controlArchitecture_count
The declaration establishes that exactly five control architectures exist for robots at configuration dimension five. Researchers in autonomous systems would cite the count to anchor architecture taxonomies. The proof is a one-line decision procedure that exhausts the finite inductive type.
claimThe cardinality of the type of robot control architectures is five, consisting of the reactive, deliberative, hybrid, behavior-based, and learning-based variants: $5 = |$reactive, deliberative, hybrid, behavior-based, learning-based$|$.
background
The module introduces five canonical robot-control architectures for configDim equal to five: reactive, deliberative, hybrid, behavior-based, learning-based. These are encoded as an inductive type deriving Fintype, DecidableEq, and related instances. The setting is the robotics depth of Recognition Science, with no axioms or sorrys in the module.
proof idea
The proof applies the decide tactic in one line. The tactic succeeds due to the Fintype derivation on the inductive type with five constructors, confirming the cardinality by enumeration.
why it matters in Recognition Science
This theorem supplies the five_architectures field in the controlArchitecturesCert definition. It completes the enumeration step in the robotics application of the framework. The result stands apart from the core forcing chain and physical constants.
scope and limits
- Does not derive the architectures from the J-function or Recognition Composition Law.
- Does not prove that these five exhaust all possible control schemes.
- Does not link to spatial dimensions or the eight-tick octave.
formal statement (Lean)
23theorem controlArchitecture_count : Fintype.card ControlArchitecture = 5 := by decide
proof body
24