NashEquilibriumCert
plain-language theorem explainer
The NashEquilibriumCert structure records that the NashType inductive contains exactly five constructors, fixing the count of equilibrium refinements at configDim five. Game theorists embedding Recognition Science into economic models cite it to anchor the dimension of credible strategy concepts. The definition is a one-field structure that directly invokes the cardinality count over the enumerated NashType constructors.
Claim. The structure $NashEquilibriumCert$ asserts that the finite type $NashType$ satisfies $|NashType| = 5$, where $NashType$ is the inductive type whose constructors are pure-strategy Nash equilibrium, mixed-strategy Nash equilibrium, subgame-perfect equilibrium, trembling-hand perfect equilibrium, and proper equilibrium.
background
The module sets configDim to five so that five successive refinements of Nash equilibrium fit inside the Recognition Science game layer. NashType is defined as an inductive type with exactly those five constructors: purePsy, mixedStrategy, subgamePerfect, tremblingHandPerfect, and properEquilibrium. This cardinality certificate sits downstream of the J-cost certificates imported from Decision.NashEquilibriumFromJCost and GameTheory.NashEquilibriumFromJCost, which require Jcost 1 = 0 together with strict positivity of Jcost r for all r ≠ 1.
proof idea
The declaration is a structure definition whose single field five_types is supplied by the sibling lemma nashType_count, which computes Fintype.card NashType by exhaustive enumeration of the five constructors.
why it matters
The certificate supplies the dimension-five datum required by the master NashEquilibriumCert instances in Decision and GameTheory, where nashEquilibriumCert assembles joint cost nonnegativity, zero-cost iff canonical profile, and the canonical Nash property. It closes the enumeration step for the economics application of the five-tick refinement chain, consistent with the framework's use of configDim to index equilibrium strength.
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