11institutetext:  Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, France
22institutetext: Aix-Marseille Université, CNRS, I2M, France

# The algebraic $λ$-calculus is a conservative extension of the ordinary
$λ$-calculus

Axel Kerinec 11 Lionel Vaux Auclair  This work was partially supported by the
French ANR project PPS (ANR-19-CE48-0014). 22

###### Abstract

The algebraic $λ$-calculus is an extension of the ordinary $λ$-calculus with
linear combinations of terms. We establish that two ordinary $λ$-terms are
equivalent in the algebraic $λ$-calculus iff they are $β$-equal. Although this
result was originally stated in the early 2000’s (in the setting of Ehrhard
and Regnier’s differential $λ$-calculus), the previously proposed proofs were
wrong: we explain why previous approaches failed and develop a new proof
technique to establish conservativity.

##  0.1 Introduction

The algebraic $λ$-calculus was introduced by the second author [Vau07, Vau09]
as a generic framework to study the rewriting theory of the $λ$-calculus in
presence of weighted superpositions of terms. The latter feature is pervasive
in the quantitative semantics of $λ$-calculus and linear logic, that have
flourished in the past twenty years [Ehr05, Lai+13, DE11, Cas+18, _etc._] and
the algebraic $λ$-calculus is meant as a unifying syntactic counterpart of
that body of works.

The algebraic $λ$-calculus was actually obtained by removing the
differentiation primitives from Ehrhard and Regnier’s differential
$λ$-calculus [ER03], keeping only the dynamics associated with linear
combinations of terms. This dynamics is surprisingly subtle in itself: for
instance, if $1$ has an opposite in the semiring $\mathsf{R}$ of coefficients,
then the rewriting theory becomes trivial. We refer the reader to the original
paper [Vau09] for a thorough discussion, and focus on the question of
conservativity only, assuming $\mathsf{R}$ is positive — _i.e._ if $a+b=0$
then $a=b=0$. We briefly outline the main definitions, keeping the same
notations as in the former paper, so that the reader can consistently refer to
it for a more detailed account if need be.

### Overview of the algebraic $λ$-calculus.

The syntax of algebraic $λ$-terms is constructed in two stages. We first
consider raw terms, which are terms inductively generated as follows:

$\mathsf{L}_{\mathsf{R}}\ni M,N,\dotsc\Coloneqq
x\mid\lambda{x}.M\mid(M)\,N\mid\mathbf{0}\mid M+N\mid a.M$

where $a$ ranges over the semiring $\mathsf{R}$ (beware that we use Krivine’s
convention for application). We consider raw terms up to $α$-equivalence:
$\mathsf{L}_{\mathsf{R}}$ contains the set $\Lambda$ of pure $λ$-terms as a
strict subset. We then consider algebraic equality $\triangleq$ on raw terms,
which is the congruence generated by the equations of $\mathsf{R}$-module,
plus the following linearity axioms:

$\displaystyle\lambda{x}.\mathbf{0}$ $\displaystyle\triangleq\mathbf{0}$
$\displaystyle\lambda{x}.(M+N)$
$\displaystyle\triangleq\lambda{x}.M+\lambda{x}.N$
$\displaystyle\lambda{x}.(a.M)$ $\displaystyle\triangleq a.\lambda{x}.M$
$\displaystyle(\mathbf{0})\,P$ $\displaystyle\triangleq\mathbf{0}$
$\displaystyle(M+N)\,P$ $\displaystyle\triangleq(M)\,P+(N)\,P$
$\displaystyle(a.M)\,P$ $\displaystyle\triangleq a.(M)\,P$

which reflects the point-wise definition of the sum of functions. Note that,
without these equations, a term such as $(\lambda{x}.M+\lambda{x}.N)\,P$ has
no redex.

The terms of the algebraic $λ$-calculus, called algebraic terms below, are
then the $\triangleq$-classes $σ=\underline{M}$ of raw terms
$M\in\mathsf{L}_{\mathsf{R}}$. We extend syntactic constructs to algebraic
terms (e.g., $\lambda{x}.\underline{M}=\underline{\lambda{x}.M}$, which is
well defined because $\triangleq$ is a congruence). Among algebraic terms, we
distinguish the simple terms, which are intuitively those without sums at top
level: a term $σ$ is simple if $σ=\underline{x}$ for some variable $x$, or
$σ=\lambda{x}.τ$ or $σ=(τ)\,ρ$ where $τ$ is itself simple (inductively). In
particular, $\underline{M}$ is simple as soon as $M\in\Lambda$. By definition,
algebraic terms form an $\mathsf{R}$-module, and it is easy to check that it
is freely generated by the set $\Delta_{\mathsf{R}}$ of simple terms: we write
$\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$ for the module of algebraic
terms.

A seemingly natural way to extend the $β$-reduction $\to_{\Lambda}$ of
$λ$-terms to algebraic terms is to define it contextually on raw terms, and
then apply it _modulo_ $\triangleq$: among other issues with this naïve
definition, note that $M\triangleq M+0.N$ would reduce to
$M+0.N^{\prime}\triangleq M$ for any $N\to_{\Lambda}N^{\prime}$, so that the
obtained reduction would be reflexive and there would be no clear notion of
normal form. Ehrhard and Regnier’s solution is to rather consider two
relations:
$\mathord{\to}\subset\Delta_{\mathsf{R}}\times\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$
defined contextually on simple terms with $β$-reduction as a base case; and
$\mathord{\mathrel{\widetilde{\to}}}\subset\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle\times\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$
on algebraic terms, obtained by setting $σ\mathrel{\widetilde{\to}}σ^{\prime}$
iff $σ=a.τ+ρ$ and $σ^{\prime}=a.τ^{\prime}+ρ$ with $τ\to τ^{\prime}$ and
$a\not=0$. Then $\mathrel{\widetilde{\to}}$ is confluent [ER03, Vau09] and,
_provided $\mathsf{R}$ is positive_, an algebraic term is in normal form iff
it is the class of a raw term without $β$-redex.

Note that, for any fixed point combinator $Y$, by setting
$\infty_{σ}=(\underline{Y})\,\lambda{x}.(σ+\underline{x})$, we obtain
$\infty_{σ}\leftrightarrow σ+\infty_{σ}$ where $\leftrightarrow$ is the
equivalence on $\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$ generated by the
reduction relation $\mathrel{\widetilde{\to}}$. In case $1\in\mathsf{R}$ has
an opposite $-1$, we can now exhibit the above-mentioned inconsistency of the
theory: $\underline{\mathbf{0}}=\infty_{σ}+(-1).\infty_{σ}\leftrightarrow σ$
for any $σ$. From now on, we thus assume that $\mathsf{R}$ is positive.

### Contributions.

Our goal is to establish that, for any two $λ$-terms $M$ and $N\in\Lambda$, we
have $\underline{M}\leftrightarrow\underline{N}$ iff
$M\leftrightarrow_{\Lambda}N$, where $\leftrightarrow_{\Lambda}$ is the usual
$β$-equivalence on $λ$-terms. For that purpose, it is sufficient to establish
a conservativity result on reduction relations rather than on the induced
equivalences: if $\underline{M}\mathrel{\widetilde{\to}}^{*}\underline{N}$
then $M\to_{\Lambda}^{*}N$. This is our main result, theorem 0.4.3 below.

In the next section, we explain what was wrong with the previous two attempts,
first by Ehrhard and Regnier, then by the second author, to establish
conservativity, and we outline the new proof strategy we propose. The rest of
the paper is dedicated to the proof of theorem 0.4.3.111 These results were
obtained during a research internship of the first author, in the first half
of 2019; they were presented by the second author at the annual meeting of the
working group Scalp (_Structures formelles pour le Calcul et les Preuves_) in
Lyon in October 2019. This collaboration was unfortunately disrupted by the
COVID-19 pandemic in the following year, which delayed dissemination to a
wider audience.

