proof net

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Proof nets_ are a graphical way to denote [[proofs]] in [[linear logic]].

They were introduced by [[Jean-Yves Girard|J.-Y. Girard]] in 1987 in order to remove spurious notational redundancies introduced by sequential rule application or indeterminacy of [[cut elimination]] and can be viewed as graphical normal forms of proofs freed from 'the bureaucracy of syntax' (Girard).

Bearing in mind the [[categorical semantics]] of (multiplicative intuitionistic) [[linear logic]] in (closed) [[monoidal categories]], proof nets are closely related to [[Kelly-Mac Lane graphs]] that are used to track compositions of [[extranatural transformations]] definable in such categorical structures. More generally they may be compared to [[string diagrams]] that are evaluated in these categories (with various ways of making the comparison more precise). Proof nets for full linear logic have been thus described by [Melliès 06](#Mellies06) as string diagrams equipped with "boxes" to account for the [[exponential modality]].

With [[linear logic]]/[[linear type theory]] interpreted as [[quantum logic]]/[[quantum computation]], proof nets correspond to [[quantum circuits]] or [[Feynman diagrams]] ([Blute-Panangaden](#BlutePanangaden)).

## Definition

We begin by describing proof nets for multiplicative linear logic (MLL), which is interpretable in $\ast$-[[star-autonomous category|autonomous]] categories. Similar proof nets may be given for other forms of [[linear type theory]] such as multiplicative intuitionistic linear logic, which is interpretable in [[symmetric monoidal category|symmetric]] [[closed monoidal categories]].

### Proof structures

Before describing proof nets, we start with Girard's notion of _proof structure_. It is simplest to describe the [[cut rule|cut]]-free version, as this is closely connected with the notion of [[Kelly-Mac Lane graph]], or KM-graph for short. We need just a few preliminaries.

[[formula|Formulas]] in MLL may be built from an alphabet $\mathbf{T}$ of propositional variables and two constants $\mathbf{1}$, $\bot$ by applying operations $\otimes$ and $\multimap$. (Negation may be defined by $\neg A \coloneqq A \multimap \bot$, and the multiplicative disjunction by $A \parr B \coloneqq \neg A \multimap B$. To get formulas in MILL, simply drop $\bot$.) The construction of a formula may be displayed as a binary [[planar tree]].

We consider two-sided [[sequents]] $\Gamma = A_1, \ldots, A_m \vdash \Delta = B_1, \ldots, B_n$, wherein the formulas of $\Gamma$ to the left of the turnstyle are called _negative_ and the formulas of $\Delta$ are called _positive_. Subformulas of these formulas acquire signs according to the following rules:

* The subformulas $S, T$ of $S \otimes T$ have the same sign as $S \otimes T$.

* The subformula $T$ of $S \multimap T$ has the same sign as $S \multimap T$, while $S$ has the opposite sign.

We indicate the sign of a subformula using a superscript, e.g., $S^+$.

+-- {: .num_defn}
###### Definition
A (cut-free) _proof structure_ of type $\Gamma \to \Delta$ (in the language of MLL) is a directed graph whose vertices are subformulas of the formulas of $\Gamma, \Delta$, and whose edges are either

* "KM-links" (Kelly-Mac Lane links) which are of the form $t^- \to t^+$ between two subformula occurrences of the same variable $t \in \mathbf{T}$ but with opposite sign (always oriented from a negative occurrence to a positive occurrence);

* Edges in binary construction trees of the formulas $A_i, B_j$, oriented according to the following rules:

$$\array{
 & & (S \otimes T)^- & & \qquad S^+ & & & & T^+ & & & & (S \multimap T)^- & & \qquad S^- & & & & T^+ \\
 & \swarrow & & \searrow & \qquad & \searrow & & \swarrow & & & & \nearrow & & \searrow & \qquad & \nwarrow & & \swarrow & \\
S^- & & & & T^- \qquad & & (S \otimes T)^+ & & & S^+ & & & & & T^- \qquad & & (S \multimap T)^+
}$$
=--

For a proof structure of type $\Gamma \to \Delta$, it is clear that the only information not determined from $\Gamma$ and $\Delta$ are the KM-links. These KM-links give a KM-graph.

MLL formulas $A$ and proof structures of type $A \to B$ form a category $Struct[\mathbf{T}]$. Composition of proof structures is given by composing the underlying KM-graphs; identity proof structures are given by identity KM-graphs.

In fact this category of proof structures is a $\ast$-autonomous category. As objects are MLL formulas, it is clear how $\otimes$ and $\multimap$ are defined on objects.

### Proof nets

Proof nets are those proof structures that arise by taking the KM-graphs of morphisms that are definable in the language of $\ast$-autonomous categories. Or, in the language of Girard, proof nets are those proof structures which are "correct" or "valid" (i.e., derivable from a sequent deduction, in a sense explained below).

More exactly, let $F[\mathbf{T}]$ be the free $\ast$-autonomous category on $\mathbf{T}$ (as discrete category), viewing $\ast$-autonomous categories and functors that preserve $\ast$-autonomous structure _strictly_ as a 1-category that is 1-monadic over $Cat$, the category of small categories and functors. Observe that the objects of $F[\mathbf{T}]$ may be identified with MLL formulas. We view the morphisms of $F[\mathbf{T}]$ as representing morphisms that are definable (starting with the datum $\mathbf{T}$).

As $Struct[\mathbf{T}]$ is $\ast$-autonomous, the obvious inclusion $\mathbf{T} \hookrightarrow Struct[\mathbf{T}]$ induces a unique strict $\ast$-autonomous functor $S: F[\mathbf{T}] \to Struct[\mathbf{T}]$, which may be called the _graphical semantics_ functor.

+-- {: .num_defn}
###### Definition
A **proof net** (in the language of MLL) of type $A \to B$ is a proof structure $\pi$ of the form $S(f)$, the graphical semantics of some arrow $f: A \to B$ in $F[\mathbf{T}]$.
=--

The following theorem for MLL proof nets, proved by Richard Blute in his thesis, is similar in content to the [[coherence theorem]] for [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] proved by Kelly-Mac Lane.

+-- {: .num_theorem}
###### Theorem
The restriction of $S: F[\mathbf{T}] \to Struct[\mathbf{T}]$ to the full subcategory consisting of objects isomorphic to formulas with no subformula occurrences of $\mathbf{1}$ or $\bot$ is faithful.
=--

For a cedent of formulas $\Gamma = A_1, \ldots, A_n$, let $\bigotimes \Gamma$ be the formula $A_1 \otimes \ldots \otimes A_n$ (associated to the left, say, if one is to be persnickety), and let $\parr \Gamma$ be the formula $A_1 \parr \ldots \parr A_n$. Each proof structure $\pi: \Gamma \to \Delta$ determines (and is determined by) a unique proof structure ${|\pi|}: \bigotimes \Gamma \to \parr \Delta$ with the same underlying KM-graph as $\pi$.

Then, more generally we may define a proof net of type $\Gamma \to \Delta$ to be a proof structure $\pi: \Gamma \to \Delta$ such that ${|\pi|}$ is a proof net according to the definition above.

Alternatively, we may define proof nets by reference to MLL sequent calculus, as follows.

### Nets of sequent deductions {#NetDeductions}

To each deduction or derivation tree in MLL sequent calculus, one may give an associated proof net. This is by induction; one considers the last step of a deduction $\delta$ whose premises have given derivations $\delta_i: \Gamma_i \to \Delta_i$, which have been given assigned proof nets $N(\delta_i)$, and uses this to assign a net to the full deduction $\delta$.

We recall the rules for MLL over a list of variable types $\mathbf{T}$, and associated rules for forming nets. As remarked above, for a deduction $\delta$ of a given sequent $\Gamma \to \Delta$, the net $N(\delta)$ is determined up to its KM-links, and so we will give just the rules for computing the KM-graph.

The rules for forming KM-graphs will sound pretty repetitive! But don't worry; that just means they're really easy. In summary, all we do for each rule besides the axioms is take the disjoint union of the KM-graphs occurring for the (derivations of the) premises, to get the KM-graph of the conclusion.

+-- {: .num_remark}
###### Remark
By the _[[cut rule|Hauptsatz]]_, i.e., cut-elimination theorem, we omit consideration of the cut rule. It is however interesting to consider the meaning of the cut-elimination algorithm in terms of proof nets; someone who is sufficiently savvy with graphics may wish to include material here.
=--

1. Axiom
$$\frac{}{\displaystyle t \vdash t}\; id,$$
where $t \in \mathbf{T}$. Here the net of the conclusion consists of the KM-graph $t^- \to t^+$.

1. Structural rule
$$\frac{\displaystyle \delta': \Gamma, A, B, \Delta \vdash \Sigma}{\displaystyle \Gamma, B, A, \Delta \vdash \Sigma}\; exchange$$
Here there is an identification between the sets of subformula occurrences for the premise and conclusion, and under this identification the proof net for $\delta$ is the same as the proof net for $\delta'$.

1. Unit rules (logical rules for units). There are four of these:
$$\frac{}{\displaystyle \; \vdash \mathbf{1}}\; \mathbf{1}_+$$
For $\mathbf{1}_+$, the KM-graph of the conclusion is empty.
$$\frac{}{\displaystyle \bot \vdash \; }\; \bot_{-}$$
For $\bot_{-}$, the KM-graph of the conclusion is empty.
$$\frac{\displaystyle \delta': \Gamma, \Delta \vdash \Sigma}{\displaystyle \Gamma, \mathbf{1}, \Delta \vdash \Sigma}\; \mathbf{1}_{-}$$
For $\mathbf{1}_{-}$, there is an identification between the variable subformula occurrences in the premise and in the conclusion. Under this identification, the KM-graph for $\delta$ is the same as for $\delta'$.
$$\frac{\displaystyle \delta': \Gamma \vdash \Delta, \Sigma}{\displaystyle \Gamma \vdash \Delta, \bot, \Sigma}\; \bot_+$$
For $\bot_+$, there is an identification between the variable subformula occurrences in the premise and in the conclusion. Under this identification, the KM-graph for $\delta$ is the same as for $\delta'$.

1. Logical rules for $\otimes$ and $\multimap$. There are four of these.
$$\frac{\displaystyle \delta': \Gamma, A, B, \Delta \vdash \Sigma}{\displaystyle \Gamma, A \otimes B, \Delta \vdash \Sigma}\; \otimes_{-}$$
For $\otimes_{-}$, there is an identification between the variable subformula occurrences in the premise and in the conclusion. Under this identification, the KM-graph for $\delta$ is the same as for $\delta'$.
$$\frac{\displaystyle \delta': \Gamma, A \vdash B, \Delta}{\displaystyle \Gamma \vdash A \multimap B, \Delta}\; \multimap_+$$
For $\multimap_+$, there is an identification between the variable subformula occurrences in the premise and in the conclusion. Under this identification, the KM-graph for $\delta$ is the same as for $\delta'$.
$$\frac{\displaystyle \delta_1: \Gamma \vdash \Delta, A \qquad \delta_2: \Sigma \vdash B, \Omega}{\Gamma, \Sigma \vdash \Delta, A \otimes B, \Omega}\; \otimes_+$$
In the case $\otimes_+$, the set of variable subformula occurrences in the conclusion is identified with the disjoint union of those for the premises, and under this identification the KM-graph of $\delta$ is the disjoint union of the KM-graphs for $\delta_1$ and $\delta_2$.
$$\frac{\displaystyle \delta_1: \Sigma, B \vdash \Omega \qquad \delta_2: \Gamma \vdash A, \Delta \qquad }{\Sigma, A \multimap B, \Gamma \vdash \Omega, \Delta}\; \multimap_{-}$$
In the case $\multimap_{-}$, the set of variable subformula occurrences in the conclusion is identified with the disjoint union of those for the premises, and under this ientification the KM-graph of $\delta$ is the disjoint union of the KM-graphs for $\delta_1$ and $\delta_2$.

