### Sequent calculus cheat sheet

Rules for sequent calculus connectives, formatted as a cheat sheet.

The Curry-Howard correspondence is a map for moving between logic and type theory, relating propositions with types and proofs with programs. It describes a two-way street, and we can freely move between the two worlds, or perhaps merely two *perspectives*, so long as we follow the map.

Sometimes the road takes us to unexpected places. Here’s a trip I’ve been on recently.

Sequent calculi are powerful and flexible tools for studying logic and, via Curry-Howard, computation. But why, and how? Where does this power come from?

Functions of type A → B can be translated into corresponding functions of type ¬B → ¬A in continuation-passing style (CPS), where ¬A is logically negation but computationally a continuation from A.

The rules for a variety of polarized classical connectives, in a focused sequent calculus presentation to reflect a variety of dualities, and interpreted via Curry-Howard.

Nearly a century ago, Alonzo Church invented the simple, elegant, and yet elusive lambda calculus. Along with Alan Turing, he then proved the Church-Turing thesis: that anything computable with a Turing machine can also be computed in the lambda calculus. However, nearly as soon as we had digital computers, we started inventing programming languages, and with them a vast treasure of features, beautiful and terrible, many of which seem very hard to relate to the fundamental nature of computability, let alone the lambda calculus specifically.

Swift’s value types are *almost* able to represent algebraic data types. Unfortunately, they fall short of the mark when it comes to recursion, and while they’ve announced that their solution, indirect `case`

s, will ship in a later build of Swift 2, there’s still reason to want them today.