# Example 947 — The drink had no half life `the_drink_had_no_half_life.eml` - A driver knows from a painkiller that four times the dose wears off in only twice the time, applies the same rule to four drinks instead of one, and plans the drive home for two hours after the last glass. How the two substances leave the body is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). A driver knows # from a painkiller that four times the dose wears off in only twice the time, # applies the same rule to four drinks instead of one, and plans the drive home # for two hours after the last glass. How the two substances leave the body is # computed below. # # The reasoning is careful. The painkiller rule is real and was checked against # its half-life; four drinks is four times one drink; the driver waited the # doubled time exactly; and the intent is exactly 'be clear before driving'. # # A drug with a half-life loses a fixed fraction per hour, so four times as # much needs only two more halvings, while alcohol is cleared at a fixed amount # per hour regardless of how much is present - so four drinks take four times # as long as one, not twice as long, and the driver was two hours early. 100 => painkiller_dose_mg 400 => painkiller_quadruple_dose_mg 25 => painkiller_worn_off_below_mg 4 => painkiller_half_life_hours 1 => one_drink_units 4 => four_drinks_units 1 => alcohol_cleared_units_per_hour int(painkiller_dose_mg / 2) => painkiller_after_one_half_life_mg int(painkiller_after_one_half_life_mg / 2) => painkiller_after_two_half_lives_mg int(painkiller_quadruple_dose_mg / 2) => quadruple_after_one_half_life_mg int(quadruple_after_one_half_life_mg / 2) => quadruple_after_two_half_lives_mg int(quadruple_after_two_half_lives_mg / 2) => quadruple_after_three_half_lives_mg int(quadruple_after_three_half_lives_mg / 2) => quadruple_after_four_half_lives_mg 2 * painkiller_half_life_hours => painkiller_hours_to_wear_off 4 * painkiller_half_life_hours => quadruple_hours_to_wear_off int(quadruple_hours_to_wear_off / painkiller_hours_to_wear_off) => painkiller_time_multiplier_for_four_times_the_dose int(one_drink_units / alcohol_cleared_units_per_hour) => one_drink_hours_to_clear int(four_drinks_units / alcohol_cleared_units_per_hour) => four_drinks_hours_to_clear int(four_drinks_hours_to_clear / one_drink_hours_to_clear) => alcohol_time_multiplier_for_four_times_the_drink one_drink_hours_to_clear * painkiller_time_multiplier_for_four_times_the_dose => hours_the_driver_waited four_drinks_hours_to_clear - hours_the_driver_waited => hours_the_driver_was_early four_drinks_units - hours_the_driver_waited * alcohol_cleared_units_per_hour => units_still_in_the_blood_at_departure "painkiller : " + str(painkiller_dose_mg) + " mg, half-life " + str(painkiller_half_life_hours) + " hours, worn off below " + str(painkiller_worn_off_below_mg) + " mg" ^0 "painkiller, one dose : " + str(painkiller_dose_mg) + " then " + str(painkiller_after_one_half_life_mg) + " then " + str(painkiller_after_two_half_lives_mg) + " mg, " + str(painkiller_hours_to_wear_off) + " hours" ^0 "painkiller, four doses : " + str(painkiller_quadruple_dose_mg) + " then " + str(quadruple_after_one_half_life_mg) + " then " + str(quadruple_after_two_half_lives_mg) + " then " + str(quadruple_after_three_half_lives_mg) + " then " + str(quadruple_after_four_half_lives_mg) + " mg, " + str(quadruple_hours_to_wear_off) + " hours" ^0 "painkiller rule : four times the dose, " + str(painkiller_time_multiplier_for_four_times_the_dose) + " times the wait" ^0 "" ^0 "alcohol : cleared at " + str(alcohol_cleared_units_per_hour) + " unit an hour, however much is present" ^0 "one drink : " + str(one_drink_units) + " unit, " + str(one_drink_hours_to_clear) + " hour" ^0 "four drinks : " + str(four_drinks_units) + " units, " + str(four_drinks_hours_to_clear) + " hours" ^0 "alcohol rule : four times the drink, " + str(alcohol_time_multiplier_for_four_times_the_drink) + " times the wait" ^0 "" ^0 "the driver waited : " + str(hours_the_driver_waited) + " hours" ^0 "still in the blood at departure : " + str(units_still_in_the_blood_at_departure) + " units" ^0 "early by : " + str(hours_the_driver_was_early) + " hours" ^0 "" ^0 # ---- what the driver verified ---- "the borrowed rule" ^0 " source : the painkiller's real half-life, checked" ^0 " arithmetic : four times the amount, twice the time" ^0 " count : four drinks is four times one drink" ^0 " wait : the doubled time, to the minute" ^0 " intent : be clear before driving" ^0 " facts wrong : 0" ^0 " verdict : FOUR DRINKS, TWO HOURS" ^0 "" ^0 " checking the rule against the painkiller's half-life is the" ^0 " part done right here, and it is why the rule is exactly" ^0 " true of the painkiller" ^0 "" ^0 # ---- how the two substances leave ---- "two kinds of clearance" ^0 " half-life : a fixed fraction leaves each hour, so a bigger" ^0 " load shrinks faster in absolute terms; four times the dose" ^0 " is only two more halvings, " + str(painkiller_hours_to_wear_off) + " to " + str(quadruple_hours_to_wear_off) + " hours" ^0 " fixed rate : a fixed amount leaves each hour, so a bigger" ^0 " load takes proportionally longer; four drinks is four" ^0 " hours, not two" ^0 " which one alcohol is : the fixed rate; the enzyme that clears" ^0 " it is saturated at a single drink and works no faster" ^0 " with more" ^0 " what a half-life is : a property some substances have, not" ^0 " a law about leaving the body" ^0 "" ^0 # ---- what the driver got ---- "the departure" ^0 " believed : clear after " + str(hours_the_driver_waited) + " hours" ^0 " actual : " + str(units_still_in_the_blood_at_departure) + " units still present, clear at " + str(four_drinks_hours_to_clear) + " hours" ^0 " is the painkiller rule wrong : no; it is right for the painkiller" ^0 " does alcohol have a half-life : not at these amounts; it has" ^0 " a rate, and a rate is multiplied by the amount" ^0 "" ^0 # ---- null control ---- # The same four drinks timed by alcohol's own clearance instead of by the # painkiller's rule. 2 => nc_hours_by_the_borrowed_half_life_rule 4 => nc_hours_by_the_fixed_clearance_rate 2 => nc_hours_the_right_rule_adds "null control - time the drink by the drink's own clearance" ^0 " hours, by the borrowed half-life rule : " + str(nc_hours_by_the_borrowed_half_life_rule) ^0 " hours, by the fixed clearance rate : " + str(nc_hours_by_the_fixed_clearance_rate) ^0 " hours the right rule adds : " + str(nc_hours_the_right_rule_adds) ^0 " no drink and no wait was measured differently; the rule was" ^0 " taken from the substance it describes" ^0 "" ^0 # ---- the rule ---- "what a verified half-life rule guarantees" ^0 " four times this drug wears off in twice the time : exactly," ^0 " for the drug whose half-life was checked" ^0 " four drinks wear off in twice the time : not addressed;" ^0 " alcohol leaves at a fixed " + str(alcohol_cleared_units_per_hour) + " unit an hour, so four units" ^0 " take " + str(four_drinks_hours_to_clear) + " hours, and the driver left " + str(hours_the_driver_was_early) + " hours early" ^0 "" ^0 "a half-life is a fact about one substance, not a habit of the body; borrow" ^0 "the rule without the substance and the arithmetic stays perfect while the" ^0 "thing it was about walks out of the room" ^0 "" ^0 "The painkiller rule is true: " + str(painkiller_quadruple_dose_mg) + " mg wears off in " + str(quadruple_hours_to_wear_off) + " hours, twice the " + str(painkiller_hours_to_wear_off) + " of a" ^0 "single dose. But alcohol has no half-life at these amounts - it leaves at" ^0 "" + str(alcohol_cleared_units_per_hour) + " unit an hour - so four drinks take " + str(four_drinks_hours_to_clear) + " hours, and the driver who waited " + str(hours_the_driver_waited) ^0 "left with " + str(units_still_in_the_blood_at_departure) + " units still in the blood, until the rule is taken from the drink." ^0 ``` ## Python (deterministic transpilation) ```python painkiller_dose_mg = 100 painkiller_quadruple_dose_mg = 400 painkiller_worn_off_below_mg = 25 painkiller_half_life_hours = 4 one_drink_units = 1 four_drinks_units = 4 alcohol_cleared_units_per_hour = 1 painkiller_after_one_half_life_mg = int(painkiller_dose_mg / 2) painkiller_after_two_half_lives_mg = int(painkiller_after_one_half_life_mg / 2) quadruple_after_one_half_life_mg = int(painkiller_quadruple_dose_mg / 2) quadruple_after_two_half_lives_mg = int(quadruple_after_one_half_life_mg / 2) quadruple_after_three_half_lives_mg = int(quadruple_after_two_half_lives_mg / 2) quadruple_after_four_half_lives_mg = int(quadruple_after_three_half_lives_mg / 2) painkiller_hours_to_wear_off = 2 * painkiller_half_life_hours quadruple_hours_to_wear_off = 4 * painkiller_half_life_hours painkiller_time_multiplier_for_four_times_the_dose = int(quadruple_hours_to_wear_off / painkiller_hours_to_wear_off) one_drink_hours_to_clear = int(one_drink_units / alcohol_cleared_units_per_hour) four_drinks_hours_to_clear = int(four_drinks_units / alcohol_cleared_units_per_hour) alcohol_time_multiplier_for_four_times_the_drink = int(four_drinks_hours_to_clear / one_drink_hours_to_clear) hours_the_driver_waited = one_drink_hours_to_clear * painkiller_time_multiplier_for_four_times_the_dose hours_the_driver_was_early = four_drinks_hours_to_clear - hours_the_driver_waited units_still_in_the_blood_at_departure = four_drinks_units - hours_the_driver_waited * alcohol_cleared_units_per_hour print("painkiller : " + str(painkiller_dose_mg) + " mg, half-life " + str(painkiller_half_life_hours) + " hours, worn off below " + str(painkiller_worn_off_below_mg) + " mg") print("painkiller, one dose : " + str(painkiller_dose_mg) + " then " + str(painkiller_after_one_half_life_mg) + " then " + str(painkiller_after_two_half_lives_mg) + " mg, " + str(painkiller_hours_to_wear_off) + " hours") print("painkiller, four doses : " + str(painkiller_quadruple_dose_mg) + " then " + str(quadruple_after_one_half_life_mg) + " then " + str(quadruple_after_two_half_lives_mg) + " then " + str(quadruple_after_three_half_lives_mg) + " then " + str(quadruple_after_four_half_lives_mg) + " mg, " + str(quadruple_hours_to_wear_off) + " hours") print("painkiller rule : four times the dose, " + str(painkiller_time_multiplier_for_four_times_the_dose) + " times the wait") print("") print("alcohol : cleared at " + str(alcohol_cleared_units_per_hour) + " unit an hour, however much is present") print("one drink : " + str(one_drink_units) + " unit, " + str(one_drink_hours_to_clear) + " hour") print("four drinks : " + str(four_drinks_units) + " units, " + str(four_drinks_hours_to_clear) + " hours") print("alcohol rule : four times the drink, " + str(alcohol_time_multiplier_for_four_times_the_drink) + " times the wait") print("") print("the driver waited : " + str(hours_the_driver_waited) + " hours") print("still in the blood at departure : " + str(units_still_in_the_blood_at_departure) + " units") print("early by : " + str(hours_the_driver_was_early) + " hours") print("") print("the borrowed rule") print(" source : the painkiller's real half-life, checked") print(" arithmetic : four times the amount, twice the time") print(" count : four drinks is four times one drink") print(" wait : the doubled time, to the minute") print(" intent : be clear before driving") print(" facts wrong : 0") print(" verdict : FOUR DRINKS, TWO HOURS") print("") print(" checking the rule against the painkiller's half-life is the") print(" part done right here, and it is why the rule is exactly") print(" true of the painkiller") print("") print("two kinds of clearance") print(" half-life : a fixed fraction leaves each hour, so a bigger") print(" load shrinks faster in absolute terms; four times the dose") print(" is only two more halvings, " + str(painkiller_hours_to_wear_off) + " to " + str(quadruple_hours_to_wear_off) + " hours") print(" fixed rate : a fixed amount leaves each hour, so a bigger") print(" load takes proportionally longer; four drinks is four") print(" hours, not two") print(" which one alcohol is : the fixed rate; the enzyme that clears") print(" it is saturated at a single drink and works no faster") print(" with more") print(" what a half-life is : a property some substances have, not") print(" a law about leaving the body") print("") print("the departure") print(" believed : clear after " + str(hours_the_driver_waited) + " hours") print(" actual : " + str(units_still_in_the_blood_at_departure) + " units still present, clear at " + str(four_drinks_hours_to_clear) + " hours") print(" is the painkiller rule wrong : no; it is right for the painkiller") print(" does alcohol have a half-life : not at these amounts; it has") print(" a rate, and a rate is multiplied by the amount") print("") nc_hours_by_the_borrowed_half_life_rule = 2 nc_hours_by_the_fixed_clearance_rate = 4 nc_hours_the_right_rule_adds = 2 print("null control - time the drink by the drink's own clearance") print(" hours, by the borrowed half-life rule : " + str(nc_hours_by_the_borrowed_half_life_rule)) print(" hours, by the fixed clearance rate : " + str(nc_hours_by_the_fixed_clearance_rate)) print(" hours the right rule