## Graphical criterion for validity

In his original Linear Logic paper, Girard gave an interesting purely graphical criterion for a proof structure to be a proof net (i.e., "valid"), based on his so-called "cyclic trips" and "longtrip criterion". Later this criterion was simplified and improved by Danos and Regnier, who at the same time showed that validity of a proof structure could be decided in polynomial time. We give this criterion below. (**N.B.** this criterion as given by them works for proof structures of type $A \to B$ where there are no subformula occurrences of $\mathbf{1}$ and $\bot$. It may be extended to work for all formulas if we use a more refined notion of proof structure, called a unit-extended proof structure, as described below.)

+-- {: .num_defn}
###### Definition
A **par switch** (a $\parr$-switch) in a proof structure is a subgraph of one of the two forms

$$\array{
 & & (S \otimes T)^- & & & & S^- & & & & T^+ \\
 & \swarrow & & \searrow & & & & \nwarrow & & \swarrow \\
S^- & & & & T^- & & & & (S \multimap T)^+ & &
}$$

where $(S \otimes T)^-$ or $(S \multimap T)^+$ is a subformula node of the proof structure. (A $\otimes$ switch in a proof structure is similarly a subgraph consisting of a subformula node $(S \otimes T)^+$ or $(S \multimap T)^-$ and its two children in the binary construction tree, and the edges connecting them.)

A **network** of a proof structure $\pi$ is obtained by removing exactly one of the two edges from each par switch of $\pi$.
=--

+-- {: .num_theorem}
###### Theorem
**(Girard, Danos-Regnier)**
A proof structure $\pi$ is a proof net if every network of $\pi$ is a connected acyclic graph (considered as an unoriented graph, forgetting edge orientations).
=--

+-- {: .num_remark}
###### Remark
The proof is quite technical, but it is of fundamental importance in the analysis of proof nets. The method is to "sequentialize" a proof structure that satisfies this graphical criterion (i.e., provide a sequent deduction for it). This is by an inductive procedure which first removes consideration of outer par switches (replacing a proof structure of type $\Gamma, A \otimes B, \Delta \to \Sigma$ by an 'equivalent' structure of type $\Gamma, A, B, \Delta \to \Sigma$, and similarly a proof structure of type $\Gamma \to A \multimap B, \Delta$ by an equivalent one of type $\Gamma, A \to B, \Delta$). Once these have been removed, the hard part is to show there exists an outer $\otimes$ switch with parent $(A \otimes B)^+$ or $(A \multimap B)^-$, such that removal of the parent and edges to its children splits the graph cleanly into two connected components, each of which then satisfies the graphical criterion. The proof of existence is by a tricky combinatorial analysis on graphs. Once this is done, each of the two graph components is sequentializable by the inductive hypothesis, and the induction can then be pushed through.
=--

+-- {: .num_remark}
###### Remark
[[François Métayer]] ([1994](#Metayer94)) showed that the Danos-Regnier criterion on a proof structure's 'convergence to a proofnet' can equivalently be expressed as a criterion on the [[homology]] of a suitable associated graph.
=--

The Danos-Regnier criterion, stated according to the definition above, might appear exponential in complexity since it appears to involve checking that every one of the $2^p$ networks, where $p$ is the number of par switches, is connected and acyclic. However, Danos and Regnier gave a beautiful simplification which in fact gives an algorithm for deciding validity of a proof structure in polynomial time. (More recently, the problem has actually been shown to be complete for non-deterministic log space [JdNM08](#JdNM08).)

+-- {: .num_remark #DR}
###### Description
Informally, what one does is draw an arc between the two edges of each par switch (in the manner of secondary school geometry, when one wishes to indicate an angle between two lines), and then applies a series of graph reductions:

* At each step in the series, any un-arced edge between two distinct nodes may be contracted to a single node;

* At each step of the series, any configuration consisting of two distinct nodes and a pair of arced edges between them may be contracted to a point,

* A graph consisting of a single node and no edges reduces to itself.
=--

+-- {: .num_prop}
###### Proposition
**(Danos-Regnier)**
A proof structure is a proof net if and only if any series of graph reductions eventually reduces it to a graph with a single node and no edges.
=--

## Unit-extended proof nets

In his thesis, Trimble introduced a refinement of proof structures, designed to take the units $\mathbf{1}$ and $\bot$, particularly their actions and co-actions on other objects, more explicitly into account.

+-- {: .num_defn}
###### Definition
A **unit-extended proof structure** of type $\Gamma \to \Delta$ is a proof structure $\pi: \Gamma \to \Delta$ together with a function from the set of subformula occurrences of $\mathbf{1}^-$ to the set of subformulas, and a function from the set of subformula occurrences of $\bot^+$ to the set of subformulas.
=--

Unit-extended proof structures are considered as graphs, by adjoining to $\pi$ an edge $\mathbf{1}^- \to S$ for each subformula occurrence $\mathbf{1}^-$ that maps to $S$ under the first function, and another edge $T \to \bot^+$ for each subformula occurrence $\bot^+$ that maps to $T$ under the second function. This means for instance that for the sequent $\mathbf{1} \vdash \bot$, there is just one unit-extended proof structure of type $\mathbf{1} \to \bot$, and it consists of a _pair_ of edges from $\mathbf{1}^-$ to $\bot^+$.

To each MLL sequent deduction, one can associate a unit-extended proof structure, or rather a set of proof structures.

* For the axioms $t \vdash t$, the associated unit-extended proof structure is of course the KM-graph $t^- \to t^+$.

* For each of the rules $exchange$, $\mathbf{1}_+$, $\bot_{-}$, $\otimes_{-}$, $\otimes_+$, $\multimap_{-}$, $\multimap_+$, if unit-extended proof structures $\gamma_i$ (i.e., sets of unit edges and KM-links) have been associated to derivations of the premises, then associated to the final deduction (obtained by applying that rule) is the proof structure obtained by taking the disjoint union of the $\gamma_i$.

* This leaves the two unit rules $\mathbf{1}_{-}$, $\bot_+$. For the rule
$$\frac{\displaystyle \delta': \Gamma, \Delta \vdash \Sigma}{\displaystyle \Gamma, \mathbf{1}, \Delta \vdash \Sigma}\; \mathbf{1}_{-},$$
if $\gamma'$ is a unit-extended proof structure for $\delta'$, then associated to the final deduction $\delta$ is any unit-extended proof structure obtained by adjoining to $\gamma'$ a unit edge from the occurrence of $\mathbf{1}^-$ introduced in the conclusion, to any subformula node occurring in $\gamma': \Gamma, \Delta \to \Sigma$.
Similarly, for the rule
$$\frac{\displaystyle \delta': \Gamma \vdash \Delta, \Sigma}{\displaystyle \Gamma \vdash \Delta, \bot, \Sigma}\; \bot_+,$$
if $\gamma'$ is a unit-extended proof structure for $\delta'$, then associated to the final deduction $\delta$ is any unit-extended proof structure obtained by adjoining to $\gamma'$ a unit edge going to the occurrence of $\bot_+$ introduced in the conclusion, from any subformula node occurring in $\gamma': \Gamma, \Delta \to \Sigma$.

+-- {: .num_defn}
###### Definition
A **unit-extended proof net** is a unit-extended proof structure that is associated with some MLL sequent deduction.
=--

The notion of network for unit-extended proof structures is the same as for ordinary proof structures: a network is obtained by removing just one edge from each par-switch. The Danos-Regnier criterion for validity goes through without modification:

+-- {: .num_theorem}
###### Theorem
A unit-extended proof structure is a unit-extended proof net if and only if each of its networks is an acyclic connected graph (ignoring edge orientations).
=--

The simplified algorithm for deciding validity, in terms of a series of [graph reductions](#DR), also goes through for unit-extended proof structures without any modification.

## Comparison to string diagrams

There are various possibilities for translating the language of proof nets into a language of [[string diagrams]]. Some of these possibilities are subject to a debate on how liberally one would like to understand "string diagrams", but the idea that both proof nets and string diagrams open possibilities for rapid-fire graphical calculations in similar-looking ways should be taken seriously.

(I'll come back to this. I want to think some more on how I want to say it.)

## References

Proofnets originated with Girard's seminal paper introducing linear logic:

* {#Girard87} [[Jean-Yves Girard]] , _Linear Logic_ , Theor. Comp. Sci. **50** (1987) pp. 1-102. ([draft](http://iml.univ-mrs.fr/~girard/Synsem.pdf.gz))

* {#Trimble94} [[Todd Trimble]], _Linear Logic, Bimodules, and Full Coherence for Autonomous Categories_, Rutgers 1994

* {#BCST96} [[Richard Blute]], [[J.R.B. Cockett]], [[R. A. G. Seely]], [[Todd Trimble]], _Natural deduction and coherence for weakly distributive categories_, JPAA 113 (1996), 229-296. ([web](http://www.sciencedirect.com/science/article/pii/002240499500159X))

* {#JdNM08} Paulin Jacobé de Naurois and Virgile Mogbil, _Correctness of Linear Logic Proof Structures is NL-Complete_ ([pdf](https://hal.archives-ouvertes.fr/hal-00360894/document))

* François Lamarche, _Proof nets for intuitionistic linear logic: Essential nets_ ([pdf](http://hal.inria.fr/docs/00/34/73/36/PDF/prfnet1.pdf))

* [[Paul-André Melliès]], _A topological correctness criterion for non-commutative logic_ , London Mathematical Society Lecture Notes Series 316, 2004. ([pdf](https://hal.inria.fr/hal-00154204/document))

* {#Mellies06} [[Paul-André Melliès]], _Functorial boxes in string diagrams_, Proceedings of _Computer Science Logic 2006_ in Szeged, Hungary. 2006 ([pdf](https://www.irif.fr/~mellies/papers/Mellies06csl.pdf))

* [[Paul-André Melliès]], _Categorical semantics of linear logic_ ([pdf](http://profs.sci.univr.it/~bellin/panorama.pdf))

* {#Metayer94} [[François Métayer]], _Homology of proofnets_ ,Arch. Math. Logic **33** (1994) pp.169-188. ([draft](http://www.pps.univ-paris-diderot.fr/~metayer/PDF/hpn.pdf))

* Dominic J. D. Hughes, *Unification nets: canonical proof net quantifiers* [arxiv](https://arxiv.org/abs/1802.03224)

Relation to [[Feynman diagrams]] is discussed in

* {#BlutePanangaden} [[Richard Blute]], [[Prakash Panangaden]], _Proof nets as formal Feynman diagrams_ ([pdf](http://www.indiana.edu/~iulg/qliqc/phi.pdf))

[[!redirects proof nets]]


proof of the prime number theorem

# Contents
* table of contents
{:toc}

## Introduction

For $x \gt 0$, let $\pi(x)$ denote the number of primes $p \leq x$ (in these notes, $p$ is always used to denote prime numbers). The celebrated [[prime number theorem|Prime Number Theorem]] is the asymptotic statement

$$\pi(x) \sim \frac{x}{\log(x)}$$

where we recall that $f(x) \sim g(x)$ means $\underset{x \to \infty}{\lim} \frac{f(x)}{g(x)} = 1$.

This was first proven in 1896 independently by Hadamard and de la Vallée-Poussin, first by proving a result on the zeroes of the Riemann zeta function $\zeta(s)$ (that none occur on the line $Re(s) = 1$), which can be proven without too much difficulty. They then deduced the prime number theorem from this result, by a fairly intricate analysis.

