pedagogyModel_count
The theorem fixes the cardinality of the pedagogy model type at five, matching the five canonical approaches listed for education depth five. Curriculum designers and recognition theorists working in the configDim equals five setting would cite this count when enumerating the distinct channels of exposition, practice, exploration, synthesis, and enculturation. The proof is a direct decision on the Fintype instance derived from the inductive definition.
claimThe set of pedagogy models has cardinality five: $|P| = 5$, where $P$ enumerates direct instruction, mastery learning, inquiry-based learning, project-based learning, and apprenticeship.
background
The module defines an inductive type with five constructors that stand for the canonical pedagogy models under configDim equal to five. These models map one-to-one onto the recognition channels of exposition, practice, exploration, synthesis, and enculturation. The upstream inductive definition supplies the DecidableEq, Repr, BEq, and Fintype instances required for the cardinality statement.
proof idea
The proof is a one-line wrapper that invokes the decide tactic on the equality Fintype.card of the finite inductive type to five.
why it matters in Recognition Science
This cardinality populates the five_models field inside the PedagogyModelsCert definition. It anchors the E5 education depth inside the Recognition Science framework by confirming exactly five models for the five recognition channels when configDim equals five.
scope and limits
- Does not derive the five models from the J-function or forcing chain.
- Does not compare learning outcomes across the models.
- Does not extend the count to other values of configDim.
- Does not address curriculum implementation details.
formal statement (Lean)
27theorem pedagogyModel_count : Fintype.card PedagogyModel = 5 := by decide
proof body
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