Natural Deduction

Logical Properties

Like a programming language, logic has

• syntax (\(e\), \(\land\), \(\lor\), \(\lnot\), \(\Rightarrow\) \(\forall\), \(\exists\))
• meaning (if \(\phi\) is true and \(\psi\) is true, then \(\phi \land \psi\) is true), and
• “evaluation” (decision procedues).

Natural Deduction is a decision procedure for logic. The Curry-Howard correspondence says that propositions are types and proofs are programs. In this lecture, we will see that natural deduction is encoded by type checking rules.

Concretely, we will see the encoding by filling in the following table:

Native Terms

We start with the simplest case, native terms. We have already seen that native terms are encoded by refinements on unit types:

So native terms are encoded by unit types with refinements:

Conjunction

Conjunction is encoded by the pair type. If you know that \(\phi_1\) is true and \(\phi_2\) is true, then \((\phi_1, \phi_2)\) is a term that shows that \(\phi_1 \land \phi_2\) is true.

Given that logical conjunction can be encoded as pairs, we can see how type checking can encode the natural deduction proof system. When typing pairs, there are three rules: left and right elimination and introduction.

\[ \inferrule {\land\text{-L-E}} { \Gamma \vdash e : (\phi_1, \phi_2) }{ \Gamma \vdash \text{case } e \text{ of } (x_1, x_2) \rightarrow x_1 : \phi_1 } \]

\[ \inferrule {\land\text{-R-E}} { \Gamma \vdash e : (\phi_1, \phi_2) }{ \Gamma \vdash \text{case } e \text{ of } (x_1, x_2) \rightarrow x_2 : \phi_2 } \]

\[ \inferrule {\land\text{-I}} { \Gamma \vdash \text{case } e \text{ of } (x_1, x_2) \rightarrow x_1 : \phi_1 \\ \Gamma \vdash \text{case } e \text{ of } (x_1, x_2) \rightarrow x_2 : \phi_2 }{ \Gamma \vdash e : (\phi_1, \phi_2) } \]

If we ignore the expressions above we take exactly the conjuction rules of natural deduction.

** You can generate Haskell terms that provide evidence for the natural deduction rules!**

As an example, consider the natural deduction devation below

\[ \inferrule {\land\text{-I}} { \phi_1,\phi_2\land\phi_3 \vdash \phi_1 \qquad \inferrule{\land\text{-R-E}} { \phi_1,\phi_2\land\phi_3 \vdash \phi_2\land\phi_3 }{ \phi_1,\phi_2\land\phi_3 \vdash \phi_3 } }{ \phi_1, \phi_2\land\phi_3 \vdash \phi_1\land \phi_3 } \]

Filling in the terms (and applying the correspondence of propositions and types) we get:

\[ \inferrule {\land\text{-I}} { x_1:\phi_1,x_{23}:\phi_2\land\phi_3 \vdash x_1:\phi_1 \qquad \inferrule{\land\text{-R-E}} { x_1:\phi_1,x_{23}:\phi_2\land\phi_3 \vdash x_{23}: \phi_2\land\phi_3 }{ x_1:\phi_1,x_{23}:\phi_2\land\phi_3 \vdash \text{case } x_{23} \text{ of } (x_2, x_3) \rightarrow x_3: \phi_3 } }{ x_1:\phi_1,x_{23}: \phi_2\land\phi_3 \vdash (x_1, \text{case } x_{23} \text{ of } (x_2, x_3) \rightarrow x_3): \phi_1\land \phi_3 } \]

Let’s encode this proof in Liquid Haskell!

So, conjuction is encoded by the pair type.

Disjunction

Disjunction is encoded by the Either type, which is a sum type that can be either Left or Right. data Either a b = Left a | Right b.

If you know that \(\phi_1\) is true or \(\phi_2\) is true, then Left () is a term that shows that \(\phi_1 \lor \phi_2\) is true and Right () is a term that shows that \(\phi_1 \lor \phi_2\) is true.

Dually to conjunction, disjunction has three rules: left and right introduction and elimination.

\[ \inferrule {\lor\text{-L-I}} { \Gamma \vdash e : \phi_1 }{ \Gamma \vdash \text{Left } e : \text{Either } \phi_1 \ \phi_2 } \]

\[ \inferrule {\lor\text{-R-I}} { \Gamma \vdash e : \phi_2 }{ \Gamma \vdash \text{Right } e : \text{Either } \phi_1 \ \phi_2 } \]

\[ \inferrule {\lor\text{-E}} { \Gamma \vdash e : \text{Either } \phi_1\ \phi_2 \\ \Gamma, x_1 : \phi_1 \vdash e_1 : \phi \quad \Gamma, x_2 : \phi_2 \vdash e_2 : \phi }{ \Gamma \vdash \text{case } e \text{ of } \{\text{Left } x_1 \rightarrow x_1 ; \text{Right } x_2 \rightarrow x_2 \} : \phi } \]

So, disjunction is encoded by the either type.