##  0.2 Two non-proofs and a new approach

Recall that an ARS (abstract rewriting system) is a pair $(A,\leadsto)$ of a
set $A$ and binary relation $\mathord{\leadsto}\subseteq A\times A$. An
extension of $(A,\leadsto)$ is another ARS $(A^{\prime},\leadsto^{\prime})$
such that $A\subseteq A^{\prime}$ and
$\mathord{\leadsto}\subseteq\mathord{\leadsto^{\prime}}$. This extension is
conservative if, for every $a_{1},a_{2}\in A$, $a_{1}\leadsto a_{2}$ iff
$a_{1}\leadsto^{\prime}a_{2}$. An equational system is an ARS $(A,\sim)$ such
that $\sim$ is an equivalence relation. Our goal is thus to establish that the
equational system
$(\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle,\leftrightarrow)$ is a
conservative extension of $(\Lambda,\leftrightarrow_{\Lambda})$ — here we
consider the injection
$M\in\Lambda\mapsto\underline{M}\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$
as an inclusion.

In their paper on the differential $λ$-calculus [ER03], Ehrhard and Regnier
claim that this follows directly from the confluence of
$\mathrel{\widetilde{\to}}$, but this argument is not valid:
$\mathrel{\widetilde{\to}}$ does contain $\to_{\Lambda}$, and it is indeed
confluent, without any positivity assumption; but we have already stated that
$\leftrightarrow$ is inconsistent in presence of negative coefficients, so
this observation cannot be sufficient.

Ehrhard and Regnier’s mistake is certainly an erroneous application of a
general conservativity result in Terese’s textbook [Ter03], missing the fact
that Terese’s notion of extension is more demanding: for Terese,
$(A,\leadsto)$ is a sub-ARS of $(A^{\prime},\leadsto^{\prime})$ if $A\subseteq
A^{\prime}$ and, for every $a\in A$ and $a^{\prime}\in A^{\prime}$,
$a\leadsto^{\prime}a^{\prime}$ iff $a^{\prime}\in A$ and $a\leadsto
a^{\prime}$. The latter is strictly stronger than the mere inclusion
$\mathord{\leadsto}\subseteq\mathord{\leadsto^{\prime}}$, and is indeed
sufficient to deduce conservativity for the induced equational systems from
the confluence of the super-ARS [Ter03, Exercice 1.3.21 (iii)]. But
$(\Lambda,\to_{\Lambda})$ is not a sub-ARS of
$(\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle,\mathrel{\widetilde{\to}})$,
even when $\mathsf{R}$ is positive: for instance, if
$\mathsf{R}=\mathbf{Q}^{+}$ and $M\to_{\Lambda}M^{\prime}\not=M$, we have
$\underline{M}=\frac{1}{2}\underline{M}+\frac{1}{2}\underline{M}\mathrel{\widetilde{\to}}\frac{1}{2}\underline{M}+\frac{1}{2}\underline{M^{\prime}}\not\in\underline{\Lambda}$.
So one must design another approach.

Given $σ\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$, one can consider the
finite set of $λ$-terms $\Lambda(σ)\subset\Lambda$ obtained by keeping exactly
one element in the support of each sum occurring in $σ$ [Vau09, Definition
3.18]. The second author tried to establish the conservativity of
$\mathrel{\widetilde{\to}}^{*}$ over $\to_{\Lambda}^{*}$ by iterating the
following:

######  Claim 0.2.1 ([Vau09, Lemma 3.20]).

If $σ\mathrel{\widetilde{\to}}σ^{\prime}$ and
$M^{\prime}\in\Lambda(σ^{\prime})$ then there exists $M\in\Lambda(σ)$ such
that $M\to_{\Lambda}^{*}M^{\prime}$.

But the latter claim is wrong! Consider, for instance,
$σ=(\lambda{x}.(\underline{x})\,\underline{x})\,(\underline{y}+\underline{z})\mathrel{\widetilde{\to}}σ^{\prime}=(\underline{y}+\underline{z})\,(\underline{y}+\underline{z})$.
We have $M^{\prime}=({y})\,{z}\in\Lambda(σ^{\prime})$ but no term in
$\Lambda(σ^{\prime})=\\{(\lambda{x}.({x})\,{x})\,{y},(\lambda{x}.({x})\,{x})\,{z}\\}$
$β$-reduces to $M^{\prime}$. Note that, in this counter-example, there is no
$M\in\Lambda$ such that $\underline{M}\mathrel{\widetilde{\to}}^{*}σ$:
somehow, we must exploit this additional hypothesis to establish a correct
version of 0.2.1.

Reasoning on $\mathrel{\widetilde{\to}}^{*}$ directly is difficult, due to its
definition as a reflexive and transitive closure. The technique we propose
involves the definition of a mixed-type relation $M\Vdash σ$ between a
$λ$-term $M$ and a term $σ\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$:
intuitively, $M\Vdash σ$ when $σ$ is obtained by pasting together terms issued
from various reductions of $M$, and we say $M\Vdash σ$ is a mashup of such
reductions. In particular: $M\Vdash\underline{M^{\prime}}$ as soon as
$M\to_{\Lambda}^{*}M^{\prime}$; and $M\Vdash σ+τ$ as soon as $M\Vdash σ$ and
$M\Vdash τ$. We then show that $\Vdash$ is conservative over
$\to_{\Lambda}^{*}$ (lemma 0.3.4) and that $M\Vdash σ$ as soon as
$\underline{M}\mathrel{\widetilde{\to}}^{*}σ$ (lemmas 0.3.3 and 0.4.2): this
ensures the conservativity of $\mathrel{\widetilde{\to}}^{*}$ over
$\to_{\Lambda}^{*}$ (theorem 0.4.3). Our whole approach thus rests on the
careful definition of the mashup relation. Among other requirements, it must
behave well w.r.t. the structure of terms: e.g., if $M\Vdash σ$ then
$\lambda{x}.M\Vdash\lambda{x}.σ$.

##  0.3 Mashup of $β$-reductions

We define two relations
$\mathord{\vdash}\subseteq\Lambda\times\Delta_{\mathsf{R}}$ and
$\mathord{\Vdash}\subseteq\Lambda\times\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$
by mutual induction, with the rules of section 0.3. If
$σ\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$, we write
$\mathsf{Supp}(σ)\subset\Delta_{\mathsf{R}}$ for its support set.

$\displaystyle\prooftree\hypo{M\to_{\Lambda}^{*}x}\infer
1[(v)]{M\vdash\underline{x}}\qquad\prooftree\hypo{M\to_{\Lambda}^{*}\lambda{x}.N}\hypo{N\vdash
τ}\infer
2[($λ$)]{M\vdash\lambda{x}.τ}\qquad\prooftree\hypo{M\to_{\Lambda}^{*}(N)\,P}\hypo{N\vdash
τ}\hypo{P\Vdash ρ}\infer 3[(a)]$ $\displaystyle\prooftree\infer
0[($\mathbf{0}$)]{M\Vdash\mathbf{0}}\qquad\prooftree\hypo{M\vdash
σ}\hypo{M\Vdash τ}\infer 2[($+$)]{M\Vdash a.σ+τ}$

Figure 1: Inference rules for the mashup relations

$\displaystyle\prooftree\hypo{M\vdash σ}\infer 1[(s)]{M\Vdash σ}$
$\displaystyle\prooftree\hypo{M\to_{\Lambda}^{*}\lambda{x}.N}\hypo{N\Vdash
τ}\infer
2[($λ$^{\prime})]{M\Vdash\lambda{x}.τ}\qquad\prooftree\hypo{M\to_{\Lambda}^{*}(N)\,P}\hypo{N\Vdash
τ}\hypo{P\Vdash ρ}\infer
3[(a^{\prime})]{M\Vdash(τ)\,ρ}\qquad\prooftree\hypo{M\Vdash σ}\hypo{M\Vdash
τ}\infer 2[($+$^{\prime})]{M\Vdash a.σ+τ}$ Figure 2: Admissible rules for the
mashup relations

######  Lemma 0.3.1.

We have $M\Vdash σ$ iff, for every $σ^{\prime}\in\mathsf{Supp}(σ)$, $M\vdash
σ^{\prime}$.

###### Proof.

The forward implication is done by induction on the derivation of $M\Vdash σ$,
noting that if $σ^{\prime}\in\mathsf{Supp}(aτ+ρ)$ with $M\vdash τ$ and
$M\Vdash ρ$ then $σ^{\prime}=τ$ or $σ^{\prime}\in\mathsf{Supp}(ρ)$. For the
reverse implication, we can write $σ=\sum_{i=1}^{n}a_{i}.σ_{i}$ with
$σ_{i}\in\mathsf{Supp}(σ)$ for $1\leq i\leq n$, and obtain a derivation of
$M\Vdash σ$ by induction on $n$. ∎

######  Lemma 0.3.2.

The rules of fig. 2 are admissible.

