I would like to prove this in real analysis, but I’m finding it difficult to do it in Coq. Can someone help me, please.
If $f(x_j)$ is non-linear
$x_j \geq 0$, and
$f(x_j) \geq 0$, and
$f(0) = 0$, and
$f'(x_j)<0$ (negative derivative) and
$f'(x_j) = f'(x_i)$, such that $j$ and $i$ are different, and
$f'(x_1)+f'(x_2)+\dots+f'(x_J) = G(x_1+x_2+\dots+x_J) = G(X)$
Than $f(x_j)$ should be some thing like tins:
$f(x_j) = -ax_j^2 + bx_j$ such that $a$ and $b$ are positive constants and $x_j ≤ b/a$.
This is what I have so far.
Require Import Reals.
Open Scope R_scope.
(*f(xj) is non-linear*)
(*/\ represent 'and'*)
Definition additive (f : R -> R) := forall x y : R, f (x + y) = f x + f y.
Definition homogeneous_deg_one (f : R -> R) := forall x a : R, f (a * x) = a * f x.
Definition linear (f : R -> R) := additive f /\ homogeneous_deg_one f.
Definition non_linear (f : R -> R) := ~ linear f.
(*f(xj)>0*)
Definition positive_for_all (f : R -> R) := forall xj : R, f xj > 0.
(*f′(xj)<0*)
Definition derivative (f : R -> R) (x : R) := (f (x + 1/1000) - f x) / 1/1000.
Definition decreasing_at (f f' : R -> R) (xj : R) := f' xj < 0.
(*f′(xj)=f′(xi), such that j and i are different*)
Definition equal_derivatives_at_different_points (f : R -> R) (xi xj : R) := xi <> xj /\ derivative f xi = derivative f xj.
(*exist G(X)=f′(x1)+f′(x2)+⋯+f′(xJ) and X=x1+x2+⋯+xJ *)
Require Import List.
Import ListNotations.
Fixpoint sum_list (l : list R) : R :=
match l with
| [] => 0
| x :: xs => x + sum_list xs
end.
Definition sum_derivatives (f' : R -> R) (l : list R) := sum_list (map f' l).
Definition exists_G (f' : R -> R) :=
exists G : R -> R, forall l : list R, G (sum_list l) = sum_derivatives f' l.
(*Lemma*)
Parameter my_f : R -> R.
Lemma my_function_is_0 : non_linear my_f.
Lemma my_function_is_1 : positive_for_all my_f.
Lemma my_function_is_2 : decreasing_at my_f.
Lemma my_function_is_3 : equal_derivatives_at_different_points my_f.
(*Hypothesis*)
Parameter a : R.
Parameter L : R -> R.
(*Hypothesis L_is_linear : exists m b : R, forall x : R, L x = m * x + b.*)
Lemma L_function_is_linear : linear L.
Definition ff (x : R) := a * x^2 + L x.
Hypothesis my_f_equals_f : forall x : R, my_f x = f x.
