A method of defining and reasoning about potentially infinite or circular structures and behaviors by characterizing them as greatest fixed points of operators and proving properties via coinductive hypotheses such as bisimulations and observations.

Principle

Coinduction establishes membership or equivalence in a largest fixed-point set: to prove that an object satisfies a coinductive property, show it exhibits the observable behaviors that are preserved by the coinductive definition (often by demonstrating a bisimulation or a preservation condition).

Demonstration

Streams (infinite sequences) are often handled coinductively: to prove two streams s and t are equal, one defines a bisimulation relation R and shows (s,t) ∈ R and that R is preserved by taking heads and tails, thus coinductively concluding s = t.

Misapplication

Applying induction-style finite-construction arguments to inherently non-well-founded structures, or using coinduction without verifying the preservation/invariance condition (thus assuming properties that are not closed under the coalgebraic unfolding), is a misuse.

Consequence

Coinductive reasoning enables sound, compositional proofs about infinite data, reactive systems, and behaviors (e.g., streams, processes, infinite trees) and supports definitions by observable behavior rather than construction.

Reversal

Induction (least fixed-point reasoning) is the dual: it proves membership by finite construction from base cases and constructors, while coinduction proves membership by demonstrating that an element cannot be distinguished from members of the coinductive set by observable moves.

Boundary

Applicable to non-well-founded or corecursive domains modeled as coalgebras and to systems specified by greatest fixed points; not appropriate for properties that require finite termination or well-founded induction without additional justification.

Semantic Tension

Tension appears between treating objects extensionally (by observable behavior, coinduction) and intensionally (by construction, induction); some proofs may be convertible between the two only when additional finiteness or guardedness constraints hold.

Synthesis

Coinductive reasoning is the practice of proving properties of infinite or circular structures by showing preservation of observable behavior under unfolding; it complements induction by targeting greatest fixed-point characterizations and enabling reasoning about ongoing processes and infinite data.