CondWithA

CondWithA is a generalized conditional operator used in formal logic and computer science to select between two expressions based on a predicate A. Rather than a fixed true/false switch, CondWithA ties the branch choice to whether a given input satisfies A.

Formal definition can be given in a functional style. Let D be a domain, A: D → Bool a predicate, and T, F: D → C be total functions into some codomain C. Then CondWithA_A(T, F)(x) equals T(x) if A(x) is true, and F(x) if A(x) is false. Equivalently, CondWithA can be expressed as a higher-order function: CondWithA = λA. λT. λF. λx. if A(x) then T(x) else F(x). This form makes the dependency on the input explicit and separates the predicate from the outcome expressions.

Usage and applications. CondWithA is used in functional programming to enable predicate-based control flow, in program synthesis to reason about branching conditioned on properties of inputs, and in formal verification to model decision points that hinge on a property rather than a simple boolean flag. It supports constructing adaptive or data-dependent branches and can simplify reasoning about families of conditionals parameterized by A.

Variants and relationship. The concept can be extended to multi-way branches with a family of predicates or to compose CondWithA with other control structures. CondWithA reduces to a standard if-then-else when A is a fixed constant predicate, illustrating it as a generalization of traditional conditionals.

See also: if-then-else, conditional operator, generalized conditionals.