popularity_nonneg
plain-language theorem explainer
Non-negativity of style popularity as a function of scaled time follows from the piecewise definition and the AM-GM bound on J-cost. Archaeologists modeling Predynastic pottery succession via J-cost minima would cite this to anchor the lower bound of the Petrie curve. The proof splits on the sign of s, applies le_refl in the non-positive case, and reduces the positive case to div_nonneg with Jcost_nonneg_of_pos.
Claim. For every real number $s$, $0$ is less than or equal to popularity$(s)$, where popularity$(s)$ equals $0$ if $s$ is less than or equal to $0$ and equals $1/(1+J(s))$ otherwise, with $J(s)=(s+s^{-1})/2-1$.
background
The module derives Petrie sequence dating from J-cost trajectories on the design graph. Popularity at scaled time $s=t/τ$ is defined as zero for non-positive arguments and as $1/(1+Jcost(s))$ for positive arguments. Jcost_nonneg_of_pos records that Jcost is non-negative for positive $s$ by the AM-GM inequality $s+s^{-1}≥2$.
proof idea
Unfold the definition of popularity and split on the if condition. The non-positive case reduces directly to le_refl. The positive case extracts $0<s$ via lt_of_not_ge, then applies div_nonneg with norm_num on the numerator 1 and Jcost_nonneg_of_pos on the denominator, closing with linarith.
why it matters
Supplies the non-negativity field of the PotterySerialCert structure used by pottery_serial_one_statement, which asserts the peak value 1, the interval bounds [0,1], and the adjacency gap in (0.11,0.13). It completes one required property in the derivation of ceramic style succession from the J-cost functional equation, consistent with T5 J-uniqueness.
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