structure
definition
def or abbrev
ProbabilityDistribution
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view Lean formalization →
formal statement (Lean)
120structure ProbabilityDistribution (Ω : Type*) [Fintype Ω] where
121 p : Ω → ℝ
122 nonneg : ∀ ω, 0 ≤ p ω
123 sum_one : ∑ ω, p ω = 1
124
125/-! ## Entropy and Free Energy -/
126
127/-- Recognition Entropy S_R for a probability distribution p.
128 S_R(p) = -∑_ω p(ω) ln(p(ω))
129 Convention: 0 * ln(0) = 0 (handled via lim). -/
used by (31)
-
is_fault_tolerant -
coarse_graining_decreases_free_energy -
h_theorem_recognition -
rs_arrow_of_time -
EntropyAncestorCertificate -
entropy_maximizer_is_gibbs -
free_energy_gap_is_kl -
gibbs_unique -
free_energy_kl_identity -
gibbs_minimizes_free_energy_basic -
gibbs_unique_minimizer -
max_ent_subject_to_cost -
entropy_increase_under_conservation -
evolve -
evolve_equilibrium_eq -
evolve_succ -
evolve_zero -
fe_kl_id -
free_energy_antitone -
free_energy_ge_equilibrium -
free_energy_le_of_le -
free_energy_step_le -
gibbsPD -
JDescentOperator -
kl_divergence_antitone -
kl_le_of_le -
kl_step_le -
second_law -
SecondLawCert -
second_law_entropy_form