BCI combinators & Linear logic

These combinators allow you to generate structures
that resemble multiplicative operators,
but they do not seem to be enough to introduce additive operators.
To introduce those it seems to be necessary to
introduce an additional construct that introduces them.
Here's the reduction rules for the additive constructors/destructors:

{a, b} c ⇒ {a c, b c}
fst {a, b} ⇒ a
snd {a, b} ⇒ b

These constructs allow you to encode a possibility for a choice
that gets resolved through the fst and snd.
I think people have known how to do this,
but few still know how this can be done.

The structures derived this way can be duplicated and destroyed easily.
It allows church-style encoding
for numbers and other inductively defined structures,
such that they remain available for duplication and discarding.

Victor Maia's recently introduced
Symmetric Interaction Calculus
proposes you can also copy lambdas around by superpositioning.
I haven't yet figured out how to encode its rules in terms of BCI.

And of course you want to try the stripped-down W combinator.
It requires that the argument function is duplicated.
Here's how it's reduction went following rules Maia proposes in his blog:

The paper also suggests why pure lambda calculus or pure prolog must expose
the evaluation strategy before side-effectful programs make sense.
Maybe intuitionistic logic does not sufficiently express causes and effects.