###### Proof.

For (s), it is sufficient to observe that $σ=1σ+\mathbf{0}$. For the other
three rules, we reason on the support sets, using lemma 0.3.1. ∎

######  Lemma 0.3.3 (Reflexivity of $\vdash$).

For every $M\in\Lambda$, $M\vdash\underline{M}$.

###### Proof.

By a straightforward induction on $M$, using the reflexivity of
$\to_{\Lambda}^{*}$ and rule (s). ∎

######  Lemma 0.3.4 (Conservativity of $\Vdash$).

If $M\Vdash\underline{M^{\prime}}$ then $M\to_{\Lambda}^{*}M^{\prime}$.

###### Proof.

Note that $\underline{M^{\prime}}\in\Delta_{\mathsf{R}}$, hence
$M\vdash\underline{M^{\prime}}$ by lemma 0.3.1. The proof is then by induction
on $M^{\prime}$, inspecting the last rule of the derivation of
$M\vdash\underline{M^{\prime}}$:

  * (v)

If $M\to_{\Lambda}^{*}x$ and $\underline{M^{\prime}}=\underline{x}$ then we
conclude directly since $M^{\prime}=x$.

  * ($λ$)

If $M\to_{\Lambda}^{*}\lambda{x}.N$ and $N\vdash τ$ with
$\underline{M^{\prime}}=\lambda{x}.τ$, then $M^{\prime}=\lambda{x}.N^{\prime}$
with $τ=\underline{N^{\prime}}$. By induction hypothesis,
$N\to_{\Lambda}^{*}N^{\prime}$, hence $M\to_{\Lambda}^{*}M^{\prime}$.

  * (a)

If $M\to_{\Lambda}^{*}(N)\,P$, $N\vdash τ$ and $P\Vdash ρ$ with
$\underline{M^{\prime}}=(τ)\,ρ$, then $M^{\prime}=(N^{\prime})\,P^{\prime}$
with $τ=\underline{N^{\prime}}$ and $ρ=\underline{P^{\prime}}$. In particular
$ρ\in\Delta_{\mathsf{R}}$, hence $P\vdash ρ$ by lemma 0.3.1. By induction
hypothesis, $N\to_{\Lambda}^{*}N^{\prime}$ and $P\to_{\Lambda}^{*}P^{\prime}$,
hence $M\to_{\Lambda}^{*}M^{\prime}$. ∎

######  Lemma 0.3.5 (Compatibility with $\to_{\Lambda}$).

If $M\to_{\Lambda}^{*}M^{\prime}\vdash σ$ then $M\vdash σ$. Similarly, if
$M\to_{\Lambda}^{*}M^{\prime}\Vdash σ$ then $M\Vdash σ$.

###### Proof.

For the first implication, it is sufficient to inspect the last rule of the
derivation $M^{\prime}\vdash σ$, and use the transitivity of
$\to_{\Lambda}^{*}$. The second implication follows directly by induction on
the derivation of $M^{\prime}\Vdash σ$. ∎

##  0.4 Conservativity of algebraic reduction

######  Lemma 0.4.1 (Substitution lemma).

If $M\Vdash σ$ and $P\Vdash ρ$ then $M[P/x]\Vdash σ[ρ/x]$.

###### Proof.

We prove the result, together with the variant assuming $M\vdash σ$ instead of
$M\Vdash σ$, by induction on the derivations of those judgements.

  * (v)

If $M\to_{\Lambda}^{*}y$ and $σ=\underline{y}$ then:

    * –

if $y=x$, then $M[P/x]\to_{\Lambda}^{*}x[P/x]=P\Vdash ρ$ and we obtain
$M[P/x]\Vdash ρ=σ[ρ/x]$ by lemma 0.3.5;

    * –

otherwise, $M[P/x]\to_{\Lambda}^{*}y[P/x]=y$, hence
$M[P/x]\Vdash\underline{y}=σ[ρ/x]$ by (v).

  * ($λ$)

If $M\to_{\Lambda}^{*}\lambda{y}.N$ and $N\vdash τ$ with $σ=\lambda{y}.τ$
(choosing $y\not=x$ and $y$ not free in $P$ nor in $ρ$), then
$M[P/x]\to_{\Lambda}^{*}\lambda{y}.N[P/x]$ and, by induction hypothesis
$N[P/x]\Vdash τ[ρ/x]$: we obtain $M[P/x]\Vdash\lambda{y}.τ[ρ/x]=σ[ρ/x]$ by
$($λ${}^{\prime})$.

  * (a)

If $M\to_{\Lambda}^{*}(N_{1})\,N_{2}$, $N_{1}\vdash τ_{1}$ and $N_{2}\Vdash
τ_{2}$, with $σ=(τ_{1})\,τ_{2}$, then we have
$M[P/x]\to_{\Lambda}^{*}(N_{1}[P/x])\,N_{2}[P/x]$ and, by induction
hypothesis, $N_{1}[P/x]\Vdash τ_{1}[ρ/x]$ and $N_{2}[P/x]\Vdash τ_{2}[ρ/x]$:
we obtain $M[P/x]\Vdash(τ_{1}[ρ/x])\,τ_{2}[ρ/x]=σ[P/x]$ by $(a^{\prime})$.

  * ($\mathbf{0}$)

If $σ=\mathbf{0}$ then $σ[ρ/x]=\mathbf{0}$ and we conclude directly, by
($\mathbf{0}$).

  * ($+$)

If $σ=a.τ_{1}+τ_{2}$ with $M\vdash τ_{1}$ and $M\Vdash τ_{2}$, then, by
induction hypothesis, $M[P/x]\Vdash τ_{1}[ρ/x]$ and $M[P/x]\Vdash τ_{2}[ρ/x]$,
hence $M[P/x]\Vdash a.τ_{1}[ρ/x]+τ_{2}[ρ/x]=σ[ρ/x]$ by $(a^{\prime})$. ∎

Note that, by positivity, if $σ=a.τ+ρ$ with $τ\in\Delta_{\mathsf{R}}$ and
$a\not=0$, then $τ\in\mathsf{Supp}(σ)\supseteq\mathsf{Supp}(ρ)$.

######  Lemma 0.4.2 (Compatibility with $\mathrel{\widetilde{\to}}$).

Let $M\in\Lambda$ and
$σ^{\prime}\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$. For every
$σ\in\Delta_{\mathsf{R}}$ such that $M\vdash σ\to σ^{\prime}$ (resp. every
$σ\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$ such that $M\Vdash
σ\mathrel{\widetilde{\to}}σ^{\prime}$), we have $M\Vdash σ^{\prime}$.

###### Proof.

The proof is by induction on the definition of the reduction $σ\to σ^{\prime}$
or $σ\mathrel{\widetilde{\to}}σ^{\prime}$.

  * •

If $σ=(\lambda{x}.τ)\,ρ\in\Delta_{\mathsf{R}}$ and $σ^{\prime}=τ[ρ/x]$, then
the derivation of $M\vdash σ$ must be of the form

$\prooftree\hypo{M\to_{\Lambda}^{*}(N)\,P}\hypo{N\to_{\Lambda}^{*}\lambda{x}.N^{\prime}}\hypo{N^{\prime}\vdash
τ}\infer 2[(λ)]{N\vdash\lambda{x}.τ}\hypo{P\Vdash ρ}\infer
3[(a)]{M\vdash(\lambda{x}.τ)\,ρ}\;.$

By lemma 0.4.1, we have $N^{\prime}[P/x]\Vdash σ^{\prime}$. Moreover,
$M\to_{\Lambda}^{*}(N)\,P\to_{\Lambda}^{*}(\lambda{x}.N^{\prime})\,P\to_{\Lambda}N^{\prime}[P/x]$
and we obtain $M\Vdash σ^{\prime}$ by lemma 0.3.5.

  * •

If $σ=\lambda{x}.τ$ and $σ^{\prime}=\lambda{x}.τ^{\prime}$ with $τ\to
τ^{\prime}$, then the derivation of $M\vdash σ$ must be of the form

$\prooftree\hypo{M\to_{\Lambda}^{*}\lambda{x}.N}\hypo{N\vdash τ}\infer
2[(λ)]{M\vdash\lambda{x}.τ}\;.$

We obtain $N\Vdash τ^{\prime}$ by induction hypothesis, and we conclude by
($λ$’).

