Progress in Computer-Assisted Inductive Theorem Proving by Human-Orientedness and Descente Infinie?

In this short position paper we briefly review the development history of automated inductive theorem proving and computer-assisted mathematical induction. We think that the current low expectations on progress in this field result from a faulty narrow-scope historical projection. Our main motivation is to explain–on an abstract but hopefully sufficiently descriptive level–why we believe that future progress in the field is to result from human-orientedness and descente infinie.

💡 Research Summary

The paper offers a concise historical overview of automated inductive theorem proving (AITP) and computer‑assisted mathematical induction, arguing that the modest expectations for future progress stem from a narrow, historically biased projection. Early efforts in the 1970s focused on mechanically applying structural and mathematical induction, but they quickly ran into difficulties when dealing with complex data structures, multiple induction variables, and non‑linear recursive definitions. The main obstacle was the generation of appropriate induction hypotheses, a step that required human intuition and could not be reliably automated.

In the late 1970s and 1980s, the community introduced proof‑planning and strategy‑based approaches. These systems attempted to encode high‑level proof strategies and to automate the selection of tactics, yet they still depended heavily on user‑provided hints. The authors label this situation “projection bias”: researchers have been fixated on the ideal of full automation, thereby overlooking the potential of human‑machine collaboration and producing tools that are difficult for mathematicians and engineers to adopt in practice.

To overcome this bias, the authors propose two complementary concepts: human‑orientedness and descente infinie. Human‑orientedness is not merely a nicer user interface; it is a design philosophy that makes the proof process transparent, interactive, and amendable at any stage. The paper outlines concrete design principles: (1) visual representation of the proof tree and rewrite rules, (2) editable induction hypotheses that users can insert, modify, or delete on the fly, (3) an interactive loop where automated steps and manual interventions seamlessly alternate, and (4) comprehensive logging for reproducibility and educational reuse. This approach contrasts sharply with the traditional “black‑box” provers that hide internal reasoning from the user.

Descente infinie, originally introduced by Fermat, is reinterpreted as a systematic reduction technique that drives the goal toward ever‑smaller sub‑goals without explicitly separating base and inductive cases. The authors argue that this method is especially powerful when a direct induction hypothesis is hard to discover, such as in proofs involving intricate recursion or mutual induction. They propose a framework that couples rewrite and reduction rules: the prover automatically decomposes the target theorem into a sequence of smaller lemmas, applies descente steps, and then presents the intermediate sub‑goals to the user for validation or adjustment. This creates a feedback‑driven cycle where the machine handles routine reductions while the human supplies strategic insight.

From an implementation standpoint, the paper analyses the inductive modules of three major proof assistants—ACL2, Isabelle/HOL, and Coq—and suggests how to retrofit them with the proposed human‑oriented UI and descente‑infinie engine. The suggested extensions include: a real‑time graphical UI displaying the proof tree, click‑to‑edit induction hypotheses, an automatically generated list of descente sub‑goals with user‑controlled prioritisation, and a standardized plugin API that allows these features to be added without rewriting the core logic of the existing systems.

In conclusion, the authors contend that the pursuit of “complete automation” is a misguided goal that limits the field’s impact. Instead, fostering a collaborative environment where humans and provers co‑operate—leveraging human intuition through a transparent interface and exploiting descente infinie for systematic goal reduction—offers a realistic path toward substantial advances. By rebalancing the emphasis from pure automation to human‑oriented, reduction‑driven proof development, the paper envisions a new generation of inductive provers that are both powerful and practically usable in complex mathematical and engineering domains.