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An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)

Peter B. Andrews

Chapter 5

Type Theory - all with Video Answers

Educators

Section 1

Introduction

Problem 1

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_\tau=x_q$

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Problem 2

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_\tau=y_\tau \supset . S_{x_\tau}^{x_\tau} A \supset S_{x_\tau}^{z_\tau} A$ provided $x_\tau$ and $y_\tau$ are free for $z_\tau$ in $A$.

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Problem 3

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_\tau=y_{\mathrm{r}} \supset y_{\mathrm{r}}=x_{\mathrm{r}}$

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Problem 4

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{\mathrm{t}}=y_{\mathrm{t}} \supset \cdot y_{\mathrm{t}}=z_{\mathrm{r}} \supset x_{\mathrm{r}}=z_{\mathrm{r}}$

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Problem 5

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{(\mathrm{r})}=y_{(\mathrm{r})} \supset \forall z_{\mathrm{r}} \cdot x z \supset y z$

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Problem 6

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{(\mathrm{f})}=y_{(\mathrm{f})} \supset \forall z_q \cdot y z \supset x z$

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03:50

Problem 7

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{(\mathrm{f})}=y_{(\mathrm{t})} \equiv \forall z_{\mathrm{\tau}} \cdot x z \equiv y z$

Clarissa Noh
Clarissa Noh
Numerade Educator
03:50

Problem 8

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{\mathrm{r}}=y_{\mathrm{r}} \equiv \forall p_{(\mathrm{t} t)}\left[\forall z_{\mathrm{t}} p_{(\mathrm{t} \mathrm{t})} z_{\mathrm{r}} z_{\mathrm{t}} \supset p_{(\mathrm{t} t)} x_{\mathrm{t}} y_{\mathrm{t}}\right]$

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 9

One formulation of the Axiom of Choice says that for each collection $u$ of nonempty pairwise disjoint sets (whose elements have type $\tau$ ), there is a set $r$ containing exactly one element from each set in $u$. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
& A C_1^\tau: \forall u_{(\mathrm{r}))} \cdot \forall p_{(\mathrm{t})}\left[u_{(\mathrm{t}(\mathrm{r})} p_{(\mathrm{t})} \supset \exists x_{\mathrm{r}} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\
& \wedge \forall p_{(\mathrm{r})} \forall q_{(\mathrm{r})}\left[u_{(\mathrm{r})} p_{(\mathrm{r})} \wedge u_{(\mathrm{t})} q_{(\mathrm{r})} \wedge p_{(\mathrm{t})} \neq q_{(\mathrm{t})}\right. \\
& \left.\supset \forall x_{\mathrm{\tau}} \sim p_{(\mathrm{r})} x_{\mathrm{r}} \wedge q_{(\mathrm{t})} x_{\mathrm{r}}\right] \\
& \supset \exists r_{(\mathrm{t})} \forall p_{(\mathrm{t})} \bullet u_{(\mathrm{t})} p_{(\mathrm{t})} \supset \exists_1 x_{\mathrm{t}}=p_{(\mathrm{t})} x_{\mathrm{t}} \wedge r_{(\mathrm{t})} x_{\mathrm{t}} \\
&
\end{aligned}
$$
A second formulation of the Axiom of Choice says that there is a function (a special sort of binary relation) $f$ which chooses from each nonempty set $p$ (whose elements have type $\tau$ ) a member of that set. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
A C_2^{\mathrm{\tau}} & : \exists f_{(\mathrm{t} \mathrm{t})} \forall p_{(\mathrm{t})} \cdot\left[\exists x_{\mathrm{t}} p_{(\mathrm{t})} x_{\mathrm{t}} \supset \exists_1 x_{\mathrm{t}} f_{(\mathrm{t}) \mathrm{r})} p_{(\mathrm{t})} x_{\mathrm{r}}\right] \\
& \wedge \forall x_{\mathrm{t}} \cdot f_{((\mathrm{t}) \mathrm{f})} p_{(\mathrm{t})} x_{\mathrm{t}} \supset p_{(\mathrm{t})} \mathbf{x}_{\mathrm{t}}
\end{aligned}
$$
$\vdash A C_2^{\natural} \supset A C_1^{\ddagger}$

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Problem 10

One formulation of the Axiom of Choice says that for each collection $u$ of nonempty pairwise disjoint sets (whose elements have type $\tau$ ), there is a set $r$ containing exactly one element from each set in $u$. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
& A C_1^\tau: \forall u_{(\mathrm{r}))} \cdot \forall p_{(\mathrm{r} \mathrm{t}}\left[u_{(\mathrm{t}(\mathrm{r})} p_{(\mathrm{t})} \supset \exists x_{\mathrm{r}} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\
& \wedge \forall p_{(\mathrm{t})} \forall q_{(\mathrm{r})}\left[u_{(\mathrm{t})} p_{(\mathrm{r})} \wedge u_{(\mathrm{t})} q_{(\mathrm{t})} \wedge p_{(\mathrm{t})} \neq q_{(\mathrm{t})}\right. \\
& \left.\supset \forall x_{\mathrm{t}} \sim p_{(\mathrm{t})} x_{\mathrm{r}} \wedge q_{(\mathrm{t})} x_{\mathrm{t}}\right] \\
& \supset \exists r_{(\mathrm{t})} \forall p_{(\mathrm{t})} \cdot u_{(\mathrm{t})} p_{(\mathrm{r})} \supset \exists_1 x_{\mathrm{r}} \cdot p_{(\mathrm{t})} x_{\mathrm{t}} \wedge r_{(\mathrm{t})} x_{\mathrm{t}} . \\
&
\end{aligned}
$$
A second formulation of the Axiom of Choice says that there is a function (a special sort of binary relation) $f$ which chooses from each nonempty set $p$ (whose elements have type $\tau$ ) a member of that set. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
& A C_2^{\mathrm{\tau}}: \exists f_{(\mathrm{t} \mathrm{r} \tau)} \forall p_{(\mathrm{t})} \cdot\left[\exists x_{\mathrm{t}} p_{(\mathrm{t})} x_{\mathrm{t}} \supset \exists_1 x_{\mathrm{r}} f_{(\mathrm{t}(\mathrm{r}) \mathrm{r})} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\
& \wedge \forall x_\tau \cdot f_{(t) t) p} p_{(t)} x_t \supset p_{(t)} \mathbf{x}_t \\
&
\end{aligned}
$$
$\vdash A C_1^{((\mathrm{t}) \mathrm{q})} \supset A C_2^{\varepsilon}$

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