  * •

If $σ=(τ)\,ρ$ and $σ^{\prime}=(τ^{\prime})\,ρ^{\prime}$ with either $τ\to
τ^{\prime}$ and $ρ=ρ^{\prime}$, or $τ=τ^{\prime}$ and
$ρ\mathrel{\widetilde{\to}}ρ^{\prime}$, then the derivation of $M\vdash σ$
must be of the form

$\prooftree\hypo{M\to_{\Lambda}^{*}(N)\,P}\hypo{N\vdash τ}\hypo{P\Vdash
ρ}\infer 3[(a)]{M\vdash(τ)\,ρ}\;.$

We obtain $N\Vdash τ^{\prime}$ and $P\Vdash ρ^{\prime}$ by induction
hypothesis, and we conclude by (a’).

  * •

If $σ=a.τ+ρ$ and $σ^{\prime}=a.τ^{\prime}+ρ$ with $τ\to τ^{\prime}$ and
$a\not=0$, then we have already observed that $τ\in\mathsf{Supp}(σ)$ and
$\mathsf{Supp}(ρ)\subseteq\mathsf{Supp}(σ)$. Since $M\Vdash σ$, we obtain
$M\vdash τ$ and $M\Vdash ρ$ by lemma 0.3.1. The induction hypothesis gives
$M\Vdash τ^{\prime}$, hence $M\Vdash a.τ^{\prime}+ρ=σ^{\prime}$ by ($+$’). ∎

######  Theorem 0.4.3 (Conservativity of $\mathrel{\widetilde{\to}}^{*}$).

If $\underline{M}\mathrel{\widetilde{\to}}^{*}\underline{N}$ then
$M\to_{\Lambda}^{*}N$.

###### Proof.

Assume $\underline{M}\mathrel{\widetilde{\to}}^{*}\underline{N}$. By lemma
0.3.3 and (s), we have $M\Vdash\underline{M}$. By iterating lemma 0.4.2, we
deduce $M\Vdash\underline{N}$. We conclude by lemma 0.3.4. ∎

######  Corollary 0.4.4 (Conservativity of $\leftrightarrow$).

If $\mathsf{R}$ is positive then $\underline{M}\leftrightarrow\underline{N}$
iff $M\leftrightarrow_{\Lambda}N$.

###### Proof.

Assuming $\underline{M}\leftrightarrow\underline{N}$, the confluence of
$\mathrel{\widetilde{\to}}$ ensures that there exist
$σ\in\mathsf{R}\langle\Delta_{\mathsf{R}}\rangle$ and $k\in\mathbf{N}$, such
that $\underline{M}\mathrel{\widetilde{\to}}^{k}σ$ and
$\underline{N}\mathrel{\widetilde{\to}}^{*}σ$. It follows [Vau09, Lemma 3.23]
that $σ\mathrel{\widetilde{\to}}^{*}\underline{M}\mathord{\downarrow}^{k}$
where $τ\mathord{\downarrow}$ is the term obtained by reducing all the redexes
of $τ$ simultaneously. Observing that
$\underline{M}\mathord{\downarrow}=\underline{M\mathord{\downarrow}}$, we
obtain
$\underline{N}\mathrel{\widetilde{\to}}^{*}\underline{M\mathord{\downarrow}^{k}}$
hence $N\to_{\Lambda}^{*}M\mathord{\downarrow}^{k}$ by theorem 0.4.3, which
concludes the proof. ∎

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Hidden unique possibilities

of mathematical physics equations.

(Formalism of skew-symmetric forms)

L.I. Petrova

Lomonosov Moscow State University, Faculty of Computational

Mathematics and Cybernetics, e-mail<EMAIL_ADDRESS>
###### Abstract

It is shown that mathematical physics differential equations have properties
that allow describing processes such as the structures emergence, discrete
transitions, quantum jumps. The peculiarity is that such properties are
hidden. They do not follow directly from the mathematical physics equations
but are realized discretely in the solving process. This is due to the
mathematical physics equations integrability, which, as shown, can be realized
only discretely in the presence of any degrees of freedom. In this case, a
transition occurs from the original coordinate space with a solution that is
not a function (the solution derivatives do not compose a differential) to
integrable structures with a solution that is a discrete function. It is the
double solutions and spatial transitions that can describe the processes of
the emergence of any structures or phenomena.

Due to hidden properties, the mathematical physics equations have unique
possibilities in describing physical processes and phenomena that cannot be
described in the framework of other mathematical formalisms.

Such results were obtained using skew-symmetric differential forms.

Keywords: the mathematical physics equations integrability; hidden properties;
skew-symmetric differential forms; non-integrable manifolds; degenerate
transformations.

##  1 Introduction

This paper deals with partial differential equations of mathematical physics
describing continuous media (such as thermodynamic, gas-dynamic, cosmic),
charged particles systems, and so on. It turns out that the equations of
mathematical physics can not only describe evolutionary processes and changes
in physical quantities (such as, for example, energy, pressure, density), but
have the ability to describe discrete processes, such as the emergence of
various structures, quantum jumps, the emergence of observable formations (
waves, vortices, turbulent pulsations) and so on.

However, such opportunities are hidden. They appear only in the process of
equations solving. And this is due to the mathematical physics equations
integrability, which is realized discretely.

The mathematical physics equations, describing real processes, on the original
tangent space turn out to be non-integrable. In this case, the equation
solution is not a function: the derivatives of this solution do not compose a
differential.

The integrability of the equation can be realized only discretely in the
process of solving the equations in the presence of any degrees of freedom. In
this case, the resulting solution is a discrete function.

Such features of mathematical physics equations integrability, as will be
shown, reveal the unique possibilities of mathematical physics equations.

Section 2 notes the currently existing approaches to equations solving, which
depend on the equations integrability, and the questions that arise in this
case.

At the same time, the characteristic properties of integrable equations are
given, namely, the presence of differentials and relations in the
differentials, which can be directly integrated. Such properties of integrable
equations and the process of realizing integrability can only be described by
skew-symmetric differential forms.

Section 3 presents some properties and features of exterior and evolutionary
skew-symmetric differential forms necessary for studying the differential
equations integrability.

Section 4 investigated the mathematical physics differential equations
integrability, which depends on the described functions derivatives
consistency and on the equations consistency if the equations of mathematical
physics are a system of equations. (It should be noted that this is
practically not taken into account when solving differential equations.)

It is shown that the differential equations integrability is realized
discretely. In this case, the hidden properties of mathematical physics
equations are manifested, such as invariance, double solutions, discrete
transitions, and so on.

It is these properties of the mathematical physics equations, as shown in
Section 5, that reveal the unique possibilities of the mathematical physics
equations in describing discrete processes and phenomena.

Section 5 summarizes the published works of the author, which describe such
processes as the onset of vorticity and turbulence, discrete transitions, the
implementation of Hamiltonian systems, the connection of the equations of
field theory with the equations of mathematical physics, and other results
that turned out to be possible to obtain only thanks to the hidden properties
of the mathematical physics equations.

##  2 Existing approaches to solving differential equations of mathematical
physics

We can mention analytical and numerical methods for solving of mathematical
physics differential equations.

The use of analytical methods is possible if the differential equations can be
reduced to an integrable form or if the integrability conditions are
satisfied.

Since the use of analytical methods even in simple cases is not always
possible, many numerical methods were developed for the solution of
mathematical physics equations.

The question arises, what is the correspondence between the solutions obtained
by analytical and numerical methods and what is the accuracy of the studied
real phenomena description by mathematical physics equations.

As shown in this work, there is a formalism of skew-symmetric differential
forms, which allows not only to answer this question, but also to reveal the
physical meaning of analytical and numerical solutions of mathematical physics
equations.

Questions about solving of mathematical physics differential equations are
related to the problem of integrability of differential equations.

Obviously, the characteristic property of integrable equation is the presence
of differentials and the ability to reduce the equation to the identical
relation in the differentials, which will allow explicit integration of the
differential equation.

Skew-symmetric differential forms have such properties. Unlike other existing
mathematical formalisms they can take the form of differentials and
differential expressions.

As you know, closed exterior forms are differentials. In addition, as
established by the author, there are skew-symmetric differential forms that
are obtained from differential equations and that generate closed exterior
forms, i.e., differentials. Such skew-symmetric forms, which are evolutionary,
made it possible to study the integrability and properties of mathematical
physics differential equations.

