We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill- and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain A, a codomain B and a "filter" T on finite approximations of functions from A to B, a generalised form GDCABT of the axiom of dependent choice and dually a generalised bar induction principle GBIABT such that:GDCABTintuitionistically captures the strength of·the general axiom of choice expressed as ∀a∃bR(a,b) ⇒ ∃α∀aR(a,α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints,·the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set \mathbbB (for a constructive definition of prime filter),·the axiom of dependent choice if A = \mathbbN,·Weak Knig's Lemma if A = \mathbbN and B = \mathbbB (up to weak classical reasoning).GBIABTintuitionistically captures the strength ofGödel's completeness theorem in the form validity implies provability for entailment relations if B = \mathbbB (for a constructive definition of validity),·bar induction if A = \mathbbN,·the Weak Fan Theorem if A = \mathbbN and B = \mathbbB.Contrastingly, even though GDCABT and GBIABTsmoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is \mathbbB\mathbbNand B is \mathbbN.

Abstract In verified generic programming, one cannot exploit the structure of concrete data types but has to rely on well chosen sets of specifications or abstract data types (ADTs). Functors and monads are at the core of many applications of functional programming. This raises the question of what useful ADTs for verified functors and monads could look like. The functorial map of many important monads preserves extensional equality. For instance, if $$f,g \, : \, A \, \to \, B$$ are extensionally equal, that is, $$\forall x \in A$$ , $$f \, x = g \, x$$ , then $$map \, f \, : \, List \, A \to List \, B$$ and $$map \, g$$ are also extensionally equal. This suggests that preservation of extensional equality could be a useful principle in verified generic programming. We explore this possibility with a minimalist approach: we deal with (the lack of) extensional equality in Martin-Löf’s intensional type theories without extending the theories or using full-fledged setoids. Perhaps surprisingly, this minimal approach turns out to be extremely useful. It allows one to derive simple generic proofs of monadic laws but also verified, generic results in dynamical systems and control theory. In turn, these results avoid tedious code duplication and ad-hoc proofs. Thus, our work is a contribution toward pragmatic, verified generic programming.

Abstract In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely solved using Bellman’s backward induction. Textbook authors (e.g. Bertsekas or Puterman) typically give more or less formal proofs to show that the backward induction algorithm is correct as solution method for deterministic and stochastic SDPs. Botta, Jansson and Ionescu propose a generic framework for finite horizon, monadic SDPs together with a monadic version of backward induction for solving such SDPs. In monadic SDPs, the monad captures a generic notion of uncertainty, while a generic measure function aggregates rewards. In the present paper, we define a notion of correctness for monadic SDPs and identify three conditions that allow us to prove a correctness result for monadic backward induction that is comparable to textbook correctness proofs for ordinary backward induction. The conditions that we impose are fairly general and can be cast in category-theoretical terms using the notion of Eilenberg–Moore algebra. They hold in familiar settings like those of deterministic or stochastic SDPs, but we also give examples in which they fail. Our results show that backward induction can safely be employed for a broader class of SDPs than usually treated in textbooks. However, they also rule out certain instances that were considered admissible in the context of Botta et al. ’s generic framework. Our development is formalised in Idris as an extension of the Botta et al. framework and the sources are available as supplementary material.