Implication

Implication is encoded by the function type. It has two rules: introduction and elimination:

\[ \inferrule {\Rightarrow\text{-I}} { \Gamma, x : \phi_x \vdash e : \phi }{ \Gamma \vdash \lambda x. e : \phi_x \rightarrow \phi } \]

\[ \inferrule {\Rightarrow\text{-E}} { \Gamma \vdash e : \phi_x \rightarrow \phi \quad \Gamma \vdash e_x : \phi_x }{ \Gamma \vdash e\ e_x : \phi } \]

The Implication Elimination Rule for natural deduction is also known as the modus ponens rule.

Negation

Negation is just implication to False.

Forall

Forall is encoded by the quantified function type. It has two rules: introduction and elimination:

\[ \inferrule {\forall\text{-I}} { \Gamma, x:\tau \vdash e : \phi }{ \Gamma \vdash \lambda x. e : x:\tau \rightarrow \phi } \]

\[ \inferrule {\forall\text{-E}} { \Gamma \vdash e : (x:\tau \rightarrow \phi) \quad \Gamma \vdash e_x : \tau }{ \Gamma \vdash e\ e_x : \phi [x / e_x] } \]

Exists

Exists is encoded by the dependent pair. It has two rules: introduction and elimination:

\[ \inferrule {\exists\text{-I}} { \Gamma \vdash \text{fst } e : \tau \quad \Gamma, x:\tau \vdash \text{snd } e : \phi }{ \Gamma \vdash e : (x:\tau, \phi[x / \text{fst } e]) } \]

\[ \inferrule {\exists\text{-E}} { \Gamma \vdash e : (x:\tau, \phi_x) \quad \Gamma, x:\tau, y:\phi_x \vdash e' : \phi }{ \Gamma \vdash \text{case } e \text{ of } (x, y) \rightarrow e' : \phi } \]

That concludes the encoding! Let’s now see some examples that use it!

Example 1: existsAll

The first example is a proof that if there exists an \(x\) that satisfies a property forall \(y\), then forall \(y\) there exists an \(x\) that satisfies the property.

\[ \phi = \exists x. \forall y. p\ x \ y \Rightarrow \forall y. \exists x. p \ x \ y \]

Let’s prove this propery in Liquid Haskell!

Question: Can you prove this property in Liquid Haskell?

The natural deduction proof is shown below:

Figure 1.1: Natural Deduction Proof for exAll

Example 2: Distributing Qualifiers

First, let’s distribute the exists quantifier over disjunction.

\[ \phi_\exists = (\exists x. p\ x \lor q\ x) \rightarrow ((\exists x. p\ x) \lor (\exists x. q\ x)) \]

Let’s prove this propery in Liquid Haskell!

Question: Can you prove this property in Liquid Haskell?

Now, let’s distribute the forall quantifier over conjunction.

\[ \phi_\forall = (\forall x. p\ x \land q\ x) \rightarrow ((\forall x. p\ x) \land (\forall x. q\ x))\]

Let’s prove this propery in Liquid Haskell!

Question: Can you prove this property in Liquid Haskell?

Example 3: List Properties

Next, let’s prove that for all lists that are constructed by appending a list to itself, there exists an integer that is half the length of the list.

\[ \phi = \forall xs. (\exists ys. xs = ys \text{ ++ } ys) \rightarrow (\exists n. \text{length } xs = n + n )\]

Let’s prove this propery in Liquid Haskell!

Question: Can you prove the above property in Liquid Haskell? Hint: You can use the property lenAppend:

Example 4: Natural Induction

Finally, let’s prove that for all natural numbers,

\[ \phi = (p \ 0 \land (\forall n. p \ (n-1) \rightarrow p \ n))) \rightarrow \forall n. p \ n\]

Let’s prove this propery in Liquid Haskell!

Summary

In this lecture, we saw that natural deduction can be encoded in Liquid Haskell. We saw that the rules of natural deduction can be encoded by the type checking rules of Liquid Haskell. We saw that the Curry-Howard correspondence can be extended to higher order logic. We saw that we can prove higher order properties in Liquid Haskell.