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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.
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  • $\begingroup$ It is dificult to answer the question because we do not know what your level of experience with Coq is. If you are a complete beginner, you might want to try some simpler exercises before you dive into real analysis. Also, you should explain a bit more about what you have tried or where specifically you get stuck. Just saying "it is difficult, please help me" does not give us much to go with. $\endgroup$ May 11, 2023 at 20:47
  • $\begingroup$ My function (f) has too many constraints. So I think it must have a specific shape. I'd just like to prove this using coq, but this is the first time I'm using this. I'll try some more. Thanks. $\endgroup$ May 11, 2023 at 22:39
  • $\begingroup$ Do you know how to write the proof in pen and paper? $\endgroup$
    – Trebor
    May 12, 2023 at 3:35
  • $\begingroup$ @FelipeMorelli: If you've never used a proof assistant, it's going to be quite a challenge to prove your theorem in Coq. Good luck! $\endgroup$ May 12, 2023 at 5:49
  • $\begingroup$ Thant guys.I will try some easy proofs an than I come back to try this one. Like Brazilian and Portuguese say: "Don't put the cart in front of the oxen". $\endgroup$ May 14, 2023 at 12:55

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