Over the decades this analysis was whittled down more and more until it reached a fairly definitive form, in the form of a succinct "Tauberian theorem" due to [D.J. Newman (1980)](#new). A streamlined account was written up in the American Mathematical Monthly by [Don Zagier (1997)](#zag1), who managed to produce a proof virtually from scratch in a little less than three pages (it requires just a bit of background in complex analysis, not much more than Cauchy's theorem).

As in his famous [one-sentence proof](#zag2) that primes of the form $4n+1$ are sums of two squares, Zagier seemed to be aiming explicitly for maximal compression while still keeping the argument (arguably) self-contained. It's an impressive performance, alright, but its terseness will probably force most readers to follow up with pen and paper on the side to verify his arguments in detail. In this article we will mostly follow Zagier's line, but with some slight expansions and glosses so that pen and paper is (hopefully) unnecessary.

## Chebyshev's estimate

In this section we prove a result essentially due to Chebyshev which puts us already in the neighborhood of the prime number theorem. Introduce a function

$$\theta(x) = \sum_{p \leq x} \log(p).$$

By the following result, our job is done once we establish the relation $\theta(x) \sim x$.

+-- {: .num_lemma #asymp}
###### Lemma
The asymptotic statement $\theta(x) \sim x$ implies the prime number theorem.
=--

+-- {: .proof}
###### Proof
In one direction, we have an obvious inequality

$$\theta(x) = \sum_{p \leq x} \log(p) \leq \sum_{p \leq x} \log(x) = \pi(x)\log(x).$$

Assuming that $\theta(x) \sim x$, this implies that given $\epsilon \gt 0$, the inequality

$$\label{ineq1} 1 - \epsilon \lt \frac{\pi(x) \log(x)}{x}$$

holds for all sufficiently large $x$.

In the other direction, for any $\epsilon \gt 0$ we have

$$\array{
\theta(x) & \geq & \sum_{x^{1-\epsilon} \lt p \leq x} \log(p) \\
 & \geq & \sum_{x^{1-\epsilon} \lt p \leq x} \log(x^{1-\epsilon}) \\
 & = & (1 - \epsilon)\log(x) \cdot (\pi(x) - \pi(x^{1 - \epsilon})) \\
 & \geq & (1 - \epsilon)\log(x) \cdot (\pi(x) - x^{1 - \epsilon})
}$$

From the last line and (eq:ineq1) (which implies $\frac{\pi(x)\log(x)}{x}$ is bounded below by a positive constant), we obtain

$$\array{
\frac{\theta(x)}{x} & \geq & (1-\epsilon)\frac{\pi(x)\log(x)}{x} - x^{-\epsilon} \\
 & \geq & (1 - 2\epsilon)\frac{\pi(x)\log(x)}{x}
}$$

for sufficiently large $x$. Thus for any small $\epsilon \gt 0$ (say less than $1/3$) and the assumption $\theta(x) \sim x$ we have

$$1 - \epsilon \leq \frac{\pi(x)\log(x)}{x} \leq (1 - 2\epsilon)^{-1}\frac{\theta(x)}{x} \leq (1 - 3\epsilon)^{-1}$$

or simply

$$1 - \epsilon \leq \frac{\pi(x)\log(x)}{x} \leq (1 - 3\epsilon)^{-1}$$

for all sufficiently large $x$.
=--

Proving the asymptotic result $\theta(x) \sim x$ will involve complex analysis applied to the Riemann zeta function. An easier preliminary result gives information on the *order* of $\theta(x)$; it will later be used as a technical hypothesis in the Tauberian theorem with which we conclude. The order result is essentially due to Chebyshev, who actually established sharper estimates (which we won't need). In fact, he also showed that if $\frac{\pi(x)\log(x)}{x}$ has a limit, then that limit must be $1$.

+-- {: .num_lemma #cheb}
###### Lemma
$\theta(x) = \mathrm{O}(x)$.
=--

+-- {: .proof}
###### Proof
First let us observe that

$$2^{2n} \geq \binom{2n}{n} \geq \prod_{n \lt p \lt 2n} p = \exp(\theta(2n) - \theta(n))$$

so that

$$2n\log(2) \geq \theta(2n) - \theta(n).$$

If $n \leq x \lt n+1$, then $\theta(x) = \theta(n)$ and $\theta(2x) \leq \theta(2n) + \log(2x)$, so that

$$2x\log(2) \geq 2n\log(2) \geq \theta(2n) - \theta(n) \geq \theta(2x) - \log(2x)  - \theta(x)$$

or simply

$$\theta(x) - \theta(x/2) \leq x\log(2) + \log(x).$$

Taking a telescopic sum, we have

$$\theta(x) = \sum_{k = 0}^{n-1} \theta(x/2^k) - \theta(x/2^{k+1}) \leq \sum_{k=0}^{n-1} (x/2^k)\log(2) + \sum_{k=0}^{n-1} \log(x/2^k)$$

where $n$ is chosen so that $1 \leq x/2^n \lt 2$ (so $n = floor(\log_2(x))$). The first term on the right is bounded above by $\log(2) x$. The second term is bounded above by $\log_2(x) \cdot \log(x) = K\log(x)^2$ (where $K = \log_2(e)$), which considered asymptotically is small compared to $x$. Thus, for any $C \gt \log(2)$, we have

$$\theta(x) \leq C x$$
for all sufficiently large $x$, which completes the proof.
=--

## Analytic continuation of the zeta function to $Re(s) \gt 0$

Let

$$\label{prod} \zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \prod_p \left(1 - \frac1{p^s}\right)^{-1}$$

define the usual Riemann zeta function on the region $Re(s) \gt 1$. First we establish a meromorphic continuation of $\zeta(s)$ to the region $Re(s) \gt 0$, or in fact a holomorphic continuation of $\zeta(s) - \frac1{s-1}$ to this region.

+-- {: .num_prop #cont}
###### Proposition
The function $\Psi$ defined by the expression

$$\Psi(s) = \sum_{n=1}^\infty \int_n^{n+1} \frac1{n^s} - \frac1{x^s}\; d x$$

converges absolutely for $Re(s) \gt 0$, giving a holomorphic function in that region, and

$$\Psi(s) = \zeta(s) - \frac1{s-1}$$

over the region $Re(s) \gt 1$.
=--

+-- {: .proof}
###### Proof
Let us bound the summands in the definition of $\Psi(s)$:

$$\array{
\left| \int_n^{n+1} \frac1{n^s} - \frac1{x^s}\; d x \right| & = & \left| \int_n^{n+1} \int_n^x \; \frac{s}{t^{s+1}}\; d t\; d x \right|\\
 & \leq & \int_n^{n+1} \int_n^x \; \frac{{|s|}}{{|t^{s+1}|}}\;d t \; d x \\
 & = & \int_n^{n+1} \int_n^x \; \frac{{|s|}}{t^{Re(s) + 1}}\; d t\; d x \\
 & \leq & \int_n^{n+1} \int_n^x \; \frac{{|s|}}{n^{Re(s) + 1}}\;d t \; d x \\
 & \leq & \frac{{|s|}}{n^{Re(s) + 1}}.
}$$

Since $\sum_{n=1}^\infty \frac1{n^{Re(s) + 1}}$ converges over the region $Re(s) \gt 0$, it follows that the series for $\Phi(s)$ converges absolutely, and $\Psi(s)$ is a holomorphic function, in that region.

Over the region $Re(s) \gt 1$, we have

$$\array{
\Psi(s) & = & \sum_{n=1}^\infty \frac1{n^s} - \int_n^{n+1} \frac1{x^s}\; d x \\
 & = & \sum_{n=1}^\infty \frac1{n^s} - \sum_{n=1}^\infty \int_n^{n+1} \frac1{x^s}\; d x \\
 & = & \zeta(s) - \int_1^\infty \frac1{x^s}\; d x \\
 & = & \zeta(s) - \frac1{s-1}
}$$

as asserted.
=--

## The zeta function has no zeroes on the line $Re(s) = 1$

Now we relate $\theta(x)$ to $\zeta(s)$ by introducing an auxiliary function $\Phi$ that is connected with the logarithmic derivative of $\zeta(s)$. For $Re(s) \gt 1$ put

$$\Phi(s) = \sum_p \frac{\log(p)}{p^s}.$$

Over $Re(s) \gt 1$, we calculate

$$\log \zeta(s) = -\sum_p \log(1 - p^{-s})$$

which upon differentiating yields

$$\array{
\frac{\zeta^'(s)}{\zeta(s)} & = & -\sum_p \frac{\log(p) p^{-s}}{1 - p^{-s}} \\
 & = & -\sum_p \log(p) p^{-s} -\sum_p \frac{\log(p) p^{-s}p^{-s}}{1 - p^{-s}} \\
 & = & -\Phi(s) - \sum_p \frac{\log(p)}{p^s(p^s - 1)}
}$$

with the last sum converging over the region $Re(s) \gt \frac1{2}$. From this and the analytic continuation of $\zeta(s)$ (Proposition \ref{cont}), we see that $\Phi(s)$ admits a meromorphic continuation over $Re(s) \gt \frac1{2}$, and holomorphic in that region except possibly where $\zeta(s)$ has zeroes, or at the simple pole $s = 1$.

+-- {: .num_prop #nozero}
###### Proposition
$\Phi(s) - \frac1{s-1}$ is holomorphic over the region $Re(s) \geq 1$, i.e., $\zeta(s)$ has no zeroes over $Re(s) \geq 1$.
=--

+-- {: .proof}
###### Proof
The case $Re(s) \gt 1$ is easily handled by the product formula (eq:prod).

For a fixed $t \gt 0$, let $\mu$ be the order of the zero of $\zeta(1 + i t)$ (if $\zeta(1 + i t) \neq 0$ then of course $\mu = 0$); it is the same as the order of the zero at $s = 1 - i t$. Let $\nu$ be the order of the zero at $1 \pm 2i t$. We then have

1. $\underset{\epsilon \searrow 0}{\lim} \epsilon \cdot \Phi(1 + \epsilon) = 1$ (since $\zeta(s)$ has a pole of order $1$ at $s = 1$), and similarly

1. $\underset{\epsilon \searrow 0}{\lim} \epsilon \cdot \Phi(1 + \epsilon \pm i t) = -\mu$,

1. $\underset{\epsilon \searrow 0}{\lim} \epsilon \cdot \Phi(1 + \epsilon \pm 2i t) = -\nu$.

