Proofs that Modify Proofs, 1/2

This paper is a prelude and elaboration on Proofs that Modify Proofs. Here we present an ordinal analysis of a fragment of the $\mu$-calculus around the strength of parameter-free ${\mathrm{\Pi }}_2^1$-comprehension using the same approach as that paper, interpreting functions on proofs as proofs in an expanded system. We build up the ordinal analysis in several stages, beginning by illustrating the method systems at the strength of paremeter-free ${\mathrm{\Pi }}_1^1$-comprehension and full ${\mathrm{\Pi }}_1^1$-comprehension.