Coq Assistant

Write a factorial function and prove that it's always strictly positive.

Write an ADT for arithmetic expressions comprising constants, variables and binary additions. Then write an evaluator taking an expression and an environment (a function that takes a variable name and returns a number) and returns the number resulting from evaluation. Then write an optimizer tha takes an expression and returns an expression with all additions by 0 removed. Then prove that the optimizer preserves the semantics as defined by the evaluation function.

Use Search with the Coq plugin to find lemmas involving +.