This article is based on the properties of external and evolutionary skew-
symmetric forms and on the features of their mathematical apparatus, which has
non-traditional elements, such as non-identical relations, degenerate
transformations, discrete transitions. Such a mathematical formalism is given
only in the works of the author. Since this is not generally known, then for
the presentation of the results obtained with the help of this formalism, it
is necessary to repeat some of the provisions of this formalism. This is
presented in section 3. (For more details on skew-symmetric exterior and
evolutionary forms, see [1].)

##  3 Some properties and features of exterior and evolutionary skew-symmetric
differential forms

Exterior differential forms (skew-symmetric forms on integrable manifolds and
structures) have the properties, which underlie many mathematical formalisms.
Historically, however, they are primarily used in differential geometry and
topology. (It can be noted that the Frobenius theorem on the integrability
conditions of the Pfaffian system is related to the skew-symmetric forms
properties.)

The exterior differential form of the degree $p$ ($p$-form) can be written as
[1, 2]

$\theta^{p}=\sum_{i_{1}\dots i_{p}}a_{i_{1}\dots i_{p}}dx^{i_{1}}\wedge
dx^{i_{2}}\wedge\dots\wedge dx^{i_{p}}\quad 0\leq p\leq n$ $None$

Here $a_{i_{1}\dots i_{p}}$ is the function of independent variables
$x^{1},...,x^{n}$, $n$ is the space dimension, and $dx^{i}$, $dx^{i}\wedge
dx^{j}$, $dx^{i}\wedge dx^{j}\wedge dx^{k}$, … is the local basis subject to
the skew-symmetry condition:

$\begin{array}[]{l}dx^{i}\wedge dx^{i}=0\\\ dx^{i}\wedge dx^{j}=-dx^{j}\wedge
dx^{i}\quad i\neq j\end{array}$ $None$

The exterior form differential $\theta^{p}$ is expressed by the formula

$d\theta^{p}=\sum_{i_{1}\dots i_{p}}da_{i_{1}\dots i_{p}}\wedge
dx^{i_{1}}\wedge dx^{i_{2}}\dots\wedge dx^{i_{p}}$ $None$

Since exterior forms are defined on integrable manifolds or structures, the
exterior forms differential does not contain the basis differential from
(since it is equal to zero). Such skew-symmetric forms can be closed.

###  3.1 Closed exterior differential forms

In mathematical formalisms the closed differential forms with invariant
properties appear to be of greatest practical utility.

A form is ’closed’, if its differential is equal to zero:

$d\theta^{p}=0$ $None$

From condition (4) one can see that a closed form is a conservative quantity.
(This means that it corresponds to a conservation law, namely, to some
conservative physical quantity, i.e., quantity conservative under all
invariant transformations.)

If the exterior form is a closed inexact form, i.e., it is defined only on
some structure (which is a pseudostructure by its metric properties), the
closure condition is written as

$d_{\pi}\theta^{p}=0$ $None$

And the pseudostructure $\pi$ obeys the condition

$d_{\pi}{}^{*}\theta^{p}=0$ $None$

where ${}^{*}\theta^{p}$ is a dual form. I.e., the dual form describes a
pseudostructure, on which a closed inexact exterior form is defined. In this
case the interior differential rather than total one becomes equal to zero.

Closed inexact exterior form and the corresponding dual form compose a
differential-geometric structure that describes conservative object, i.e.,
this quantity is conserved on a pseudostructure. (This can also correspond to
some conservation law, i.e. to a conservative object.)

Following properties of closed exterior differential forms and features of
their mathematical apparatus is related to the integrability of partial
differential equations.

1) From the condition for the exterior form closedness

$d\theta^{p}=0$ $None$

it follows that the closed form is a differential.

The differential of the form is a closed form, since

$dd\phi=0$ $None$

where $\phi$ is an arbitrary exterior form.

A closed exterior inexact form is an interior differential (on a
pseudostructure).

2) Since the closed form is a differential, this means that the closed form
proves to be an invariant under all transforms that conserve the differential.

The closed inexact exterior form and relevant dual form, composing the
differential-geometrical structure, describe an invariant object.

3) The transformations of exterior skew-symmetric forms, which conserves the
differential, are non-degenerate transformations. The unitary transforms
(O-form), the tangent and canonical transforms (1-form), the gradient and
gauge transforms (2-form) and so on are examples of such transforms.

4) Identical relations are an important element of the mathematical formalism
of closed exterior forms. The identical relations express the fact that each
closed exterior form is a differential of some exterior form. Such an
identical relation can be written as

$d\phi=\theta^{p}$ $None$

In this relation the form in the right-hand side has to be a closed  one.

It can be noted that practically in all branches of physics, mechanics,
thermodynamics, there are such identical relations. For example, the Poincare
invariant $ds\,=\,-H\,dt\,+\,p_{j}\,dq_{j}$, the second thermodynamics
principle $dS\,=\,(dE+p\,dV)/T$, and so on..

It can be seen that closed exterior forms, firstly, are differentials, which
indicates their integrability, and, secondly, the closed exterior forms
consist of identical relations that can be integrated.

This indicates that if we understand the connection between differential
equations and closed exterior forms, then we can solve the problem of
differential equations integrability. It turns out that the evolutionary skew-
symmetric forms properties reveal the connection of differential equations
with closed exterior forms.

###  3.2 Properties and features of evolutionary skew-symmetric differential
forms.

Evolutionary skew-symmetric forms [1], in contrast to exterior forms, are
defined on nonintegrable manifolds (such as tangent manifolds of differential
equations, Lagrangian manifolds and so on).

The evolutionary form can be written in a manner similar for exterior
differential form [1]. However, in distinction from the exterior form
differential, an additional term will appear in the evolutionary form
differential. This is due to the fact that the evolutionary form basis changes
since such a form is defined on nonintegrable manifold.

The evolutionary form differential takes the form

$d\theta^{p}{=}\\!\sum_{i_{1}\dots\i_{p}}\\!da_{i_{1}\dots\i_{p}}\wedge
dx^{i_{1}}\wedge dx^{i_{2}}\dots\wedge
dx^{i_{p}}{+}\\!\sum_{i_{1}\dots\i_{p}}\\!a_{i_{1}\dots\i_{p}}d(dx^{i_{1}}\wedge
dx^{i_{2}}\dots\wedge dx^{i_{p}})$ $None$

where the second term is connected with the basis differential being nonzero:
$d(dx^{i_{1}}\wedge dx^{i_{2}}\wedge\dots\wedge dx^{i_{p}})\neq 0$. (For the
exterior form defined on integrable manifold one has $d(dx^{i_{1}}\wedge
dx^{i_{2}}\wedge\dots\wedge dx^{i_{p}})=0$).

[Hereinafter a summing symbol $\sum$ and a symbol of exterior multiplication
$\wedge$ will be omitted. Summation over repeated indices is implied.]

An evolutionary form, in contrast to exterior forms, cannot be a closed form,
because its differential is not zero, since the second term of the
evolutionary form differential associated with the basis differential is not
zero.

The second term in the expression for the skew-symmetric form differential
connected with the basis differential is expressed in terms of the metric form
commutator. For example, let us consider the first-degree form
$\omega=a_{\alpha}dx^{\alpha}$.

The differential of this form can be written as
$d\omega=K_{\alpha\beta}dx^{\alpha}dx^{\beta}$ where $K_{\alpha\beta}$ is the
evolutionary form commutator. If we use the connectednesses
$\Gamma^{\sigma}_{\beta\alpha}$ to describe a nonintegrable manifold, then the
commutator $K_{\alpha\beta}$ can be written in the form:

$K_{\alpha\beta}=\left(\frac{\partial a_{\beta}}{\partial
x^{\alpha}}-\frac{\partial a_{\alpha}}{\partial
x^{\beta}}\right)+(\Gamma^{\sigma}_{\beta\alpha}-\Gamma^{\sigma}_{\alpha\beta})a_{\sigma}$
$None$

The commutator $K_{\alpha\beta}$ is nonzero as the coefficients $a_{\mu}$ not
potential, and connectednesses of a nonintegrable manifold is not symmetric
[3]. Thus it turns out that the commutator of the form $\omega$ and its
differential does not equal to zero.

Since the evolutionary form is not closed, then it cannot be a differential,
like a closed exterior form. The evolutionary form has an unconventional
mathematical apparatus containing non-degenerate self-varying relations and
degenerate transformations.

