Proving Standard Results about Residuated Kleene Lattices using the Gentzen system
To display the proofs in a linear fashion, rather than the more usual Gentzen style,
the proof trees are displayed from the root downwards, with the depth of nodes indicated
by the indentation level. (Only the branching rules increase the indentation depth.)
Each sequent is labeled by the name of the rule that produced it. When a terminal
node is reached, the indentation level decreases again. With a bit of practice, these
proofs are actually easier to read (and write) than the standard tree-like presentation.
Here are some standard (in)equational results about residuated kleene lattices, followed by a
a list of their Gentzen system proof.