adds : " + str(nc_hours_the_right_rule_adds)) print(" no drink and no wait was measured differently; the rule was") print(" taken from the substance it describes") print("") print("what a verified half-life rule guarantees") print(" four times this drug wears off in twice the time : exactly,") print(" for the drug whose half-life was checked") print(" four drinks wear off in twice the time : not addressed;") print(" alcohol leaves at a fixed " + str(alcohol_cleared_units_per_hour) + " unit an hour, so four units") print(" take " + str(four_drinks_hours_to_clear) + " hours, and the driver left " + str(hours_the_driver_was_early) + " hours early") print("") print("a half-life is a fact about one substance, not a habit of the body; borrow") print("the rule without the substance and the arithmetic stays perfect while the") print("thing it was about walks out of the room") print("") print("The painkiller rule is true: " + str(painkiller_quadruple_dose_mg) + " mg wears off in " + str(quadruple_hours_to_wear_off) + " hours, twice the " + str(painkiller_hours_to_wear_off) + " of a") print("single dose. But alcohol has no half-life at these amounts - it leaves at") print("" + str(alcohol_cleared_units_per_hour) + " unit an hour - so four drinks take " + str(four_drinks_hours_to_clear) + " hours, and the driver who waited " + str(hours_the_driver_waited)) print("left with " + str(units_still_in_the_blood_at_departure) + " units still in the blood, until the rule is taken from the drink.") ``` ## stdout (executed) ```text painkiller : 100 mg, half-life 4 hours, worn off below 25 mg painkiller, one dose : 100 then 50 then 25 mg, 8 hours painkiller, four doses : 400 then 200 then 100 then 50 then 25 mg, 16 hours painkiller rule : four times the dose, 2 times the wait alcohol : cleared at 1 unit an hour, however much is present one drink : 1 unit, 1 hour four drinks : 4 units, 4 hours alcohol rule : four times the drink, 4 times the wait the driver waited : 2 hours still in the blood at departure : 2 units early by : 2 hours the borrowed rule source : the painkiller's real half-life, checked arithmetic : four times the amount, twice the time count : four drinks is four times one drink wait : the doubled time, to the minute intent : be clear before driving facts wrong : 0 verdict : FOUR DRINKS, TWO HOURS checking the rule against the painkiller's half-life is the part done right here, and it is why the rule is exactly true of the painkiller two kinds of clearance half-life : a fixed fraction leaves each hour, so a bigger load shrinks faster in absolute terms; four times the dose is only two more halvings, 8 to 16 hours fixed rate : a fixed amount leaves each hour, so a bigger load takes proportionally longer; four drinks is four hours, not two which one alcohol is : the fixed rate; the enzyme that clears it is saturated at a single drink and works no faster with more what a half-life is : a property some substances have, not a law about leaving the body the departure believed : clear after 2 hours actual : 2 units still present, clear at 4 hours is the painkiller rule wrong : no; it is right for the painkiller does alcohol have a half-life : not at these amounts; it has a rate, and a rate is multiplied by the amount null control - time the drink by the drink's own clearance hours, by the borrowed half-life rule : 2 hours, by the fixed clearance rate : 4 hours the right rule adds : 2 no drink and no wait was measured differently; the rule was taken from the substance it describes what a verified half-life rule guarantees four times this drug wears off in twice the time : exactly, for the drug whose half-life was checked four drinks wear off in twice the time : not addressed; alcohol leaves at a fixed 1 unit an hour, so four units take 4 hours, and the driver left 2 hours early a half-life is a fact about one substance, not a habit of the body; borrow the rule without the substance and the arithmetic stays perfect while the thing it was about walks out of the room The painkiller rule is true: 400 mg wears off in 16 hours, twice the 8 of a single dose. But alcohol has no half-life at these amounts - it leaves at 1 unit an hour - so four drinks take 4 hours, and the driver who waited 2 left with 2 units still in the blood, until the rule is taken from the drink. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done