The reverse of a string, introduced informally above, can be defined more precisely by the recursive rules ar = a, (wa)r = awr, for all a ∈ σ, w ∈ σ*. use this to prove that (uv)r = vrur, for all u, v ∈ σ+.

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Write a program that performs a simple n-body simulation, called "jumping leprechauns." this simulation involves n leprechauns, numberd 1 to n. it maintains a gold value g_i for each leprechaun i, which begins with each leprechaun starting out with a million dollars worth of gold, that is, g_i = 1000000 for each i = 1,. in addition, the simulation also maintains, for each leprachaun,i, a place on the horizon, which is represented as a double-precision floating point number, x_i. in each iteration of the simulation, the simulation processes the leprachauns in order. processing a leprachaun i during its iteration begins by computing a new place on the horizon for i, which is determined by the assignment: