Metamathematics B HW #01
Ex 1.3
Show that if a structure S has computable quantifier elimination, and the set of quantifier-free sentences that are true in S is computable, then the theory of S is computable. If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$ is a sentence in the language of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S$, let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \text{dec}(\phi)$ denote the following algorithm. If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$ has no quantifiers, then output $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$. Otherwise, do the following. First replace all universal quantifiers $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \forall$ with $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \neg \exists \neg$. Now find an existential quantifier in $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$ containing no other quantifiers (ie, an “inner-most” quantifier). Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \psi = \psi(x, \ol y)$ denote the formula directly inside this quantifier, where $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} x$ is the variable bound by the quantifier and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \ol y$ is a tuple of free variables. Note that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \psi$ is quantifier-free, so by assumption we can computably perform quantifier elimination on it. Do so to produce a formula $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \rho = \rho(\ol y)$ so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S \models \rho(\ol y) \leftrightarrow \exists x \psi(x, \ol y)$ Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi'$ be $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$ but with $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \exists x \psi(x, \ol y)$ replaced by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \rho(\ol y)$. Output $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi'$. By construction this algorithm $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \text{dec}(\phi)$ has the following properties:
$% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S \models \text{dec}(\phi)$ exactly when $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S \models \phi$
$% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \text{dec}(\phi)$ has one fewer quantifier than $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \phi$
We can use $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \text{dec}$ to compute $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \text{Th}(S)$ as follows. Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma$ be a sentence in the language of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S$. Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$ be the number of quantifiers in $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma$, and let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma' = \text{dec}^n(\gamma)$. Then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma'$ are equivalent, and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma'$ is quantifier-free. By assumption we can compute whether or not $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma'$ is true in $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S$; do so and output $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 1$ if it is true and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 0$ if it is not. Since $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma'$ are equivalent then this is the same as outputting the truthiness of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \gamma$.
Ex 1.6
Electing to skip this one
ok, i will (update: ok i did)
Ex 2.1
Show that there is an algorithm for computing a function that is not primitive recursive. Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} u$ be a universal turing machine, so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} u(n, m) = \varphi_n(m)$. Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \{p_n\}$ be an enumeration of primitive recursive functions (I am assuming this is possible; the problem hint seems to suggest so). Now define our function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ as: $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(n) = u(p_n, n) + 1$ Since every function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} p_n$ is total, then every $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} u(p_n, n)$ converges, and hence $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is total. Additionally, it is computable, since it consists only of compositions of known-computable functions. Additionally, $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is not primitive recursive: there are only countably-many primitive recursive functions, and on the $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n^\text{th}$ primitive recursive function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} p_n$ we have that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ differs from $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} p_n$ on input $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$. Hence $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ differs from every primitive recursive function, and so $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is not primitive recursive.
Ex 2.2
(Problem done partially in collaboration with Theo) 1. Show that if $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f : \bb N^k \to \bb N$ is primitive recursive, then exists $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$ so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(x_1, x_2, \dots, x_k) \leq 2 \uparrow^n ( \max_i (x_i) + 2 )$
This isn’t true. The function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(x) = 5$ does not have such an upper bound; choosing any $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$, we get that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(0) = 5 > 4 = 2 \uparrow^n 2 = 2 \uparrow^n (x + 2)$ If we amend the statement to replace $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \max_i(x_i) + 2$ by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \max_i(x_i) + 3$ then it’s probably true. I’ve only done a proof sketch, since it seems to me that doing an actual proof would actually be pretty hard. We proceeds by induction on the structure of primitive recursive functions. Take $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ a primitive recursive function.
If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is the successor function, then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n = 1$ suffices
If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is a projection, then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n = 1$ suffices
If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is a constant function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \ol x \mapsto c$, then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n = c$ suffices
If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is a composition $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f : \ol x \mapsto h(g_1(\ol x), \dots, g_k(\ol x))$, then we know that each $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$ is bounded and that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h$ is bounded; let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \{\eta_i\}$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$ be the respective witnesses to their bounds so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i(\ol x) \leq 2 \uparrow^{\eta_i} (\max \ol x + 3)$and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h(\ol x) \leq 2 \uparrow^n (\max \ol x + 3)$ Letting $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} m = \max_i \eta_i$ we get since $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is the composition of all $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$s with $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(\ol x) \leq 2 \uparrow^n (2 \uparrow^m (\max \ol x + 3) + 3)$ It’s probably possible to bound this again with some large witness $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} N$ so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(\ol x) \leq 2 \uparrow^N (\max \ol x + 3)$ but I wasn’t able to easily prove it
If $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f : \bb N^{k+1} \to \bb N$ is a primitive recursion of functions $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g : \bb N^k \to \bb N$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h : \bb N^{k+2} \to \bb N$, then we know that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g, h$ are bounded; let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} m$ be the respective witnesses to their bounds. Since $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ is a primitive recursion then we know it has semantics % shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \begin{align*} f(\ol x, 0) &= g(\ol x) \\ f(\ol x, y + 1) &= h(\ol x, y, f(\ol x, y)) \end{align*} If we expand this definition we get % shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \begin{align*} & f(\ol x, y) \\ &= h(\ol x, y - 1, h(\ol x, y - 2, h(\ol x, y - 3, h(\ {\cdots}\ g(\ol x)\ {\cdots}\ )))) \end{align*} We know that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h$ is bounded by some witness $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$, and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ is bounded by some witness $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} m$ so we can bound this entire expression by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 2 \uparrow^n (2 \uparrow^n (\ {\cdots}\ 2 \uparrow^n (2 \uparrow^m (\max \ol x + 3) + 3) \ {\cdots}\ ) + 3)$ Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} N = \max(n, m) + 1$. Then that expression is probably itself bounded by: $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 2 \uparrow^N (2 \uparrow^N (\ {\cdots}\ 2 \uparrow^N (2 \uparrow^N \max \ol x + 3)) \ {\cdots}\ ))$ Which is again probably bounded by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 2 \uparrow^{N+1} (y + \max \ol x + 3)$ which is probably bounded by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 2 \uparrow^{N+{10000}} (\max \ol x + 3)$ Hence $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ can be bounded in the way desired.
