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Proving Standard Results about Residuated Lattices using the Gentzen system`

You can also test yourself by deciding the following
randomly generated RL equations.

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.
The symbol !- can be interpreted as *not less or equal*, in which case
each line implies the one below it (if at the same indentation level) or one
of the two indented alternatives (in the case of a branching rule). With this
interpretation, the terminal nodes represent contradictions, reminiscent of
a tableau style proof.

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Here are some standard (in)equational results about residuated lattices, followed by a
a list of their Gentzen system proof.

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Here is a random inequality in the language of residuated lattices, followed by a
proof or proof attempt.

If the inequality does not hold then the steps show all of the
potential proof-tree that was checked to reach this conclusion. On the
other hand, a successful proof only shows those branches which
demonstrate that the result holds.