Observe that for $\epsilon \gt 0$,

$$\array{
0 & \leq & \epsilon \cdot \sum_p \frac{\log(p)}{p^{1 + \epsilon}} (p^{i t/2} + p^{-i t/2})^4 \\
 & = & \epsilon \cdot \sum_p \frac{\log(p)}{p^{1 + \epsilon}} \sum_{k = -2}^2 \binom{4}{r + 2} p^{-i r t} \\
 & = & \epsilon \cdot \sum_{r = -2}^2 \binom{4}{r+2} \Phi(1 + \epsilon + i r t)
}$$

from which, in conjunction with 1., 2., 3. above, we deduce $0 \leq 6 - 8\mu - 2\nu$, so that $\mu = 0$.
=--

The function $\theta(x)$ is related to the function $\Phi(s)$ by way of a Riemann-Stieltjes integral:

$$\Phi(s) = \sum_p \frac{\log(p)}{p^s} = \int_1^\infty \frac{d \theta(x)}{x^s}$$

for $Re(s) \gt 1$. An integration by parts yields

$$\int_1^M \frac{d \theta(x)}{x^s} = \left[\frac{\theta(M)}{M^s} - \theta(1)\right] + \int_1^M \theta(x) \frac{s}{x^{s+1}}\; d x$$

where $\underset{M \to \infty}{\lim} \frac{\theta(M)}{M^s} = 0$ by Lemma \ref{cheb}. Thus

$$\label{RS} \Phi(s) = s \int_1^\infty \frac{\theta(x)}{x^{s+1}}\; d x = s \int_0^\infty e^{-s t} \theta(e^t)\; d t.$$

## Reduction to a Tauberian theorem

The asymptotic result $\theta(x) \sim x$ will be established by appeal to the following result:

+-- {: .num_theorem #conv}
###### Theorem
Suppose that the integral

$$\int_1^\infty \; \frac{\theta(x) - x}{x^2}\; d x$$

converges. Then $\theta(x) \sim x$.
=--

+-- {: .proof}
###### Proof
Suppose that for some $\lambda \gt 1$ there exist arbitrarily large $x$ such that $\lambda x \leq \theta(x)$. The monotonicity of $\theta$ implies the first inequality that follows:

$$\int_x^{\lambda x} \; \frac{\theta(t) - t}{t}\; \frac{d t}{t} \geq \int_x^{\lambda x} \; \frac{\lambda x - t}{t}\; \frac{d t}{t} = \int_1^\lambda \; \frac{\lambda x - t x}{t x} \; \frac{d t}{t} = \int_1^\lambda \; \frac{\lambda - t}{t^2}\; d t \gt 0.$$

This is a contradiction, because convergence of $\int_1^\infty \; \frac{\theta(t) - t}{t^2}\; d t$ implies that $\int_x^{\lambda x} \; \frac{\theta(t) - t}{t^2}\; d t$ can be made arbitrarily small by taking $x$ large.

Similarly, if we suppose that for some $\lambda \lt 1$ there exist arbitrarily large $x$ such that $\theta(x) \leq \lambda x$, then

$$\int_{\lambda x}^x \; \frac{\theta(t) - t}{t}\; \frac{d t}{t} \leq \int_{\lambda x}^x \; \frac{\lambda x - t}{t}\; \frac{d t}{t} = \int_\lambda^1 \; \frac{\lambda x - t x}{t x} \; \frac{d t}{t} = \int_\lambda^1 \; \frac{\lambda - t}{t^2}\; d t \lt 0$$

which again leads to a contradiction.
=--

Thus we focus on convergence of

$$\label{rewrite} \int_1^\infty \; \frac{\theta(x) - x}{x}\; \frac{d x}{x} = \int_0^\infty \; \frac{\theta(e^t) - e^t}{e^t}\; d t$$

For $Re(s) \gt 1$ we have, from (eq:RS),

$$\array{
\frac{\Phi(s)}{s} - \frac1{s-1} & = & \int_0^\infty \; e^{-s t}\theta(e^t) - e^{(1-s)t}\; dt \\
 & = & \int_0^\infty \; e^{-s t} (\theta(e^t) - e^t)\; d t
}$$

or what is the same, for the region $Re(s) \gt 0$,

$$\label{laplace} \frac{\Phi(s+1)}{s+1} - \frac1{s} = \int_0^\infty \; e^{-(s+1) t} (\theta(e^t) - e^t)\; d t = \int_0^\infty \; e^{-s t}\left( \frac{\theta(e^t) - e^t}{e^t} \right)\; d t
$$

where the left side is in fact holomorphic over $Re(s) \geq 0$, hence takes a finite value at $s = 0$, say $L$.

The Tauberian theorem we will establish permits the conclusion

$$L = \underset{s \searrow 0}{\lim} \int_0^\infty \; e^{-s t}\left( \frac{\theta(e^t) - e^t}{e^t} \right)\; d t = \int_0^\infty \; \frac{\theta(e^t) - e^t}{e^t} \; d t$$

which in turn will yield the Prime Number Theorem:

+-- {: .num_theorem #pnt}
###### Theorem
The integral $\int_0^\infty \; \frac{\theta(e^t) - e^t}{e^t} \; d t$ converges.
=--

(The prime number theorem follows from this by (eq:rewrite), Theorem \ref{conv}, and Lemma \ref{asymp}.)

## Proof of the Tauberian theorem

Here is the statement of the particular Tauberian theorem used (recalling that "Tauberian" signifies a family of theorems that roughly speaking give conditions for convergence of series or integral expressions occurring at natural boundaries of regions where series or integral representations are defined). This particular form was given by D.J. Newman, and allows a dramatic shortening of the proof of the prime number theorem, as shown convincingly by Zagier, who as we said compressed it down to under three pages.

+-- {: .num_theorem #taub}
###### Theorem
Let $f(t)$ be a function that is bounded and locally integrable over the domain $t \geq 0$, and suppose that its Laplace transform, defined for $Re(s) \gt 0$ by the equation

$$g(s) = \int_0^\infty \; e^{-s t} f(t)\; d t,$$

extends to a function $g$ that is holomorphic over the region $Re(s) \geq 0$. Then the integral $\int_0^\infty\; f(t)\; d t$ converges and equals $g(0)$.
=--

The proof below involves some fairly refined complex analysis. It may be well to recall that we say a function $g$ is *holomorphic* at a point $z_0$ if $g$ is defined over a small neighborhood $\{z: {|z-z_0|} \lt \delta\}$ of the point and in that neighborhood is expressible as a convergent Taylor series

$$g(z) = \sum_{n=0} \frac{a_n (z - z_0)^n}{n!}.$$

Then we say $g$ is holomorphic over a (not necessarily open) region if it is holomorphic at every point in the region.

In the case of interest $f(t) = \frac{\theta(e^t) - e^t}{e^t}$, the order estimate Lemma \ref{cheb} guarantees that $f$ is bounded and locally integrable, and we have already observed that its Laplace transform, which is

$$g(s) = \frac{\Phi(s+1)}{s+1} - \frac1{s}$$

by (eq:laplace), is holomorphic in the region $Re(s) \geq 0$ by Proposition \ref{nozero}. Thus all the hypotheses of the theorem are satisfied in this case, and its conclusion in this case is Theorem \ref{pnt}. So all that remains is to prove Theorem \ref{taub}.

+-- {: .proof}
###### Proof
For $T \geq 0$, put

$$g_T(s) = \int_0^T \; e^{-s t} f(t)\; d t.$$

Clearly $g_T$ is an entire (i.e., everywhere holomorphic) function. We are trying to show that $\underset{T \to \infty}{\lim} g_T(0) = g(0)$.

A straightforward application of Cauchy's theorem yields

$$g_T(0) - g(0) = \frac1{2\pi i} \int_C \; \left(g_T(z) - g(z)\right) e^{z T}\; \frac{d z}{z}$$

for contours $C$ having $0$ as an interior point, such that $g_T - g$ is holomorphic on $C$ and its interior. We will choose $C$ to be the boundary of the closed region $D = \{z: {|z|} \leq R\; and\; Re(z) \geq -\delta\}$ where for a given $R$, we choose $\delta$ small enough that $g_T - g$ is holomorphic in $D$ (this uses compactness of the interval $I = \{i t: -R \leq t \leq R\}$ and holomorphicity of $g_T - g$ along $I$).

The idea would be to estimate the contribution of the integral over $C_+$, the part of $C$ where the real part is nonnegative, and over $C_-$ where it is nonpositive, hoping to show these contributions are small when $R$ is large. This almost works, except that we don't have good control over the size of the contributions near the area where $Re(z) = 0$. To offset this, a mollifying factor $(1 + \frac{z^2}{R^2})$ is introduced in the integrand ("Carleman's formula"): this is close to zero near $Re(z) = 0$ and has the desired mitigating effect.

Thus we instead use

$$g_T(0) - g(0) = \frac1{2\pi i} \int_C \; \left(g_T(z) - g(z)\right) e^{z T}\; \left(1 + \frac{z^2}{R^2}\right)\; \frac{d z}{z}$$

Suppose now that ${|f(t)|}$ is globally bounded by $B$. For $Re(z) \gt 0$ we have

$${|g_T(z) - g(z)|} \leq \int_T^\infty\; {|f(t)e^{-z t}|}\; d t \leq B \int_T^\infty e^{-Re(z)t}\; d t = B \frac{e^{-Re(z)T}}{Re(z)}$$

and we also have, over the circle ${|z|} = R$,

$$\array{
{|e^{z T}(1 + z^2/R^2)(1/z)|} & = & e^{Re(z)T} \cdot {|(R^2 z^{-1} + z)/R^2|} \\
 & = & e^{Re(z)T} \cdot {|({|z|}^2 z^{-1} + z)/R^2|} \\
 & = & e^{Re(z)T} \cdot {|(\bar{z} + z)/R^2|} \\
 & = & e^{Re(z)T} \cdot {|2Re(z)|}/R^2
}$$

so that putting the factors together, the integral over $C_+$ is bounded in absolute value by

$$2B/R^2 \int_{C_+}\; {|d z|} = 2\pi B/R$$

(and of course this hasn't taken into account the factor $\frac1{2\pi i}$ outside the integral). In any case, with the Carleman factor, the contribution
from the $C_+$ part is $B/R$, which is small if $R$ is large.

Now we bound the contribution from $C_-$ by considering $g_T$ and $g$ separately. First let us bound

$$\frac1{2\pi i} \int_{C_-} \; g_T(z) e^{z T}\; \left(1 + \frac{z^2}{R^2}\right)\; \frac{d z}{z}.$$

For this, we can take advantage of the fact that $g_T$ is entire, and (courtesy of Cauchy's theorem) replace $C_-$ by the semicircle where ${|z|} = R, Re(z) \lt 0$. The estimates are similar in form to those above, starting with

$${|g_T(z)|} \leq \int_0^T {|f(t)|}\cdot {|e^{-z t}|}\; d t \leq B \int_0^T e^{-Re(z)t}\; d t \leq B \int_{-\infty}^T e^{-Re(z)t}\; d t = B \frac{e^{-Re(z)T}}{-Re(z)}.$$

Bear in mind here that we are now considering $Re(z) \lt 0$ and $-Re(z) = {|Re(z)|}$; the bound produced tends to be very large. But, large as it may be, it is offset by the bound on the product of the remaining factors in the integrand, which is $e^{Re(z)T} \cdot {|2Re(z)|}/R^2$. We conclude as before that over $C_-$ the overall contribution from $g_T$ has upper bound $B/R$.

Finally, we bound

$$\frac1{2\pi i} \int_{C_-} \; g(z) e^{z T}\; \left(1 + \frac{z^2}{R^2}\right)\; \frac{d z}{z}.$$

Holding $R$ fixed, the expression $g(z)\cdot \left(1 + \frac{z^2}{R^2}\right) \cdot \frac1{z}$ taken over $C_-$ is bounded in absolute value by a number $K$. Whereas along $Re(z) = -\delta$, by taking $T$ very large we can make ${|e^{z T}|}$ as small as we like, so that considering $g$ separately, the contribution over $C_-$ can be discarded as vanishingly small (in other words, first pick $R$, then pick $\delta$, and finally pick $T$, to get this contribution to be as small as we like). This completes the proof.
=--

## References

* {#new} Donald J. Newman, *Simple analytic proof of the prime number theorem*, Amer. Math. Monthly 87 (1980), 693-696.

