Slides of a (very) informal lecture of mine on the Curry-Howard correspondence
Changed the notation for propositions
Also added a link to a talk on the idea of propositions as types
Also added decorative formulas to the last slide
4 files changed, 43 insertions, 27 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | examples/example.lean | 4 | 2 | 2 |
| Modified | examples/types.lean | 12 | 6 | 6 |
| Modified | examples/types.py | 10 | 5 | 5 |
| Modified | main.tex | 44 | 30 | 14 |
diff --git a/examples/example.lean b/examples/example.lean @@ -3,5 +3,5 @@ lemma ex : or p q → or q p := by match h with | inl h1 => exact inr h1 - | inr h1 => - exact inl h1 + | inr h2 => + exact inl h2
diff --git a/examples/types.lean b/examples/types.lean @@ -1,7 +1,7 @@ --- a e b <-> and a b -inductive and (a b : Prop) : Prop -| intro : a → b → and a b +-- p e q <-> and p q +inductive and (p q : Prop) : Prop +| intro : p → q → and p q -inductive or (a b : Prop) : Prop -| inl : a → or a b -| inr : b → or a b +inductive or (p q : Prop) : Prop +| inl : p → or p q +| inr : q → or p q
diff --git a/examples/types.py b/examples/types.py @@ -1,5 +1,5 @@ -class AndAB: - """A e B <-> AndAB""" - def __init__(self, a: A, b: B): - self.left = a - self.right = b +class AndPQ: + """P e Q <-> AndPQ""" + def __init__(self, h1: P, h2: Q): + self.left = h1 + self.right = h2
diff --git a/main.tex b/main.tex @@ -203,9 +203,9 @@ \begin{frame}{A Correspondência de Curry-Howard} \begin{center} \begin{tabular}{rcl} - Provar que vale \(A\) + Provar que vale \(P\) & \(\leftrightsquigarrow\) - & Encontrar \(a \in \{ \text{provas de}\ A\}\) + & Encontrar \(h \in \{ \text{provas de}\ P\}\) \end{tabular} \end{center} @@ -215,18 +215,18 @@ \item Proposições como tipos \begin{center} \begin{tabular}{rcl} - \(A\) + \(P\) & \(\leftrightsquigarrow\) - & \(\{\text{provas de}\ A \}\) \\ - \(A\ \text{e}\ B\) + & \(\{\text{provas de}\ P \}\) \\ + \(P\ \text{e}\ Q\) & \(\leftrightsquigarrow\) - & \(\{\text{provas de}\ A \} \times \{\text{provas de}\ B \}\) \\ - \(A\ \text{ou}\ B\) + & \(\{\text{provas de}\ P \} \times \{\text{provas de}\ Q \}\) \\ + \(P\ \text{ou}\ Q\) & \(\leftrightsquigarrow\) - & \(\{\text{provas de}\ A \} \cup \{\text{provas de}\ B \}\) \\ - \(A \implies B\) + & \(\{\text{provas de}\ P \} \cup \{\text{provas de}\ Q \}\) \\ + \(P \implies Q\) & \(\leftrightsquigarrow\) - & \(\{\text{provas de}\ A \} \to \{\text{provas de}\ B \}\) \\ + & \(\{\text{provas de}\ P \} \to \{\text{provas de}\ Q \}\) \\ \end{tabular} \end{center} @@ -255,12 +255,15 @@ \begin{center} \begin{tabular}{rcl} - Provar que vale \(A\) + Provar que vale \(P\) & \(\leftrightsquigarrow\) - & Construir \(a : \{ \text{provas de}\ A\}\) \\ + & Construir \(h : \{ \text{provas de}\ P\}\) \\ Checar a prova & \(\leftrightsquigarrow\) - & Checar os tipos + & Checar os tipos \\ + Inconsistência na prova + & \(\leftrightsquigarrow\) + & Tipos não batem \end{tabular} \end{center} \end{frame} @@ -317,7 +320,7 @@ \end{itemize} \end{frame} -\begin{frame}{Provadores de Teoremas} +\begin{frame}{Proof Checkers} \begin{itemize} \item Coq (\url{https://coq.inria.fr/}) @@ -442,9 +445,22 @@ {Advancing mathematics by guiding human intuition with AI} \item + \href{https://www.youtube.com/watch?v=IOiZatlZtGU} + {Propositions as Types} + + \item \href{https://www.hedonisticlearning.com/posts/understanding-typing-judgments.html} {Understanding typing judgments} \end{enumerate} + \begin{align*} + \frac{\Gamma \vdash_\Sigma a : A} + {\Gamma \vdash_\Sigma \operatorname{inl} a : A \vee B} + \rightarrow D_1 + & & + \frac{\Gamma \vdash_\Sigma b : B} + {\Gamma \vdash_\Sigma \operatorname{inr} b : A \vee B} + \rightarrow D_2 + \end{align*} \end{frame} \end{document}