2. Show that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(x) = 2 \uparrow^x x$ is not primitive recursive
The function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(x) = 2 \uparrow^x x$ cannot be bounded above by $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g(x) = 2 \uparrow^n (x + 3)$ for any fixed $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$, and hence is not primitive recursive.
Ex 2.7
Show that the partial computable functions $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \Sigma^{<\infty} \to \Sigma^{<\infty}$ are closed under composition We can compose two partial computable functions by sequencing their turing machines. That is, if $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f, g$ are both partial computable on the same alphabet, we may define their composition $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g \circ f$ as a ‘disjoint union’ of sorts, where the turing machine for $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g \circ f$ has as its state set the disjoint union of the state sets of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$, has as its starting state the starting state of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$, as its halting state the halting state of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$, and has as its transition function the union of the transition functions of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$, plus extra transitions to take the halting state of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ to the starting state of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ without modifying the tape or moving.
Ex 2.8
Prove that every partial recursive function $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \bb N^k \to \bb N$ is computable by a turing machine This is possible because each of the closure conditions for partial recursive functions are satisfied by turing machines:
Constant functions can be expressed
The successor function can be expressed
Projection functions can be expressed, for instance by representing tuples of naturals by sequencing them delimited by the space character
Turing machines are closed under composition. First assume we’re working with turing machines which map $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \bb N^k$ to $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \bb N$, and also assume that we’re working with alphabet $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} S = \{ 0,1,\_ \}$. Now to encode the composition $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h : n \mapsto f(g_1(n), \dots, g_k(n))$ we can do the following. WLOG assume that each $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$ uses only the subset $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k\bb N + i \subseteq \bb Z$ of the tape (this does not impact turing machine power). Now to run $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} h$, do the following. Accept an input number in binary format. Now ‘smear’ each input digit into eight copies along the substrates $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k\bb N + i$ we’ve allotted for the composees $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$. Now run $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_1$, then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_2$, up through $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_k$. By restricting each $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$ to the substrate $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k\bb N + i$ we guarantee that no composee $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$ will interfere with another. The result will be a tape empty on the left side and containing $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k$ interlaced numbers on the right side. Read off the first number, copying it to the left side of the tape. Read off the second number, copying it, placing it next to the first number separated by a blank character. Repeat $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k$ times, and then zero out the right side of the tape. Move these numbers back to the right side of the tape, and zero out the left side. Now we have $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k$ numbers—the results of the $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g_i$s—on the tape; these form the input for $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$. Run $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$.
Turing machines are closed under minimization: take $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f : \bb N^{k + 1} \to \bb N$. To produce its minimization $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g : \bb N^k \to \bb N$, create a turing machine that does the following. WLOG assume $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$ only runs on the right side of the tape. Have $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ accept $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} k$ numbers. The first thing that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ does is copy them to the left side of the tape alongside a distinguished number initialized to zero. Then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ proceeds as follows. Repeatedly it copies the numbers, including the distinguished number, from the left of the tape to the right and then executes $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f$. If the result is zero, $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ clears the right side of the tape, copies the distingushed number, clears the left side of the tape, and halts. Otherwise, $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} g$ increments the distinguished number and repeats.
Ex 2.16
Yes, such an algorithm exists. Note that if $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$ consecutive $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 7$s are present in the decimal expansion of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \pi$, then $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n{-1}$ consecutive $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 7$s are present, and $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n{-2}$, etc. Now proceed by cases. Either the digits of $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \pi$ contain arbitrarily-long digit-$% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 7$ sequences, or some $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} N \in \bb N$ is the maximum length of a digit-$% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} 7$ sequence present in $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \pi$. In the first case, our algorithm is given by outputting “yes” and immediately halting. In the second case, our algorithm is given by taking in a number $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} m$ and outputting “yes” if $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} m \leq N$ and “no” otherwise.