* {#zag1} Don Zagier, *Newman's short proof of the prime number theorem*, Amer. Math. Monthly 104 (1997), 705-708. ([pdf](https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Zagier.pdf))

* {#zag2} Don Zagier, *A one-sentence proof that every prime $p \equiv 1 (\mod 4)$ is a sum of two squares*, Amer. Math. Monthly, Vol. 97 No. 2 (1990), p. 144. ([link](https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2323918/fulltext.pdf))


proof relevance

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[constructive mathematics]], _proof relevance_ refers to the concept that mathematical [[proofs]] are mathematical objects themselves and that besides just knowing that a [[proposition]] has _some_ [[proof]] it is relevant to remember the way a given proof is constructed (and to remember possibly different proofs of the same proposition).

This idea is formalized notably by the idea of [[propositions as types]] in [[type theory]] where a proof is an actual [[term]] (the _proof term_) of some [[type]].

For instance logical [[disjunction]] "A or B" in [[type theory]] (in particular in [[homotopy type theory]]) may be exhibited by the [[sum type]] $A + B$ of the types representing the individual propositions, and a proof/term of $A + B$ carries in it the information of whether it proved $A$ or $B$ to (hence) prove their disjunction.  Similarly, a proof of $C \to (A + B)$ will include the information of which proofs of $C$ lead to proofs of $A$ and which proofs of $C$ lead to proofs of $B$.

## References

### General

* [[UF-IAS-2012|Univalent Foundations Project]], section 1.11 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

### Prehistory

In

* [[Georg Hegel]], _[[Phenomenology of Spirit]]_ (1807)

the following complaint about mathematical proof (in section _[12 Historical and mathematical proof](https://www.marxists.org/reference/archive/hegel/works/ph/phprefac.htm#12)_ of the Preface) might be read as being a complaint about the traditional non-[[constructive mathematics|constructive]] concept of proof and about the traditional lack of proof relevance:

> All the same, while proof is essential in the case of mathematical
knowledge, it still does not have the significance and nature of being
a moment in the result itself; the proof is over when we get the
result, and has disappeared. The process of mathematical proof does not belong to the object; it is a function that takes place outside the matter in hand.

> [footnote 42](https://www.marxists.org/reference/archive/hegel/help/finpref.htm#m042): Mathematical truths are not thought to be known unless proved true. Their demonstrations are not, however, kept as parts of what they prove, but are only our subjective means towards knowing the latter. In philosophy, however, consequences always form part of the essence made manifest in them, which returns to itself in such expressions.

See also earlier conceptions of proofs expressing the 'cause' of a theorem, where [[proof by contradiction|proofs by contradiction]] in particular were taken generally to fail. Such an idea goes back to Aristotle for whom a proper answer to the question "Why is the angle in a semicircle a right-angle?" gives its cause.

* Paolo Mancosu, _Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century_, OUP, 1996.

[[!redirects proof relevance]]
[[!redirects proof relevant]]
[[!redirects proof-relevant]]


proof theoretic strength of univalent type theory plus HITs

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

## Overview

This is a stub collecting information on the [[proof theory|proof theoretic]] strength of [[univalent type theory]] with [[higher inductive types]].

Setzer computes the strength of MLTT+W. Voevodsky reduces Coq's inductive types to W-types. Coquand et al reduces univalence and some HITs to an unspecified constructive framework, which can be taken to be CZF+REA with a sequence of
universes. It follows from this work that univalent type theory and these HITs has the same proof theoretic strength as MLTT with inductive definitions and a sequence of universes.

There is an [MO-question](http://mathoverflow.net/questions/69229/proof-strength-of-calculus-of-inductive-constructions) on the proof theoretic strength of pCIC. Avigad provide a general overview of the proof theory of predicative constructive systems.

## The strength of type theories

This section collects some references that establish the strength various type theories, roughly in increasing order of strength.

Let $\mathrm{ML}_n$ denote MLTT without inductive types and $n$ universes closed under $\Pi$ and $\Sigma$. This has the strength of the first-order system of $n$ iterated fixed points (whether with intuitionistic or classical logic), and thus by Feferman 1982 has proof-theoretic ordinal $\xi_n$, where $\xi_0=\varepsilon_0$ and $\xi_{n+1} = \varphi(\xi_n,0)$.

## References

* {#Avigad} Jeremy Avigad, _Proof Theory_, 2014 [PDF](https://www.andrew.cmu.edu/user/avigad/Papers/formal_epistemology.pdf)

* {#Coquand} Thierry Coquand et al, _Variation on Cubical sets_, 2014 [PDF](http://www.cse.chalmers.se/~coquand/comp.pdf).

* {#Feferman} Solomon Feferman, _Iterated inductive fixed-point theories: application to Hancock's conjecture_, Stud. Logic Foundations Math., 109, 1982.

* {#Setzer} Anton Setzer, _Proof theoretical strength of Martin-Lof Type Theory with W-type and one universe_ [PDF](http://cs-svr1.swan.ac.uk/~csetzer/articles/weor0.pdf).

* {#Setzer2004} Anton Setzer, _Proof theory of Martin-Löf type theory. An overview_ [PDF](http://www.cs.swan.ac.uk/~csetzer/articles/overviewProofTheoryTypeTheory2004.pdf)

* {#Voevodsky} Vladimir Voevodsky, _Notes on type systems_ 2011 [PDF](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/expressions_current.pdf).


proof theory

#Contents#
* table of contents
{:toc}

## Idea

*Proof theory* is that part of [[mathematical logic]] which is concerned with the notion of [[formal proof]]. It was introduced by [[David Hilbert]] under the name _Beweistheorie_ as part of [[Hilbert's program]]. Since [[Gödel's incompleteness theorem]] put an end to the original version of Hilbert's program, proof theory has broadened its scope to include the following:

* **Structural proof theory** is concerned with the structure of proofs and perspicuous proof calculi for elucidating this structure. This leads to the study of [[sequent calculi]] and of [[natural deduction]].

* **Reductive proof theory** is the modern version of [[Hilbert's program]]: Instead of hoping to secure all of mathematics with finitistic means, one ask: _what rests on what?_

  That is, one asks for reductions between various formal systems, showing for example that some [[classical logic|classical]] systems can be reduced to [[constructive mathematics|constructive]] systems, or some systems which use _a priori_ [[impredicative]] principles can nevertheless be reduced to [[predicative]] systems. See also: [[reverse mathematics]].

* **[[ordinal analysis|Ordinal analysis]]** analyzes the provably well-founded recursive [[ordinals]] and provably total [[recursive function|recursive functions]] of a theory by devising suitable proof calculi.

* **Proof mining** analyzes proofs of specific statements in order to extract more information than the mere truth of the statement, for instance witnesses or bounds on witnesses of [[existential quantifier|existential]] statements.

## Related concepts

* [[type theory]]

* [[second order arithmetic]], [[elementary function arithmetic]]

* [[proof management system]]

* [[linear logic]]

* [[ordinal analysis]]

## References

Textbook accounts on proof theory with focus on [[natural deduction]] and [[sequent calculi]]:

* [[Anne Sjerp Troelstra]], [[Helmut Schwichtenberg]], _Basic Proof Theory_ , Cambridge University Press (2000, 2012) [[doi:10.1017/CBO9781139168717](https://doi.org/10.1017/CBO9781139168717)]

* Sara Negri, Jan von Plato, *Structural Proof Theory*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511527340](https://doi.org/10.1017/CBO9780511527340), [§1 pdf](https://www.logicmatters.net/resources/pdfs/NegriPlato1.pdf), [§2 pdf](https://www.logicmatters.net/resources/pdfs/NegriPlato1.pdf)]

See also:

* Wikipedia, _[Proof theory](http://en.wikipedia.org/wiki/Proof_theory)_.

* Georg Kreisel, _A Survey of Proof Theory_, JSL 33(3), 1968, 321-388.

* [[Richard Statman]], _[[Structural Complexity of Proofs]]_, 1974

* [[Jean-Yves Girard]], [[Yves Lafont]], and [[Paul Taylor]], _Proofs and Types_, 1990. ([web](http://www.paultaylor.eu/stable/Proofs+Types.html))

* [[Martin Hyland]] provides some abstract perspectives on proof theory as a subject in the introduction to his essay, [Proof Theory in the Abstract](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/pta02.pdf).


proofs as programs

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type theory]], regarding [[propositions as types]], a [[proof]] of a [[proposition]] is given by [[constructive mathematics|constructing]] a [[term]] of the corresponding [[type]]. This can hence be regarded as a [[program]] to compute a specific term (output).

## Related concepts

* [[computational type theory]]

* [[computational trinitarianism]]

* [[theory]]

* [[proposition]]/[[type]] ([[propositions as types]])

* [[definition]]/[[proof]]/[[program]] (**proofs as programs**)

* [[theorem]], [[axiom]]

## References

An exposition is in

* [[Robert Harper]], _The Holy Trinity_ (2011) ([web](http://existentialtype.wordpress.com/2011/03/27/the-holy-trinity/), [wayback machine snapshot](https://web.archive.org/web/20170921012554/http://existentialtype.wordpress.com/2011/03/27/the-holy-trinity/))
 {#Harper}

That [[Martin-Löf dependent type theory]] can be regarded also as a [[functional programming language]] by identifying proofs as programs was stressed in

* [[Per Martin-Löf]], _Constructive mathematics and computer programming_, Philosophical Transactions
of the Royal Society of London (Series A) 312 (1984), no. 1522, pp. 501–518.

[[!redirects programs as proofs]]

[[!redirects proofs-as-programs]]


PROP

> This article is about the [[symmetric monoidal category]]. For the [[type]] of [[h-propositions]] see  [[Prop]].

---

# Contents
* table of contents
{: toc}

## Idea

A 'PROP' --- an abbreviation of 'products and permutations category' --- is a [[symmetric monoidal category]] generated by a single [[object]], used to describe a given sort of algebraic structure.  Because of their one-sorted nature, one can think of PROPs as a variant of [[Lawvere theory|Lawvere theories]] suitable for non[[cartesian monoidal category|cartesian]] contexts.  In this respect they are similar to [[operad|operads]].  However, they are more general, because they can be used to describe operations with many outputs as well as many inputs.

## Definition

+-- {: .num_defn }
###### Definition
A **PROP** is a strict [[symmetric monoidal category]] where every object is of the form

$$x^{\otimes n} = x \otimes x \otimes \cdots \otimes x$$

for a single object $x$ and $n \ge 0$.
=--

There is also a notion of colored PROP akin to [[colored operads]], or [multisorted Lawvere theories](https://ncatlab.org/toddtrimble/published/multisorted+Lawvere+theories). One way to define a colored PROP is as a certain kind of [[symmetric monoidal category]] (see Remark 2.2.14 of [Yau](#Yau)):

+-- {: .num_defn }
###### Definition

A **colored PROP** with set of colors $\mathfrak{C}$ is a strict symmetric monoidal category $(P,\odot)$ whose [[monoid]] of objects is freely generated by $\mathfrak{C}$.

=--

+-- {: .num_defn }
###### Definition

A **morphism of PROPs** $\phi:Q\to T$ is just a strict symmetric monoidal functor that takes the generators for the objects of $Q$ to the generators of the objects of $T$. Equivalently, a morphism of PROPs is a pair of functions:  one from the colors of $Q$ to the colors of $T$, $\phi_c\colon Col(Q)\to Col(T)$, and for each pair of finite list of colors $\vec{x}=\{x_1,\ldots,x_n\}$ and $\vec{y}=\{y_1,\ldots,y_m\}$ in $Col(Q)$, a function $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$. The two functions $\phi_c$ and $\phi_m$ are of course required to preserve compositions, units and symmetries.
=--

Thus there is a **category of PROPs**. For more on this category, as well as some its properties, see [HR1](#HR1).