### Non-identical self-varying relations.

It was shown that identical relations lie at the basis of exterior
differential forms the mathematical apparatus. In contrast to this,
nonidentical relations lie at the basis of evolutionary differential forms
mathematical apparatus. Nonidentical relation appear in descriptions of any
processes.

Nonidentical relation may be written as well as the identical relation

$d\phi=\theta^{p}$ $None$

However, on the right side of this relation there is an evolutionary skew-
symmetric form, which is not a differential.

The evolutionary relation turns out to be non-identical, since there is a
differential on the left, and a skew-symmetric on the right, which is not a
differential.

An evolutionary nonidentical relation is self-varying, because, it is an
evolutionary relation, and contains two members, one of which is immeasurable
and cannot be compared with the other in the evolutionary process.

### Degenerate transformation.

One of the main elements of the mathematical formalism of the evolutionary
forms theory is a degenerate transformation.

Unlike a non-degenerate transformation of closed exterior forms, which
conserves the differential, a degenerate transformation is a transformation
that does not conserve the differential. (It can be noted that an example of a
degenerate transformation is the Legendre transformation. However, it is
practically applied only tacitly.)

The degenerate transformation reveals the unique property of the evolutionary
form. An evolutionary form can generate closed exterior forms that have
invariant properties and correspond to the differential equations
integrability.

As will be shown below, the evolutionary forms formalism, which has non-
traditional elements, such as non-identical self-varying relations and
degenerate transformations, makes it possible to discover the hidden
properties of the mathematical physics equations, solve the problem of
differential equations integrability and reveal their unique capabilities.

##  4 On integrability of partial differential equations

As noted above, the characteristic property of integrable equations is the
existence of differentials and identity relations in differentials, which
allows us to explicitly integrate the differential equation. As has been
shown, such property is possessed by closed exterior forms. The differential
equations integrability problem can be solved if one finds the connection of
differential equations with closed exterior forms and understands how such a
connection is realized. It is possible to solve this with the help of
evolutionary skew-symmetric forms, which, as has been shown, can generate
closed exterior forms.

A qualitative study of differential equations has shown that the integrability
of mathematical physics differential equations depends on the consistency of
the derivatives of described functions and on the consistency of equations if
the equations of mathematical physics are the system of equations.

###  4.1 Investigation of the correspondence between derivatives in
differential equations.

The correspondence between the derivatives of the sought-for functions in
differential equations can be traced by the example of a first-order partial
differential equation:

$F(x^{i},\,u,\,p_{i})=0,\quad p_{i}\,=\,\partial u/\partial x^{i}$ $None$

Obviously, derivatives are consistent if they constitute a differential, i.e.,
the following relation is fulfilled

$du\,=\,\theta$ $None$

where $\theta\,=\,p_{i}\,dx^{i}$ (summation over repeated indices is meant).

In the general case, it does not follow (explicitly) from Eq. (13) that the
derivatives $p_{i}\,=\,\partial u/\partial x^{i}$, satisfying the equation,
constitute a differential.

Derivatives $p_{i}$ can constitute a differential only if the conditions for
the closedness of the form $\theta\,=\,p_{j}dx^{j}$ and the corresponding dual
form are satisfied (in this case, the functional $F$ plays the role of dual
form for $\theta$):

$\cases{dF(x^{i},\,u,\,p_{i})\,=\,0\cr d(p_{i}\,dx^{i})\,=\,0\cr}$ $None$

If we expand the differentials, then we get a system of homogeneous equations
with respect to $dx^{i}$ and $dp_{i}$ (in the $2n$ -dimensional cotangent
space):

$\cases{\displaystyle\left({{\partial F}\over{\partial x^{i}}}\,+\,{{\partial
F}\over{\partial u}}\,p_{i}\right)\,dx^{i}\,+\,{{\partial F}\over{\partial
p_{i}}}\,dp_{i}\,=\,0\cr dp_{i}\,dx^{i}\,-\,dx^{i}\,dp_{i}\,=\,0\cr}$ $None$

The solvability condition of this system (i.e., vanishing of the determinant
composed of coefficients at $dx^{i}$, $dp_{i}$) has the form:

${{dx^{i}}\over{\partial F/\partial p_{i}}}\,=\,{{-dp_{i}}\over{\partial
F/\partial x^{i}+p_{i}\partial F/\partial u}}$ $None$

This condition determine an integrable structure, namely, a pseudostructure
(in its metric properties), on which the form $\theta\,=\,p_{i}\,dx^{i}$ turns
out to be closed, i.e. it becomes a differential.

It turns out that conditions (15) are satisfied only on integrable structures
of the cotangent space. In this case, on integrable structures
(pseudostructures) derivatives of the differential equation constitute the
differential $\delta u\,=\,p_{i}dx^{i}\,=\,du$. That is, the solution becomes
a function (discrete, since it is defined only on structures). On the
cotangent space differential equation (13) turns out to be an integrable
equation.

But on the original the tangent space the differential equation (13) turns out
to be non-integrable. In this case, the derivatives of the differential
equation do not constitute the differential. Relation (14) turns out to be
non-identical. On the right is a skew-symmetric form that is not a
differential. Such a relation cannot be exactly integrated. In this case, the
solution of a differential equation is not a function.

It turns out that equation (13) have double solutions, namely, on the original
coordinate space and on the integrable structures. It can be seen that the
transition from the first solution to the second is associated with the
transition of the differential equation from the tangent space to the
cotangent one. And this is due to the implementation of the integrability of
the differential equation. Such a transition is described by a degenerate
transformation.

###  4.2 On the integrability of mathematical physics equations.

Equations of mathematical physics are practically a system of equations, i.e.,
they consist of several differential equations. Examples of mathematical
physics equations are equations describing material media, such as
thermodynamic, gas-dynamic, cosmic, and so on, which consist of the energy,
momentum, angular momentum, and mass conservation laws equations.

The system of equations integrability, an example of which are mathematical
physics equations, depends on the consistency of the equations that make up
the given system of equations.

The integrability mathematical physics equations was studied using the example
of the system of equations consisting of conservation laws equations.

### On the consistency of the conservation laws equations. Evolutionary
relation.

When investigating the consistency of the equations that form the system of
equations, two coordinate systems were used: inertial and accompanying -
associated with the manifold formed by the particles trajectories. (An example
can be the Euler and Lagrangian coordinate systems.)

In addition, the conservation laws equations were transformed into equations
for functionals that characterize the described medium state. Functionals such
as wave function, entropy, action functional, Poynting vector, Einstein
tensor, etc. (which are also field theory functionals) are such state
functionals [4].

When investigating the consistency of the conservation laws, we obtain an
evolutionary relation in skew-symmetric forms for the state functional, which
reveals the features of mathematical physics equations integrability.

Let consider the consistency of the energy and momentum equations. In the
accompanying coordinate system, the energy and momentum equations can be
written in the form:

${{\partial\psi}\over{\partial\xi^{1}}}\,=\,A_{1}$ $None$
${{\partial\psi}\over{\partial\xi^{\nu}}}\,=\,A_{\nu}$ $None$

Here $\psi$ is the state functional, $\xi^{1}$ and $\xi^{\nu}$ are the
coordinates along the trajectory and in the direction normal to the
trajectory, $A_{1}$ is a quantity that depends on specific features described
material medium and on external energy actions onto the medium, and $A_{\nu}$
are the quantities that depend on the specific features of material medium and
external force actions.

Here you can see the specific feature. For the same quantity (functional
$\psi$), there are two equations. How can such a system be investigated?

Equations (18) and (19) can be folded into a relation

$d\psi\,=\,A_{\mu}\,d\xi^{\mu}$ $None$

Relation (19) can be written as

$d\psi\,=\,\omega$ $None$

where $\omega\,=\,A_{\mu}\,d\xi^{\mu}$ is the skew-symmetric differential form
of the first degree.

Since the conservation laws equations are evolutionary, the resulting relation
and the skew-symmetric form $\omega$ will also be evolutionary.

If we also take into account the angular momentum and mass conservation laws,
then the evolutionary relation takes the form

$d\psi\,=\,\omega^{p}$ $None$

where the degree of the form $p$ takes the values $p\,=\,0,1,2,3$.

(The article [5] describes the derivation of the evolutionary relation for a
gas-dynamic medium. This evolutionary relation is given in section 5.)

