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3 posts tagged with "function convergence"

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· 6 min read

In Tezos, the skip-list data structure is used in several places. Verification of skip-list required many days of painstaking work, and this work is not yet finished. In this post, we will talk about the techniques we used. In particular, we describe the verification of function mem - one of the functions operating in this data structure. More information about formal verification of skip list can be found here.

· 5 min read

Formal verification of the skip list of Tezos includes the work on verification of the following properties :

  • Validity of a cell (containing content and a list of back pointers),
  • Validity of a dereferencing operator,
  • Validity of encoding functions for cell and pointer,
  • Validity of backpointers,
  • Verification of equality definition,
  • Verification of group of functions and definitions serving the organization of data-structure : next, genesis, best_skip, mem, pp and others,
  • Validity of a back_path (the list of backpointers),
  • Uniqueness of express line (back_path, list of pointers, generated with usage of best_skip function).

· 4 min read
Natasha Klaus

As you may already know from the article Simulations - dependently-typed version, in the Proto_alpha folder we keep code, generated by the coq-of-ocaml. Many of the fixpoints here are preceded by the annotation #[bypass_check(guard)], which means that guard checking is locally disabled. And this means that we can not consider these functions to be total. In such cases, proof engineers usually feel irritation, because "total" is one of their favorite words. We know that total functions never crush and they always return a well-typed result within a finite time.


Thanks to Alan Turing we know that the halting problem - the difficulty of determining whether a specific program terminates or not - is undecidable. Coq can't determine if a function is total in general, according to the original definition of totality. Instead, it analyzes a function's syntax and makes a conservative approximation.

Our current goal is that the doubles defined in the folder Simulations, whose equivalence with their prototypes from Proto_alpha should be proved, would be total (have enabled Coqs guard checking on fixpoint). We already mentioned the Guard Checking question here.

Coq is analyzing the function's syntax to consider it to be total. One way to convince Coq that a function is total is to show that there is a decreasing argument that converges to the base case.

Let's consider the function parse_ty_aux from Proto_alpha and its clone dep_parse_ty_aux from the Simulations folder. parse_ty_aux is having {struct node_value} annotation. The point of the {struct ident} annotation is to tell the Coq, which argument decreases along the recursive calls. node_value : Alpha_context.Script.node is designed in such a way that it is not trivial to show that the function converges. Coq can not detect that each and every recursion call is done on the direct subterm of a given node_value argument (and it is indeed actually not the case).