semiconductorType_count
plain-language theorem explainer
The theorem states that the finite type enumerating semiconductor configurations has exactly five elements. Researchers modeling band structures from configDim in Recognition Science would cite this cardinality when assembling the semiconductor certification. The proof is a direct decide computation on the Fintype instance induced by the five-constructor inductive definition.
Claim. The cardinality of the set of semiconductor types is five: $|$SemiconductorType$| = 5$.
background
The module treats semiconductor band structure as determined by configDim equal to five, with five canonical types: intrinsic, n-type doped, p-type doped, compensated, and degenerate. Band-gap energies sit on the phi-ladder. The upstream inductive definition of SemiconductorType automatically derives Fintype, DecidableEq, and Repr, supplying the finite set whose cardinality is asserted here.
proof idea
The proof is a one-line wrapper that applies the decide tactic to evaluate Fintype.card on the inductive type with exactly five constructors.
why it matters
This result supplies the five_types field inside the semiconductorCert definition that certifies the full band-structure model. It realizes the configDim = 5 enumeration required by the Recognition Science solid-state layer, with band gaps positioned on the phi-ladder. The declaration closes the type-count step for downstream certification without introducing new hypotheses.
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