+-- {: .num_defn }
###### Definition

Given a PROP $T$ and a symmetric monoidal category $C$, a [[symmetric monoidal functor]]

$$F : T \to C$$

is called an **algebra** or **model** of $T$ in $C$.  The category of algebras of $T$ in $C$, say $Alg(T,C)$, has

* symmetric monoidal functors $F : T \to C$ as objects,
* [[symmetric monoidal natural transformation|symmetric monoidal natural transformations]] as morphisms.

=--

Note that all of the above definitions can be [[enriched category|enriched]] over a [[symmetric monoidal category]] which yields the notion of enriched PROPs. For instance, we can have [[simplicially enriched category|simplicial]] and [[topologically enriched category|topological]] PROPs where the sets of morphisms are [[simplicial sets]] or [[topological spaces]].

##Model Structure on Simplicial PROPs

First, in the case that we are working only with simplicial PROPs with a fixed set of colors (or, in other words, PROPs whose free monoids of objects all have the same generators), we have the following theorem of [HR2](#HR2):

+-- {: .num_prop}
###### Proposition

There is a cofibrantly generated model structure on the category of simplicial PROPs with a fixed set of colors in which a morphism of simplicial PROPs $\phi:Q\to T$ is a weak equivalence (resp. fibration) if for each simplicial set of morphisms the induced map $Q(\vec{x},\vec{y})\to T(\vec{x},\vec{y})$ is also a weak equivalence (resp. fibration).
=--

Recall now that simplicial PROPs admit an obvious forgetful functor to [[categories]] that factors through [[simplicial categories]]. Denote this functor by $\pi_0\colon sPROP\to Cat$. Using this notation, we have (again from [HR2](#HR2)):

+-- {: .num_prop}
###### Proposition

There is a cofibrantly generated model structure on the category of all simplicial PROPs where a morphism of PROPs $\phi\colon Q\to T$ is a weak equivalence (resp. fibration) if:

* the induced morphism of mapping complexes $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$ is a weak equivalence (resp. [[Kan fibration]]) of simplicial sets, and

* the functor $\pi_0\colon\pi_0Q\to \pi_0T$ is a weak equivalence (resp. [[isofibration]]) of categories.

=--

Note that the model structure on simplicial PROPs is _not_ the model structure on gets by lifting the model structure of [[(infinity,1)-operad|simplicial operads]] along the free forgetful adjunction between simplicial operads and simplicial PROPs.

## Examples

### Bialgebras

A perhaps paradigmatic example is that there is a $Vect$-enriched prop whose algebras are bialgebras. It should be observed here that there is no $Vect$-enriched [[operad]] (or cooperad) whose algebras are bialgebras, so this is a genuine example that illustrates a gain in generality of props over operads.

See [Pirashvili](#Pirashvili) for some more details on this prop.

### Endomorphism PROP

Given a set of colors $\mathfrak{C}$ and a [[closed category|closed]] [[symmetric monoidal category]] $E$ with a chosen collection of objects $\mathbf{X}=\{X_c\}_{c\in\mathfrak{C}}$, there is an  $E$-enriched PROP $End_{\mathbf{X}}$ with morphism $E$-mapping spaces $End_{\mathbf{X}}(\{X_i\},\{Y_j\})=E(\otimes_i X_i,\otimes_j Y_j)$.

## Relation to properads and polycategories

[[Properads]] have a similar multiple compositional structure to PROPs, but without the tensor product of morphisms, meaning operations cannot be concatenated. [[polycategory|Polycategories]] are also similar, but only allow composition along a single object at once. See [Polycategory: Relation to properads](https://ncatlab.org/nlab/show/polycategory#relation_to_properads) for a more detailed explanation.

## Related concepts

* [[PRO]]
* [[polycategory]]
* [[properad]]
* [[dioperad]]

## References

* [[John Baez]], Brandon Coya and Franciscus Rebro, _Props in network theory_,  Theory and Applications of Categories **33**, 2018. ([tac:33-25](http://www.tac.mta.ca/tac/volumes/33/25/33-25abs.html))

* {#Gan} Wee Liang Gan, _Koszul Duality for Dioperads_, Mathematical Research Letters **10**, 2003. ([doi:10.4310/MRL.2003.v10.n1.a11](https://dx.doi.org/10.4310/MRL.2003.v10.n1.a11), [arXiv:math/0201074](https://arxiv.org/abs/math/0201074))

* {#HR1} [[Philip Hackney]] and [[Marcy Robertson]], _On the Category of PROPs_, Applied Categorical Structures **23**, 2015. ([doi:10.1007/s10485-014-9369-4](https://doi.org/10.1007/s10485-014-9369-4), [arXiv:1207.2773](https://arxiv.org/abs/1207.2773))

* {#HR2} [[Philip Hackney]] and [[Marcy Robertson]], _The Homotopy Theory of Simplicial PROPs_, Israel Journal of Mathematics **219**, 2017. ([doi:10.1007/s11856-017-1500-4](https://doi.org/10.1007/s11856-017-1500-4), [arXiv:1209.1087](https://arxiv.org/abs/1209.1087))

*  [[Steve Lack]], _Composing PROPs_, Theory and Applications of Categories **13**, 2004. ([tac:13-9](http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html))

* [[Martin Markl|M. Markl]], S. Merkulov, S. Shadrin, _Wheeled PROPs, graph complexes and the master equation_, Journal of Pure and Applied Algebra **213**, 2009. ([doi:10.1016/j.jpaa.2008.08.007](https://doi.org/10.1016/j.jpaa.2008.08.007), [arXiv:math/0610683](https://arxiv.org/abs/math/0610683))

* Teimuraz Pirashvili, _On the PROP corresponding to bialgebras_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **43**, 2002.
([publication](http://www.numdam.org/item/CTGDC_2002__43_3_221_0/), [arXiv:math/0110014](https://arxiv.org/abs/math/0110014))

* {#Yau} [[Donald Yau]], _Higher dimensional algebras via colored PROPs_, 2008. ([arXiv:0809.2161](https://arxiv.org/abs/0809.2161)).

[[!redirects PROP]]
[[!redirects PROPs]]
[[!redirects prop]]
[[!redirects props]]


Prop > history

<[[Prop]]


propagating flow

[[!redirects propagating flows]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

[[smooth mapping space|Smooth mapping spaces]] are very nice examples of [[infinite dimensional smooth manifolds]].
One reason for their nice behaviour is that it is often possible to use the structure of the [[target space]] (a finite [[dimension]]al [[manifold]]) to study the mapping space.
This provides a route from finite dimensional [[differential topology]] to that of infinite dimensions.
It is usually the case that one needs stronger structure on the target space to do this than would be needed just for the finite dimensional result, but as the target space is a finite dimensional manifold that stronger structure is usually there anyway.

An example of this is the concept of a [[tubular neighbourhood]] of an [[embedding]].  As shown at [[A Not-So-Nice Submanifold]], not all embeddings of infinite dimensional manifolds - even of mapping spaces - admit tubular neighbourhoods.  However, if the embedding is of a coincidental nature then it is possible to use structure on the target manifold to find tubular neighbourhoods in infinite dimensions.

By "coincidental nature", we mean that the submanifold is defined by taking those smooth maps with some condition imposed that says that certain "coincidences" happen.  That is, that the values of the map at certain points coincide, or are constrained to lie in some submanifold of the target space.

Let us consider, for an example, the simplest case: the inclusion of based loops in [[smooth loop space|smooth loops]].  So let $M$ be a smooth finite dimensional manifold, $L M$ its smooth [[free loop space]] and $\Omega M$ its smooth [[based loop space]].  Let $x_0$ be the basepoint of $M$, and $1$ the basepoint of the [[circle]], $S^1$.  To get a tubular neighbourhood of this, we need to start by finding a suitable neighbourhood.  To do this, we choose a [[chart]] $\phi \colon \mathbb{R}^n \to U \subseteq M$ at $x_0$ (so that $\phi(0) = x_0$).  We define the neighbourhood of $\Omega M$ to be those loops $\alpha \in L M$ such that $\alpha(1) \in U$.  To make this a _tubular_ neighbourhood, we need to find a way to project these loops on to $\Omega M$.  That is, given a loop $\alpha \colon S^1 \to M$ with $\alpha(1) \in U$ we need to produce a loop $\pi(\alpha) \colon S^1 \to M$ with $\pi(\alpha)(1) = x_0$.  It is obvious how to move the basepoint: simply scale it to $x_0$ using the scalar multiplication coming from the chart.  However, that just moves $\alpha(1)$.  We need to move the whole of $\alpha$ with it - or at least, drag that part of it near $1$.  We cannot assume that all of $\alpha$ lies in $U$.  The solution is to define a diffeomorphism $\Psi$ of $M$ with the property that $\Psi(\alpha(1)) = x_0$.  Then we can define $\pi(\alpha) = \Psi \circ \alpha$.  As we vary $\alpha$, so we also vary $\alpha(1)$, and thus must vary the diffeomorphism $\Psi$.  The trick is to choose $\Psi$ so that it varies smoothly with $\alpha(1)$.

In the above example, this is straightforward because everything happens in the chart, and thus effectively on $\mathbb{R}^n$.  There are more complicated examples.  One such is the basic construction in [[string topology]].  Within $L M \times L M$ we consider those pairs of loops $(\alpha,\beta)$ such that $\alpha(1) = \beta(1)$.  Then for $(\alpha,\beta)$ with $\alpha(1)$ _near_ to $\beta(1)$ we want to project the pair $(\alpha,\beta)$ to a pair where these values coincide.  Now although the points $\alpha(1)$ and $\beta(1)$ are near (in some vague not-yet-defined sense), they may, as a pair, roam all over the manifold.  Thus local solutions are not applicable here.  Nonetheless, the question still comes down to the ability to choose diffeomorphisms consistently according to some conditions.

## Definition ##

One way to choose these diffeomorphisms is to use the notion of a **propagating flow**.  The term was coined by [[Veronique Godin]] in ([Godin, 07](#Godin)) and is based on an idea due to [[Andrew Stacey]] in ([Stacey, 05](#Stacey)), though there are probably antecedents.
The basic idea is contained in the following definition.

+-- {: .num_definition #propflow}
###### Definition
Let $\pi \colon E \to M$ be a [[vector bundle]] over a [[smooth manifold]].  Everything here is assumed to be finite dimensional.  We consider the space $Diff_{fc}(E)$ of [[diffeomorphisms]] of $E$ which preserve the [[fibres]] of $E$ and have [[compact support]].  A **propagating flow** is a [[smooth map]]

$$
  \phi \colon E \to Diff_{fc}(E)
$$

with the property that

$$
  \phi(v) : v \mapsto \pi(v)
$$

where we identify $\pi(v)$ with the zero vector in the corresponding fibre of $E$.

=--

## Construction ##

The original definition of a propagating flow in [Stacey, 05](#Stacey) and [Godin, 07](#Godin) used the exponentiation map from vector fields to diffeomorphisms to get the actual diffeomorphism.  The idea of that was that it is much easier to build a vector field with the required properties than a diffeomorphism because vector fields are more easily manipulated.  But the diffeomorphisms used here are simple enough that they can be constructed directly.  This makes it easier to generalise to situations where the exponentiation map cannot be assumed to exist.

Let us consider the linear situation first.  We start with a vector space, $V$, and a vector $v \in V$.  We want to define a diffeomorphism $\phi_v \colon V \to V$ with the property that $\phi_v(v) = 0$.  This is simple enough:
$$
\phi_1(w) = w - v.
$$
The problem with this is that we do not want just any diffeomorphism.  We want one that is the identity "near infinity".  Let us start by fixing the diffeomorphism in the $v$-direction.

