s1209.py

from research import *
s=begin(1209,'State the exact cap-round commutation condition','When does rounding commute with the cap?',{},['C1208','C1131 inherited cap-round theorem'])
def Q(x):return 5*((x+2)//5)
proof='For any nondecreasing Q, if L<=c then Q(L)<=Q(c), and both Q(min(L,c)) and min(Q(L),Q(c)) equal Q(L); reverse the roles when c<=L.'
witness={'L':12,'c':11,'round_after_cap':Q(min(12,11)),'cap_after_round_with_unrounded_c':min(Q(12),11),'cap_after_round_with_rounded_c':min(Q(12),Q(11))}
a=artifact('model/cap_round_commutation_scope.json',json.dumps({'law':'Q(min(L,c))=min(Q(L),Q(c))','proof':proof,'unrounded_capacity_warning':'min(Q(L),c) is a different construction; no commutation claim follows','witness':witness},indent=2)+'\n')
finish(s,{'commutation_scope':a},'Cap and rounding commute when both quantities are rounded by the same monotone rule. Keeping the capacity exact on only one route changes the operation; it is not a counterexample to that theorem.','Test whether rounding preserves the strict fact that a life exceeded capacity.',{'monotone_checks':all(Q(min(l,c))==min(Q(l),Q(c)) for l in range(26) for c in range(26)),'witness_values':witness['round_after_cap']==10 and witness['cap_after_round_with_unrounded_c']==10})