functional programming - How to prove uniqueness of a function in Coq given a specification? -

given specification of function, e.g., specification_of_sum, how prove in coq 1 such function exists?

i studying mathematics, , prove in hand, skills in coq limited (proving using rewrite , apply).

i found code snippet below, i've been struggling time now.

i try unfold specification in proof, using old friend rewrite doesn't seem let me go further.

can explain how approach problem using simple syntax?

definition specification_of_sum (sum : (nat -> nat) -> nat -> nat) :=   forall f : nat -> nat,     sum f 0 = f 0     /\     forall n' : nat,       sum f (s n') = sum f n' + f (s n').  (* ********** *)  theorem there_is_only_one_sum :   forall sum1 sum2 : (nat -> nat) -> nat -> nat,     specification_of_sum sum1 ->     specification_of_sum sum2 ->     forall (f : nat -> nat)            (n : nat),       sum1 f n = sum2 f n. proof.   abort. 

the following start ejgallego described.

intros sum1 sum2 h1 h2 f n. (* introduce hypotheses *)                                      unfold specification_of_sum in *. (* unfold definition in places *)                             specialize h1 (f := f). (* narrow statement f *)                                   specialize h2 (f := f). (* narrow statement f *)                                   inversion_clear h1.  (* split , statements *)                                              inversion_clear h2.                                                                                 (* induction on n, , rewrites *) 

i have included few more basic commands make slower simpler. rest of proof requires rewrite , reflexivity.