11165번 - Dimensions 스페셜 저지다국어

You and your friend Christian have decided to take a vacation year and travel all around the world to see magnificent places, meet wonderful people, and experience new cultures. Unfortunately, with different cultures come difficult differences. And the hardest differences engineers like you know of are unit differences. Why are people using miles, firkins, microfortnights, candlepowers, boisseaux, foes, pints, kWh and not to mention degrees Celsius when they instead would be much happier using the standard and beloved SI units of metres, kilograms, seconds, amperes, kelvins and candelas?

Table 1: List of SI units

In fact, you love the SI units so much that you refuse to use any other units. Derived units like the joule (J), the newton (N), and the ohm (Ω) are perfectly expressible in their equivalent SI units of kg m² / s², kg m / s², and kg m² / s³ A², respectively. So during your travel, you record all units you come across, along with their definitions. Of course, some definitions are depending on other definitions, like Pa = N / m².

With your definitions ready at hand, you don’t have to put up with such nonsense as 60 firkins / microfortnights or 63 km / h anymore, since you can always convert them SI units. Even calculations like 100 m + 1.3 km and 7 N · 8 Ω become a breeze to you.

The following syntax is given (’?’ denotes “zero or one”, ’+’ denotes “one or more”, and ’*’ denotes “zero or more”):

          power ::= { integer larger than 1 }
           unit ::= { upper or lower case English letter }+
      dimension ::= unit [’^’ power ]?
           size ::= {any floating-point number} [’ ’ dimension]* [’/’ [’ ’ dimension]+]?
       operator ::= ’+’ OR ’-’ OR ’*’
     expression ::= size ’ ’ operator ’ ’ size
unit definition ::= unit ’=’ size

All units have lenghts less than 10, and in a size, no units are repeated.

The first line of the input consists of a single integer U, the number of new units. Then follow U lines with new unit definitions. After the unit definitions follows a line with a single integer N, and then N lines with either an expression or a size.

In the input, all power s are less than 5, and all units are SI unit symbols or previously defined units.

• 0 < U ≤ 100
• 0 < N ≤ 1000
• Units are case sensitive.
• No lines are longer than 140 characters.
• No input, output or part of any computation have size of absolute value above 10¹⁰⁰.
• No part of any computation will yield division by 0.
• Any output with a relative or absolute error of 10⁻⁶ is accepted.

For each computation, output one line with the answer to the expression, or the size itself, converted to SI units. If the answer cannot be computed, output “Incompatible” (without the quotes). The answer should be formatted as a size, with the following clarifications:

• Units can be written in any order, but keep them on the correct side of the division sign.
• Do not output a unit if its exponent is 0, or its exponent if it is equal to 1.
• Output one space between each unit and between units and the division sign, \.
• Do not output any spaces before or after the exponent sign, ^.