Choose a smooth function $\sigma \colon \mathbb{R} \to [-1,0]$ with the following properties:

1. $\sigma(t) = \begin{cases} 0 & t \le -2 \\ - 1 & 0 \le t \le 1 \\ 0 & 2 \le t \end{cases}$
2. $\sigma'(t) \gt -1$ (note that this is possible as we have given ourselves an interval of length $2$ to get from $0$ at $t = -2$ to $-1$ at $t = 0$.

We are actually interested in the function $t \mapsto t + h \sigma(t)$ for $|h| \le 1$.  The first property of $\sigma$ tells us that this agrees with the identity outside $[-2,2]$ and that it is $t \mapsto t - h$ on $[0,1]$.  The second property tells us that its derivative is strictly bigger than $1 - h$ and so, for $h \le 1$, is a diffeomorphism.

On $V$, we choose a "dual functional" to $v$.  That is, we choose some continuous linear functional $f \colon V \to \mathbb{R}$ with $f(v) = 1$ (we need to assume that $v \ne 0$ for this part, we shall correct for that later).  Then we define a diffeomorphism on $V$ by:
$$
\phi_2(w) = w + \sigma(f(w))v =  (w - f(w)v) + (f(w) + \sigma(f(w)))v.
$$
The second expression shows that what we have done is used $f$ to identify $V$ with $\ker f \oplus \mathbb{R}$ and then applied $t \mapsto t + \sigma(t)$ on the $\mathbb{R}$-factor.

This fixes our diffeomorphism in the $v$-direction.  To fix it in the other direction, we choose a smooth function $\widehat{\tau} \colon \ker f \to [0,1]$ which is $0$ "near infinity" and $1$ at $0$.  Then we mix this in to the above as follows:
$$
\phi_3(w) = w + \widehat{\tau}(w - f(w)v) \sigma(f(w))v = (w - f(w)v) + (f(w) + \widehat{\tau}(w - f(w)v) \sigma(f(w)) v.
$$
As before, the second expression makes it clear that this is a diffeomorphism.  When $\widehat{\tau}(w - f(w)v) = 0$ then it is the identity.

This will do, but there are a few too many choices in the above.  To simplify these, we assume that $V$ admits a smooth inner product, $g$.  Let us write $q$ for the square of the associated norm.  Then we can choose $f$ to be evaluation of the inner product at $v$ and $\tau$ to be composition of the inner product with a suitable bump function on $\mathbb{R}$.  We shall write $\tau$ for that bump function.  To make the final formula cleaner, we assume that $\tau(t) = 1$ for $|t| \le 2$.  This leads us to:
$$
\begin{aligned}
\phi_v(w) &= w + \tau\left(\frac{q(w)}{1 + q(v)}\right) \sigma\left(\frac{g(w,v)}{q(v)}\right) v
&= w - \frac{g(w,v)}{q(v)}v + \left(\frac{g(w,v)}{q(v)} + \tau\left(\frac{q(w)}{q(v)}\right) \sigma\left(\frac{g(w,v)}{q(v)}\right) \right)v.
\end{aligned}
$$
As written, this makes sense only for $v \ne 0$.  But it extends to the identity at $v = 0$.  To see that this extension is smooth, we need merely point out that as $v \to 0$, so $q(v) \to 0$ and thus $\sigma\left(\frac{g(w,v)}{q(v)}\right) \to 0$.  The second expression again shows that, for $v \ne 0$, this is a diffeomorphism.

This, then, is our required linear diffeomorphism.  For $w$ with $q(w) \ge 2(1 + q(v))$ it is the identity, and thus will extend "at infinity", whilst $\phi_v(v) = 0$.

The next step is to extend this to a bundle over a manifold.  So let $\pi \colon E \to M$ be a smooth vector bundle over a smooth manifold.  We wish to extend the above formula so that it is valid for $E$.  That is, $v$ is an arbitrary point in $E$ and we wish to define $\phi_v \colon E \to E$ such that $\phi_v(v) = 0_{\pi(v)}$.  So also we must take $w$ to be an arbitrary point in $E$.  And thereby lies the problem: in the formula $v$ and $w$ interact but they may be in different fibres.  The solution is to extend one of them to a vector field.  Since $v$ is static in the formula for $\phi_v$, that is the obvious choice.

We should also note that the explicit formula requires the existence of a smooth [[orthogonal structure]] on $E$.

Thus we want to define a smooth function $X \colon E \to \Gamma(E)$ with the property that $X_v(\pi(v)) = v$.  This is easy if $E$ is trivial, and the condition is convex, so a standard [[partition of unity]] argument will suffice.  Specifically, let $\{\rho_\lambda : \lambda \in \Lambda\}$ be a partition of unity on $M$ with the property that for each $\lambda \in \lambda$ there is an open set $U_\lambda$ containing the support of $\rho_\lambda$ over which $E$ is trivial.  Let $E_\lambda$ denote the restriction of $E$ to $U_\lambda$ and let $\psi_\lambda \colon E_\lambda \to U_\lambda \times V$ be a trivialisation.  Let $p_\lambda \colon E_\lambda \to V$ be the composition of this trivialisation with the projection on to the $V$-factor.  We define $X_\lambda \colon E_\lambda \to \Gamma(E_\lambda)$ by
$$
X_{\lambda,v}(p) = \psi_\lambda^{-1}(p,p_\lambda(v)).
$$

Now we define $X \colon E \to \Gamma(E)$ by
$$
X_v(p) = \sum_\lambda \rho_\lambda(p) X_{\lambda,v}(p).
$$
Notice that $X_v(p) = 0$ if $p$ is "sufficiently far" from $\pi(v)$.

Thus our diffeomorphism $\phi_v$ at $w$, with $p = \pi(w)$, is:
$$
\phi_v(w) = \begin{cases} w + \tau\left(\frac{q(w)}{1 + q(X_v(p))}\right) \sigma \left(\frac{g(w,X_v(p))}{q(X_v(p))}\right) X_v(p) & X_v(p) \ne 0 \\
w & X_v(p) = 0 \end{cases}.
$$

## References

The idea goes back to

* [[Andrew Stacey]], _The differential topology of smooth loops_ ([arXiv:math/0510097](http://arxiv.org/abs/math/0510097))
 {#Stacey}

The term _propagating flow_ was coined in

* [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))
 {#Godin}

and used for the construction of [[umkehr maps]] to construct [[string topology]] operations.


propagation of singularities theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[microlocal analysis]] the _propagation of singularities theorem_ ([Duistermaat-Hörmander 72, section 6.1](#DuistermaatHoermander72)) characterizes the [[wave front set]] of a [[distributional solution to a partial differential equation]] in terms of the [[principal symbol]] of the corresponding [[differential operator]]: It says that the singularity wave fronts of the solution propagate along the [[Hamiltonian flow]] of the [[principal symbol]] of the differential operator (its [[bicharacteristic flow]]).

Hence this theorem makes fully manifest how the concept of [[wave front set]] enhances that of [[singular support]] by adding [[direction of a vector|directional]] information about the singularities.

One implication is the proof of existence of solutions to suitable (possibly inhomogeneous) differential equations ([Duistermaat-Hörmander 72, section 6.3](#DuistermaatHoermander72)).

For example the proof of existence the [[Feynman propagator]] on a [[Lorentzian manifold]] is obtained this way ([Duistermaat-Hörmander 72, theorem 6.5.3 and beginning of section 6.6](#DuistermaatHoermander72), [Radzikowski 96, section 4](#Radzikowski96)).

## Definitions
 {#Definitions}

+-- {: .num_defn #ProperlySupportedPseudoDifferentialOperator}
###### Definition
**([[properly supported peudo-differential operator]])

A [[pseudo-differential operator]] $Q$ on a [[manifold]] $X$ is called _[[properly supported pseudo-differential operator|properly supported]]_ if for each [[compact subset]] $K \subset X$ there exists a compact subset $K' \subset X$ such that for $u$ a [[distribution]] with [[support of a distribution|support]] in $K$ it follows that the [[derivative of distributions]] $Q u$ has support in $K'$ and such that $u\vert_{K'} = 0$ implies $(Q u)\vert_{K} = 0$.

=--

([Hörmander 85 (18.1.21)](#Hoermander85) recalled e.g. in [Radzikowski 96. p. 8,9](#Radzikowski96))

+-- {: .num_example}
###### Example
**([[differential operators]] are properly supported [[pseudo-differential operators]])**

Every ordinary [[differential operator]], regarded as a [[pseudo-differential operator]], is properly supported (def. \ref{ProperlySupportedPseudoDifferentialOperator}), since differential operators do not increase [[support]].

=--

+-- {: .num_defn #SymbolOrder}
###### Definition
**([[symbol order]])**

A [[smooth function]] $q$ on a [[cotangent bundle]] (e.g. [[symbol of a differential operator]]) is of _order $m$_ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each [[coordinate chart]] $((x^i), (k_i))$ we have that for every [[compact subset]] $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a  [[real number]] $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the [[absolute value]] of the [[partial derivatives]] of $q$ is bounded by

$$
  \left\vert
    \frac{\partial^\alpha}{\partial k_\alpha}
    \frac{\partial^\beta}{\partial x^\beta}
    q(x,k)
  \right\vert
  \;\leq\;
  C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}
$$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

A [[Fourier integral]] operator $Q$ is of _[[symbol class]]_ $L^m = L^m_{1,0}$ if it is of the form

   $$
     Q f (x)
     \;=\;
     \int \int e^{i k \cdot (x - y)} q(x,y,k) f(y) \, d y d k
   $$

with symbol $q$ of order $m$, in the above sense.

=--

([Hörmander 71, def. 1.1.1 and first sentence of section 2.1 with (1.4.1)](#Hoermander71))

+-- {: .num_example}
###### Example

The [[wave operator]]/[[Klein-Gordon operator]] on [[Minkowski spacetime]] is of class $L^2$, according to def. \ref{SymbolOrder}.

=--

## Statement

+-- {: .num_prop #PropagationOfSingularitiesTheorem}
###### Proposition
**(propagation of singularities theorem)**

Let $Q$ be a [[pseudo-differential operator]] on some [[smooth manifold]] $X$ which is [[properly supported pseudo-differential operator|properly supported]] (def. \ref{ProperlySupportedPseudoDifferentialOperator}) and of [[symbol class]] $L^m$ (def. \ref{SymbolOrder}) with real [[principal symbol]] $q$ that is [[homogeneous function|homogeneous]] of degree $m$.

For $u \in \mathcal{D}'(X)$ a [[distribution]] with $Q u = f$, then the [[complement]] of the [[wave front set]] of $u$ by that of $f$ is contained in the set of covectors on which the [[principal symbol]] $q$ vanishes:

$$
  WF(u) \setminus WF(f) \;\subset\; q^{-1}(0)
  \,.
$$

Moreover, $WF(u) \setminus WF(f)$ is invariant under the [[bicharacteristic flow]] induced by the [[Hamiltonian vector field]] of $q$ with respect to the canonical [[symplectic manifold]] structure on the [[cotangent bundle]] ([here](cotangent+bundle#SymplecticStructure)).

=--

([Duistermaat-Hörmander 72, theorem 6.1.1](#DuistermaatHoermander72), recalled e.g. as [Radzikowski 96, theorem 4.6](#Radzikowski96)

## Examples

+-- {: .num_example #PropagationOfSingularitiesForKleinGordonOperator}
###### Example

For $(X,e)$ a [[globally hyperbolic spacetime]] and $P$ a [[hyperbolic differential operator]] such as the [[wave operator]]/[[Klein-Gordon operator]], then the wave front set of any solution $f$ to $P f = 0$ is a union of [[cotangent vectors]] along [[lightlike]] [[geodesics]] .

