Infinite Compression Explained

Because most of you don't read my blog,(i don't read random blogs as well) and have so much questions about what i'm doing with my programs, i'll write it here:
All of the programs(about 6 developed so far) despite wildly varying routines are targeted to generate large integers.
These integers are not files. Following transformations occurs:
encode
1.file is converted to large integer X(arbitrary length).
2.X multiplied by some scale factor e.g. 10e1000 to get a lower bound
2.(X+1)by some scale factor e.g. 10e1000 to get an uppper bound
3.a search is performed for finding numbers inside that range which are easy to represent via formula.
4.if number(s) found its saved to a file.
decode:
1.a formula is supplied with number(s) and filesize
2.the formula generates an integer/float, which is then divided by scale factor(e.g. 10e1000).
3.the first filesize bytes are then written to output.

the proces isn't perfected yet, because the formulas currently used in my programs either too slow to search or cuttoff at float precision(for non-integer parameters)
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consider a compression algorithm to be a function from the set of non-negative integers onto the set of non-negative integers

suppose there exists a compression algorithm f such that for all x less than some positive integer k, 0 <= f(x) < x. then consider S = { x in Z | 0 <= x <= k }. Then f(S) = { f(x) : x in Z | 0 <= x <= k } and for any f(x) in f(S), 0 <= f(x) < x <= k, 0 <= f(x) < k.  Then, by counting, |f(S)| < k+1, but |f(S)| = k, so f cannot be one-to-one and f is not invertible.

any compression algorithm that reduces the size of every input of less than a certain size is not invertible, i.e. it must be lossy.