Comment by xorcist

1. Sledgehammer/CoqHammer are automated proof-search/assistant tools that bridge interactive proof assistants (Isabelle/Coq) to external SMT provers, they aren't about concolic testing in a PL sense. They translate goals and context to formats the external provers understand, call those provers, and replay/translate returned proofs or proof fragments back into the ITP, that's search and translation of a complete proof, not running concrete+symbolic program executions like concolic testing.

2. there is no exact analogue of fuzz-testing bytecode for "theorem fuzzing", but arguably the closest match is counterexample generators - model finders, finite-model search, SMT instantiation with concrete valuations, all serve a role similar to fuzzers by finding invalid conjectures quickly.

these are weaker than fuzzing for finding execution bugs because mathematical statements are higher-level and provers operate symbolically.

3. here's how proof frameworks are constructed, at the high-level:

a. start by picking a logical foundation: e.g., first-order logic (FOL), higher-order logic (HOL), dependent type theory (Martin-Löf/CIC).

b. define syntax (terms, types, formulas) and typing/judgment rules (inference rules or typing rules).

c. define proof objects or proof rules: natural deduction, sequent calculus, Hilbert-style axioms, or type-as-proofs (Curry-Howard).

d. pick or design the kernel: small, trusted inference engine that checks proof objects (reduction rules, conversion, rule admissibility).

e. add automation layers: tactics, decision procedures, external ATP/SMT, rewriting engines.

f. provide libraries of axioms/definitions and extraction/interpretation mechanisms (code extraction, models).

g. implement proof search strategies and heuristics (backtracking, heuristics, lemma databases, proof-producing solvers).

this is the standard engineering pipeline behind modern ITPs and automated systems.

4. yes, many proof assistants and automated provers treat computation inside proofs as:

a. term rewriting / reduction (beta-reduction, delta-unfolding, normalization) for computation oriented parts; this is the "computation" layer.

b. a separate deductive/logic layer for reasoning (inference rules, quantifier instantiation, congruence closure).

the combined model is: terms represent data/computation; rewriting gives deterministic computation semantics; logical rules govern valid inference about those terms. Dependent type theories conflate computation and proof via conversion (definitional equality) in the kernel.

5. here's how a proof step's semantics is defined:

proof steps are applications of inference rules transforming sequents/judgments. formally:

a. a judgment form J (e.g., Γ ⊢ t : T or Γ ⊢ φ) is defined.

b. inference rules are of the form: from premises J1,...,Jn infer J, written as (J1 ... Jn) / J.

c. a proof is a finite tree whose nodes are judgments; leaves are axioms or assumptions; each internal node follows an inference rule.

d. for systems with computation, a reduction relation → on terms defines definitional equality; many rules use conversion: if t →* t' and t' has form required by rule, the rule applies.

e. in type-theoretic kernels, the step semantics is checking that a proof object reduces/normalizes and that constructors/eliminators are used respecting typing/conversion; the kernel checks a small set of primitive steps (e.g., beta, iota reductions, rule applications).

operationally: a single proof step = instantiate a rule schema with substitutions for metavariables, perform any required reductions/conversions, and check side conditions (freshness, well-formedness).

equational reasoning and rewriting: a rewrite rule l → r can be applied to a term t at a position p if the subterm at p matches l under some substitution σ; result is t with that subterm replaced by σ(r). Confluence/termination properties determine global behavior.

higher-level tactics encode sequences of such primitive steps (rule applications + rewrites + searches) but are not part of kernel semantics; only the kernel rules determine validity.

relevant concise implications for mapping to PL semantics:

treat proof state as a machine state (context Γ, goal G, local proof term), tactics as programs that transform state; kernel inference rules are the atomic instruction set; rewriting/normalization are deterministic evaluation semantics used inside checks.

automation ≈ search processes over nondeterministic tactic/program choices; model finders/SMTs act as external oracles producing either counterexamples (concrete) or proof certificates (symbolic).

These questions are hard to understand.

1. Yes, the Sledgehammer suite contains three testing systems that I believe are concolic (quickcheck, nitpick, nunchaku), but they only work due to hand-coded support for the mathematical constructs in question. They'd be really inefficient for a formalised software environment, because they'd be operating at a much lower-level than the abstractions of the software environment, unless dedicated support for that software environment were provided.

2. Quickcheck is a fuzz-testing framework, but it doesn't help to construct anything (except as far as it constructs examples / counterexamples). Are you thinking of something that automatically finds and defines intermediary lemmas for arbitrary areas of mathematics? Because I'm not aware of any particular work in that direction: if computers could do that, there'd be little need for mathematicians.

3. By thinking really hard, then writing down the mathematics. Same way you write computer programs, really, except there's a lot more architectural work. (Most of the time, the computer can brute-force a proof for you, so you need only choose appropriate intermediary lemmas.)

4. I can't parse the question, but I suspect you're thinking of the meta-logic / object-logic distinction. The actual steps in the term rewriting are not represented in the object logic: the meta-logic simply asserts that it's valid to perform these steps. (Not even that: it just does them, accountable to none other than itself.) Isabelle's meta-logic is software, written in a programming language called ML.