### Properties and features of the evolutionary skew-symmetric form and
evolutionary relation.

An evolutionary skew-symmetric form, unlike exterior forms, cannot be closed.

As shown, the evolutionary relation (21) and evolutionary form $\omega$ were
obtained in the accompanying coordinate system associated with the manifold
formed by the described medium particles trajectories. The manifold formed by
the particles trajectories is deformable, i.e., a non-integrable manifold. As
shown in subsection 3.2 evolutionary form defined on the non-integrable
manifold cannot be closed because its commutator (which contains a non-zero
metric form commutator), and, consequently, the differential is not equal to
zero.

Since the evolutionary form $\omega$ is not a differential, the evolutionary
relation turns out to be non-identical (the differential is on the left, and
the skew-symmetric form, which is not a differential, is on the right).

In a similar way, it can be shown that relation (22) will also be
nonidentical. In this case, the evolutionary forms of degrees $1-3$ will be
non-closed, since the commutator of these evolutionary forms will contain
nonzero commutators of the metric form of the first, second, and third
degrees, which determine the torsion, rotation, and curvature,
respectively.(It should be noted that the resulting evolutionary relation
remains nonidentical regardless of the accuracy with which were written the
conservation laws equations. And it has a physical meaning.)

Nonidentical evolutionary relation turns to be self-changing relation as it
evolutionary and comprises members, which do not agree between themselves.

### Non-integrability of mathematical physics equations on the original
tangent space.

The nonidentity of the evolutionary relation points out to the fact that the
conservation law equations appear to be inconsistent. This means that the
initial set of equations of mathematical physics proves to be nonintegrable.

The nonidentical evolutionary relation cannot be explicitly integrated, since
there is a not closed form on the right, which is not a differential.
Mathematical physics equations cannot be folded into the identical relation
and explicitly integrated. It turns out that on the original tangent space the
mathematical physics equations turn out to be non-integrable.

In this case, the solutions of mathematical physics equations will depend not
only on variables. They will depend on the evolutionary form commutator. The
solutions will not be analytical solutions, i.e., functions. The derivatives
of such solutions do not form a differential. (As will be shown below, such
solutions have a physical meaning.)

### Realization of the mathematical physics equations integrability.

The mathematical physics equations integrability can be realized only if the
identical relation (which can be integrated) is obtained from the non-
identical relation. To obtain the identical relation from an evolutionary non-
identical relation, it is necessary to obtain the closed exterior form from
the evolutionary form included in the evolutionary relation.

Since the differential of the evolutionary form is non-zero, and the
differential of the closed exterior form is equal to zero, the transition from
the evolutionary form to the closed exterior form is possible only with a
degenerate transformation, i.e., with a transformation that does not conserve
the differential.

Degenerate transformations can be done under additional conditions, which are
resulted from the implementation of any degrees of freedom. Vanishing of
functional expressions such as determinants, Jacobians, Poisson brackets,
residues, and others correspond to these additional conditions. Vanishing of
these functional expressions is a condition for the closedness of the dual
form describing the implemented integrable structure.

Additional conditions can be realized upon self-change of a non-identical
evolutionary relation. This can only happen discretely.

The conditions for the degenerate transformation define integrable structures
(pseudostructures): a characteristic (the determinant vanishes), singular
points (the Jacobian is zero), etc.

The degenerate transformation is realized as a transition from the original
tangent, non-integrable space to a cotangent, integrable manifold.

An example of the degenerate transformation is the Legendre transformation.
However, the Legendre transform is usually performed implicitly and does not
take into account the discrete nonequivalent transition from one space to
another. (An example of such a transition is the transition from a Lagrangian
manifold to a Hamiltonian one.)

### Realization of a closed inexact exterior form. Obtaining an identical
relation from a nonidentical.

When the condition of a degenerate transformation is fulfilled, the following
transition is realized

$d\omega^{p}\neq 0\to$ (degenerate transformation) $\to d_{\pi}\omega^{p}=0$,
$d_{\pi}{}^{*}\omega^{p}=0$

Condition $d_{\pi}{}^{*}\omega^{p}=0$ defines a closed dual form that
describes some integrable structure $\pi$ (a pseudostructure in its metric
properties). The condition $d_{\pi}\omega^{p}=0$ defines the closed inexact
(on an integrable structure) exterior form.

(The following should be emphasized here. The realization of a closed external
form, which is a differential and, therefore, an invarant, indicates that the
mathematical physics equations have invariant properties.)

On an integrable structure, from evolutionary relation (22) implies relation

$d\psi_{\pi}=\omega^{p}_{\pi}$ $None$

which turns out to be identical since the form $\omega^{p}_{\pi}$ is a
differential (interior).

Since the identity relation can be integrated, this indicates that the
equation of mathematical physics turns out to be integrable on the integrable
structure. I.e., the mathematical physics equation solution is a function. But
such an analytical solution is a discrete function, since it is determined
only on structure. This is the so-called generalized solution.

So, it turns out that mathematical physics equations have double solutions,
namely, on the original coordinate space and on the integrable structure.
Moreover, the solution on the original coordinate space is not a function. And
the solution on the integrable structure is a discrete function.

As shown, there is a connection between these solutions, which is implemented
discretely. When the conditions for the degenerate transformation are
satisfied, integrable structures are realized and a transition occurs from the
original tangent space with a solution that is not a function to the
integrable structures of the cotangent manifold with a generalized solution,
which is a discrete function.

### Physical meaning of double solutions.

It was shown that mathematical physics equations have double solutions: on a
non-integrable initial coordinate space and on integrable structures. It makes
physical sense. Inexact solutions describe the nonequilibrium state of the
medium under study. Exact solutions (which are discrete functions) describe
the local-equilibrium state of the medium.

This follows from the evolutionary relation $d\psi\,=\,\omega^{p}$ and is
connected with the fact that the evolutionary relation contains the functional
$\psi$, which characterizes the state of the medium.

The presence of the state functional differential indicates the presence of a
state function, which corresponds to the equilibrium state of the medium. But
there is a subtlety here. Since the evolutionary relation is not identical, it
is impossible to obtain the differential of the state functional $d\psi$ from
it. This indicates the absence of a function of state and means that the state
of the described medium is nonequilibrium. Inexact solutions describe such a
nonequilibrium state of the medium. Obviously, the internal forces that cause
non-equilibrium should be described by a commutator of the open evolutionary
form $\omega^{p}$.

From the identical relation $d\psi_{\pi}=\omega^{p}_{\pi}$, which is realized
from the evolutionary relation, one can obtain the differential $d\psi_{\pi}$
(interior, on an integrable structure). And this indicates the presence of a
state function and the transition of the medium to a locally equilibrium
state. (In this case, the general state of the medium remains non-
equilibrium.) The locally-equilibrium state of the medium is described by a
discrete function.

The transition of medium from the nonequilibrium state to the local-
equilibrium state means that an immeasurable quantity, which was described by
the evolutionary form commutator and acted as an internal force, becomes a
measurable quantity (potential, corresponding to the given medium). This
manifests itself as the emergence of any discrete structures and observable
formations, an example of which are waves, vortices, turbulent pulsations. It
should be emphasized once again that mathematical physics equations solutions
on integrable structures are related to solutions on the original coordinate
space. This must be taken into account when describing real physical processes
by mathematical physics equations.

(It should be noted that inconsistency of the conservation law equations
reflects the non-commutativity of conservation laws. It is with the non-
commutativity of conservation laws that the processes occurring in material
environments are associated.)

So, it turns out that mathematical physics equations have double solutions: on
a non-integrable initial coordinate space and on integrable structures.

How can you get these solutions?

Since mathematical physics equations on the original coordinate space have
solutions that are not functions, such solutions can only be obtained by
numerical methods.

And on integrable structures, solutions that are functions can be obtained
using analytical methods. Such solutions can also be obtained by numerical
methods. But in this case, the equations are modeled on the integrable
manifold, and not on the original space. I.e., other, nonequivalent,
coordinate systems are used. (It is seen that double solutions cannot be
obtained using a unified continuous simulation of mathematical physics
equations.)

As is known, at present there are both analytical and numerical methods for
solving these equations. In this case, one and the same phenomenon is
sometimes described both by numerical methods on the original coordinate
space, and by analytical methods on integrable spaces, such as phase
(covariant, cotangent) spaces (which are obtained from the integrability
conditions imposed on the equations).

