1 (AppExtD k f v le),
And (Derivable v k f le :: d2 :: nil))
| v1 :: v2 :: nil =>
let '(d1,d2) := nth (Cst 0,Never) ld v1 in
let '(d3,d4) := nth (Cst 0,Never) ld v2 in
(Binary Eplus (Binary Emult d1 (AppExtD k f v1 le)) (Binary Emult d3 (AppExtD k f v2 le)),
And (Derivable2 v1 v2 k f le :: d2 :: Derivable v2 k f le :: d4 :: nil))
| _ => (Cst 0, Never)
end
| AppExtD k f v le => (Cst 0, Never)
| App f e => (Cst 0, Never)
| Subst f e => (Cst 0, Never)
| Binary b e1 e2 =>
let '(a1,b1) := D e1 n in
let '(a2,b2) := D e2 n in
match b, is_const e1 n, is_const e2 n with
| Eplus, true, _ => (a2, b2)
| Eplus, _, true => (a1, b1)
| Eplus, _, _ => (Binary Eplus a1 a2, And (b1::b2::nil))
| Emult, true, _ => (Binary Emult e1 a2, b2)
| Emult, _, true => (Binary Emult a1 e2, b1)
| Emult, _, _ => (Binary Eplus (Binary Emult a1 e2) (Binary Emult e1 a2), And (b1::b2::nil))
end
| Unary u e =>
let '(a,b) := D e n in
match u with
| Eopp => (Unary Eopp a, b)
| Einv => (Binary Emult (Unary Eopp a) (Unary Einv (Binary Emult e e)), And (b:: (Partial (fun x => x <> 0) e) :: nil))
| Efct f f' H => (Binary Emult a (AppExt 1 f' [:: e]), b)
| Efct' f f' df H => (Binary Emult a (AppExt 1 f' [:: e]), And (b :: (Partial df e) :: nil))
end
| Int f e1 e2 =>
let '(a1,b1) := D e1 n in
let '(a2,b2) := D e2 n in
let '(a3,b3) := D f (S n) in
match is_const f (S n), is_const e1 n, is_const e2 n with
| true, true, _ =>
(Binary Emult a2 (App f e2),
And (b2::(Integrable f e1 e2)::(Forone e2 (Locally 0 (Continuous 0 f)))::nil))
| true, false, true =>
(Unary Eopp (Binary Emult a1 (App f e1)),
And (b1::(Integrable f e1 e2)::(Forone e1 (Locally 0 (Continuous 0 f)))::nil))
| true, false, false =>
(Binary Eplus (Binary Emult a2 (App f e2)) (Unary Eopp (Binary Emult a1 (App f e1))),
And (b1::b2::(Integrable f e1 e2)::(Forone e1 (Locally 0 (Continuous 0 f)))::(Forone e2 (Locally 0 (Continuous 0 f)))::nil))
| false, true, true =>
(Int a3 e1 e2,
And ((ForallWide n e1 e2 b3)::(Locally n (Integrable f e1 e2))::
(Forall e1 e2 (Continuous2 (S n) 0 a3))::nil))
| false, false, true =>
(Binary Eplus
(Unary Eopp (Binary Emult a1 (App f e1)))
(Int a3 e1 e2),
And ((Forone e1 (Locally2 (S n) 0 (Continuous2 (S n) 0 a3)))::
(Forall e1 e2 (Continuous2 (S n) 0 a3))::
b1::(Forone e1 (Locally 0 (Continuous 0 f)))::
ParamIntegrable n f e1 e2::LocallyParamIntegrable n f e1::
ForallWide n e1 e2 b3::nil))
| false, true, false =>
(Binary Eplus
(Binary Emult a2 (App f e2))
(Int a3 e1 e2),
And ((Forone e2 (Locally2 (S n) 0 (Continuous2 (S n) 0 a3)))::
(Forall e1 e2 (Continuous2 (S n) 0 a3))::
b2::(Forone e2 (Locally 0 (Continuous 0 f)))::
ParamIntegrable n f e1 e2::LocallyParamIntegrable n f e2::
ForallWide n e1 e2 b3::nil))
| false, false, false =>
(Binary Eplus
(Binary Eplus
(Binary Emult a2 (App f e2))
(Unary Eopp (Binary Emult a1 (App f e1))))
(Int a3 e1 e2),
And ((Forone e1 (Locally2 (S n) 0 (Continuous2 (S n) 0 a3)))::
(Forone e2 (Locally2 (S n) 0 (Continuous2 (S n) 0 a3)))::
(Forall e1 e2 (Continuous2 (S n) 0 a3))::
b1::(Forone e1 (Locally 0 (Continuous 0 f)))::
b2::(Forone e2 (Locally 0 (Continuous 0 f)))::
ParamIntegrable n f e1 e2::LocallyParamIntegrable n f e1::LocallyParamIntegrable n f e2::
ForallWide n e1 e2 b3::nil))
end
end.

(* Specifications: *)
Lemma D_correct :
forall (e : expr) l n,
let '(a,b) := D e n in
interp_domain l b ->
is_derive (fun x => interp (set_nth R0 l n x) e) (nth R0 l n) (interp l a).