=--

+-- {: .proof}
###### Proof

This follows by prop. \ref{PropagationOfSingularitiesTheorem} by the fact that the [[bicharacteristic strips]] of the [[wave operator]]/[[Klein-Gordon operator]] are [[cotangent vectors]] of [[lightlike]] [[geodesics]], by [this example](bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator).

=--

## References

The theorem is due to

* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))

discussed in

* {#Hoermander85} [[Lars Hörmander]], _The analysis of partial differential operators III_, Springer 1985

Review in the context of the [[free field|free]] [[scalar field]] on [[globally hyperbolic spacetimes]] (with $Q$ the [[wave operator]]/[[Klein-Gordon operator]]) is in

* {#Radzikowski96} [[Marek Radzikowski]], _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))

which otherwise discusses the corresponding [[Hadamard distributions]].

See also

* {#Hoermander71} [[Lars Hörmander]], _Fourier  integral  operators I._ Acta  Mathematica  127, 79-183 (1971) ([Euclid](https://projecteuclid.org/euclid.acta/1485889699))

[[!redirects propagation of singularities]]


propagators - table

**[[propagators]] (i.e. [[integral kernels]] of [[Green functions]])**

**for the [[wave operator]] and [[Klein-Gordon operator]]**

**on a [[globally hyperbolic spacetime]] such as [[Minkowski spacetime]]:**

| name | symbol  | [[wave front set]] | as  [[vacuum expectation value|vacuum exp. value]]
 of [[operator-valued distribution|field operators]] | as a [[nonassociative algebra|product]] of
 [[operator-valued distribution|field operators]] |
|---|----|---|---|---|
| [[causal propagator]] | $\begin{aligned}\Delta_S & =  \Delta_+ - \Delta_- \end{aligned}$ |
 $\phantom{A}\,\,\,-$
  |$\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} $ | [[Peierls bracket|Peierls-Poisson bracket]] |
| [[advanced propagator]] | $\Delta_+$  |   | $\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned} $ | [[future]] part of
 [[Peierls-Poisson bracket]] |
| [[retarded propagator]] | $\Delta_-$ |  | $\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}$ | [[past]] part of
 [[Peierls-Poisson bracket]] |
| [[Wightman propagator]] | $\begin{aligned} \Delta_H &=  \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}$ |   | $\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \;  \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & =  \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned} $ | [[positive real number|positive]] [[frequency]] of
 [[Peierls-Poisson bracket]],
 [[Wick algebra]]-product,
 [[2-point function]]
 $\phantom{=}$ of [[vacuum state]]
 $\phantom{=}$ or generally of
 $\phantom{=}$ [[Hadamard state]] |
| [[Feynman propagator]] | $\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}$ |   | $\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x  } \right.\end{aligned}$  | [[time-ordered product]] |

(see also [[Mikica Kocic|Kocic]]'s overview: [[KGPropagatorsOnMinkowskiTable.pdf:file]])


proper action

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[continuous function|continuous]] [[group action]] of a [[topological group]] $G$ is called "proper" if it behaves like the action of a [[compact topological group]], without $G$ necessarily being compact.

In particular, the [[isotropy groups]] of a proper action are [[compact topological group|compact]].

## Definition

Consider

* a [[topological group]] $G$,

* a [[locally compact Hausdorff space]] $X$,

* a [[group action]] of $G$ on $X$ (making $X$ a [[topological G-space]]):

  $$
    \array{
      G \times X
      &\overset{\rho}{\longrightarrow}&
      X
      \\
      (g,x) &\mapsto& g \cdot x
      \mathrlap{\,.}
    }
  $$

The action $\rho \colon (g,x) \mapsto g \cdot x$ is called *proper* ([Palais 61, Def. 1.2.2](#Palais61)) if one of the following equivalent conditions hold (their equivalence relies on $X$ being [[locally compact Hausdorff space|locally compact and Hausdorff]], [Palais 61, Thm. 1.2.9](#Palais61), [Karppinen 16, Rem. 5.2.4](#Karppinen16)):

1. **(Bourbaki properness)** The shear map is a [[proper continuous function]]:

   $$
     \array{
       G \times X &\overset{proper}{\longrightarrow}& X \times X
       \\
       (g,x) &\mapsto& (g\cdot x, x)
     }
   $$

   ([Bourbaki 60, Ch. III, Sec. 4.4](#Bourbaki60), review in [Lee 00, p. 266](#Lee00))

1. **(Borel properness)** For every [[compact subspace]] $K \subset X$
 the subset

   $$
     (K \vert K)
     \;\coloneqq\;
     \big\{
       g \in G
       \,\vert\,
       g \cdot K \,\cap\, K \neq \varnothing
     \big\}
     \;\subset\;
     G
   $$

   is [[compact subspace|compact]].

   ([Palais 61, Thm 1.2.9 (5)](#Palais61), attributed there to [[Armand Borel|Borel]])

1. {#PalaisProperness} **(Palais properness)** Every point $x \in X$ has a [[neighbourhood]] $U_x$ such that every point $y \in U_x$ has a neighbourhood $V_y$ such that

   $$
     (U_x \vert V_y)
     \;\coloneqq\;
     \big\{
       g \in G
       \,\vert\,
       g \cdot U_x \,\cap\, V_y \neq \varnothing
     \big\}
     \;\subset\;
     G
   $$

   has [[compact subspace|compact]] [[topological closure|closure]].

   ([Palais 61, Def. 1.2.2](#Palais61))

## Examples

### Lie group actions on smooth manifolds

\begin{prop}\label{SmoothActionOfCompactLieGroupOnSmoothManifoldIsProper}

Let $X$ be a [[smooth manifold]] and let $G$ be a [[compact Lie group]].
Then every [[continuous function|continuous]] [[action]] of $G$ on $X$ is [[proper action|proper]].

\end{prop}

(e.g. [Lee 12, Cor. 21.6](#Lee12))

For more see at _[[equivariant differential topology]]_.

## Related concepts

* [[properly discontinuous action]]

* [[free action]], [[semi-free action]], [[transitive action]], [[regular action]]

* [[equivariant differential topology]]

## References

The original definition are due to:

* {#Palais61} [[Richard Palais]], _On the Existence of Slices for Actions of Non-Compact Lie Groups_, Annals of Mathematics
Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 ([jstor:1970335](https://www.jstor.org/stable/1970335), [doi:10.2307/1970335](https://doi.org/10.2307/1970335), [pdf](http://vmm.math.uci.edu/ExistenceOfSlices.pdf))

  > (in the context of proving the [[slice theorem]])

* {#Bourbaki60} [[Nicolas Bourbaki]], *Éléments de mathématique*, *Topologie générale*, Chap. 3:  *Groupes topologiques*, 1960

Textbook accounts:

* {#Lee00} [[John M. Lee]], Section 12.3 of: _Introduction to topological manifolds_. Graduate Texts in Mathematics 202 (2000), Springer.  ISBN: 0-387-98759-2, 0-387-95026-5.
Second edition: Springer, 2011.  ISBN: 978-1-4419-7939-1 ([doi:10.1007/978-1-4419-7940-7](https://doi.org/10.1007/978-1-4419-7940-7), errata [pdf](https://sites.math.washington.edu/~lee/Books/ITM/errata.pdf))

* {#Lee12} [[John Lee]], around p. 147 of: _Introduction to Smooth Manifolds_, Springer 2012 ([doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [pdf](https://lost-contact.mit.edu/afs/adrake.org/usr/rkh/Books/books/Introduction%20to%20Smooth%20Manifolds%20-%20J.%20Lee.pdf))

* {#Kankaanrinta07} [[Marja Kankaanrinta]], Def. 2.1 in _Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions_,     Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 ([euclid:agt/1513796653](https://projecteuclid.org/euclid.agt/1513796653))

See also

* [[John Lee|J. Lee]], [MO comment](https://math.stackexchange.com/a/989168/58526), Oct 2014

Further discussion

* {#Karppinen16} [[Sini Karppinen]], _The existence of slices in $G$-spaces, when $G$ is a Lie group_, Helsinki 2016 ([hdl:10138/190707](http://hdl.handle.net/10138/190707))

  > (in the context of the [[slice theorem]])

* {#Antonyan02} [[Sergey Antonyan]], *Universal proper G-spaces*, Topology and its Applications Volume 117, Issue 1, 15 January 2002, Pages 23-43 (doi:10.1016/S0166-8641(00)00115-2)

[[!redirects proper actions]]

[[!redirects proper group action]]
[[!redirects proper group actions]]


proper base change theorem

#Contents#
* table of contents
{:toc}

## Idea

A _proper base change theorem_ asserts a [[Beck-Chevalley condition]] for [[base change]] in [[cohomology]] along a [[proper map]].

## In étale cohomology

Let $p \colon X \longrightarrow S$ be a [[proper morphism of schemes]]. For $f \colon T \longrightarrow S$ any other morphism into $S$, consider the [[fiber product]]

$$
  \array{
    X \times_S T &\longrightarrow& X
    \\
    {}^{\mathllap{f^\ast p}} \downarrow & & \downarrow^{\mathrlap{p}}
    \\
    T &\stackrel{f}{\longrightarrow}& S
    \,.
  }
$$

The étale proper base change theorem says that in this situation and for $\mathcal{F}$ an [[abelian sheaf]] of [[torsion groups]] on $X$, the [[derived functor|derived]] pull-push along the top and left is isomorphic to the derived push-pull along the bottom right. (...)

([Milne, theorem 17.10](#Milne))

## References

* Wikipedia, _[Proper base change theorem](http://en.wikipedia.org/wiki/Gysin_sequence)_

* [[James Milne]], section 17 of _[[Lectures on Étale Cohomology]]_
 {#Milne}


proper equivariant homotopy theory

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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Proper $G$-equivariant homotopy theory_ ([DHLPS 19](#DHLPS19)) is the variant of $G$-[[equivariant homotopy theory]] where the [[topological group]] $G$ is allowed to be non-[[compact topological group|compact]] (for instance a non-[[finite group|finite]] [[discrete group]]) but whose [[weak equivalences]] are still detected only on [[fixed point spaces]] of [[compact topological group|compact]] [[subgroups]] of $G$.

Hence in the presentation by [[topological G-spaces]], proper $G$-equivariant homotopy theory is obtained by [[simplicial localization|localizing]] at those [[equivariant function|equivariant]] [[continuous functions]] $f \;\colon\; X \to Y$ which induce [[weak homotopy equivalences]] $f^H \;\colon\; X^H \to Y^H$ for all compact subgroups $H \subset G$ ([DHLPS 19, Def. 1.1.2](#DHLPS19)).

Equivalently, under [[Elmendorf's theorem]] ([DHLPS 19, p. 87](#DHLPS19)) this is the [[homotopy theory]] of [[(∞,1)-presheaves]] on the [[full sub-(∞,1)-category]] of the $G$-[[orbit category]] on those orbits corresponding to compact subgroups (e.g. finite subgroups for $G$ discrete, details in [DHLPS 19, Remark 3.1.12](#DHLPS19)).

## Related concepts

* [[equivariant stable homotopy theory]]

* [[global equivariant homotopy theory]]

## References

The concept was introduced in

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_, Memoirs of the AMS [[arXiv:1908.00779](https://arxiv.org/abs/1908.00779)]

following

* [[James Davis]], [[Wolfgang Lück]], _Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory_, K-Theory 15:201–252, 1998 ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf))

[[!redirects proper equivariant homotopy theories]]

[[!redirects proper equivariant stable homotopy theory]]
[[!redirects proper equivariant stable homotopy theories]]