But, in this case, the solutions of mathematical physics equations describing
the same phenomenon but obtained by different methods (numerical methods on
the original coordinate space or analytical methods on integral structures),
will be different. This does not mean that the solutions are wrong. The
solutions of mathematical physics equations, obtained by both numerical and
analytical methods, are correct, but in each case they describe only one of
the sides of the physical phenomenon. Therefore, in order to obtain a complete
description of a physical phenomenon, it is necessary to simultaneously use
both methods for solving equations of mathematical physics and take into
account the relationship between these approaches.

(It can be noted that in the continuum mechanics, describing the change in
quantities characterizing material media, numerical methods are usually used
in the initial coordinate space, and in the continuum physics, describing
conserved (under non-degenerate transformations) physical quantities or
objects, usually mathematical physics equations are solved by analytical
methods or numerical methods, but on integrable manifolds. This can be
explained by the fact that in continuum mechanics they are interested in
evolutionary processes described by numerical solutions, and in continuum
physics they are interested in conserved objects described by analytical
solutions.)

Investigation of the integrability of differential equations describing real
processes and phenomena showed that the differential equations integrability
is realized discretely in the presence of any degrees of freedom. At the same
time, unique hidden properties of mathematical physics equations are
manifested, such as invariance, double solutions, discrete transitions, and so
on, which makes it possible to describe evolutionary processes, the emergence
of physical structures, observed formations, and so on. It should be
emphasized once again that such results were obtained using skew-symmetric
evolutionary forms, which are defined on non-integrable manifolds and have an
unconventional mathematical apparatus containing nonidentical relations and
degenerate transformations.

Such properties of differential equations reveal the unique capabilities of
mathematical physics equations in describing physical processes and phenomena
that cannot be described within the framework of other mathematical
formalisms.

##  5 Unique hidden possibilities of the mathematical physics equations

In the author‘s published works, which are briefly presented below, the
properties and possibilities of some mathematical physics equations are given.
The features of the described phenomena are shown, which are due to the hidden
properties of the mathematical physics equations.

### Double Solutions of the Euler and Navier-Stokes Equations. Process of
Origination the Vorticity and Turbulence

The Euler and Navier-Stokes equations (which describe a gas-dynamic medium,
i.e., the ideal and viscous gas flow) are examples of mathematical physics
equations.

As is known, the proof of the existence and smoothness of the Navier-Stokes
equations solution was declared as a millennium problem.

But as follows from this article, and also as shown in [5], there is no smooth
analytical solution. The Euler and Navier-Stokes equations, as well as the
mathematical physics equations, have double solutions, namely, the solution on
the original coordinate space and solution on integrable structures. In this
case, the transition from the first solution to the second describes the
transition of a gas-dynamic medium from a nonequilibrium state to a local-
equilibrium state, which is accompanied by the emergence of vorticity (in an
ideal gas) or the emergence of turbulence (in a viscous gas).

This follows from the evolutionary relation, which is obtained from the Euler
and Navier-Stokes equations. The properties of the Euler and Navier-Stokes
equations can be found in [5].

It can be noted that the evolutionary relation obtained from the Euler and
Navier-Stokes equations can appear as follows [5]:

$ds\,=\,A_{\mu}d\xi^{\mu}$

Here the coefficients $A_{1}$ obtained from the energy conservation law
equation, respectively, are equal to $A_{1}=0$ for the ideal gas, and

$A_{1}\,=\,{1\over{\rho}}{{\partial}\over{\partial x_{i}}}\left(-{{q_{i}}\over
T}\right)\,-\,{{q_{i}}\over{\rho T}}\,{{\partial T}\over{\partial
x_{i}}}\,+{{\tau_{ki}}\over{\rho}}\,{{\partial u_{i}}\over{\partial x_{k}}}$

for the viscous gas. And the coefficients $A_{\nu 1}$ obtained from the
momentum conservation law equation are [6]:

$A_{\nu}={{\partial
h_{0}}\over{\partial\xi^{\nu}}}+{({u_{1}}^{2}+{u_{2}}^{2})}^{1/2}\zeta-
F_{\nu}+{{\partial U_{\nu}}\over{\partial t}}$

It should be emphasized here that the entropy $s$, which is included in the
evolutionary relation, depends on the space-time coordinates $\xi^{\mu}$, in
contrast to the thermodynamic entropy, which depends on thermodynamic
variables. It is this entropy (and not thermodynamic) that describes the gas-
dynamic medium state.

### Connection of the field theory equations with the mathematical physics
equations.

At the end of the 19th and the beginning of the 20th centuries, a situation
arose that Henri Poincare called the ”crisis of physics”. A lot of phenomena
and structures have accumulated that turned out to be impossible to describe
by the mathematical physics equations. Therefore, in different areas of
physics, new equations- the field theory equations were obtained (such as the
equations of Einstein, Maxwell, Schr?dinger, Dirac, Heisenberg). These
equations are based on the invariance and covariance properties, which are
necessary to describe measurable physical structures and observable phenomena,
and which, it was assumed, do not have the mathematical physics equations.

It turns out that the equations of mathematical physics have invariant
properties. But, as shown, such properties are hidden. They are realized
discretely and this is to the realization of the mathematical physics
equations integrability, which is described by the realization of closed skew-
symmetric exterior and dual forms that have invariant and covariant
properties. Invariant and covariant properties of mathematical physics
equations reveal a connection of the field theory equations with the
mathematical physics equations [7].

The evolutionary relation that is obtained from the mathematical physics
equations, has the properties of the field theory equations.

The work [7] shows the correspondence between the evolutionary relation and
the field theory equations, which indicates the connection of the field theory
equations with the of mathematical physics equations.

As already been mentioned, the evolutionary relation is a relation for
functionals such as the action functional, entropy, wave function, Lagrangian,
Einstein tensor, Poynting vector, etc., which are also functionals of field
theory [4].

The field theory equations also have the form of relations written in terms of
skew-symmetric forms or their tensor or differential analogs. So

\- the Einstein equation is a relation in skew-symmetric forms (or a tensor
relation);

\- the Maxwell equations have the form of tensor relations;

\- the Schrodinger equations have the form of relations expressed in terms of
derivatives and their analogs.

From the evolutionary relation closed inexact exterior forms can be obtained,
which are solutions to field theory equations.

Evolutionary relation of the mathematical physics equations unifies the field
theory equations, reveals their internal connection, reveals properties common
to all field theory equations. This can serve as an approach to a unified and
general field theory.

### Duality of conservation laws.

The mathematical physics equations reveal the conservation laws duality [8].

The hidden properties of the mathematical physics equations make it possible
to understand the features of conservation laws, which have different meanings
in thermodynamics, physics and mechanics.

In mechanics and continuous media physics the concept of “conservation laws”
relates to the conservation laws for energy, linear momentum, angular
momentum. These are conservation laws for material media. They are described
by differential equations.

In areas of physics related to the field theory and in the theoretical
mechanics “the conservation laws” is conservation laws, according to which the
conservative physical quantities or objects exist. Such conservation laws are
described by closed exterior and dual forms, which, as noted, are conserved
quantities or objects. Conservation laws for physical fields are such
conservation laws.

The first principle of thermodynamics is an example of evolutionary relation.
It is obtained from two conservation laws, namely, from the energy
conservation law and the linear momentum conservation law. The second
principle of thermodynamics, which determines the thermodynamic entropy, is an
example of the identity relation that is obtained from the evolutionary
relation (the first principle of thermodynamics) when implementing the
integrating factor: 1 / T, where T is the temperature.

The realization of closed exterior forms from the mathematical physics
equations, which consist of equations for energy, linear momentum, angular
momentum, reveals the connection between the conservation laws for physical
fields and the conservation laws for material media. This connection is
realized discretely in the evolutionary process [8]. Realization of closed
exterior forms describing the conservation laws for physical fields from the
mathematical physics equations describing material media, indicates that
physical fields are generated by material media.

It should be emphasized here that there is no exact law of conservation of
energy. There is no law of conservation of energy as a conserved quantity,
which is described by a closed external form and corresponds to the
conservation law for physical fields. But there is a balanced (differential)
law of conservation of energy, which is described by the differential equation
[8].

### Hamiltonian systems realization process.

The mathematical physics equations reveal the integrable Hamiltonian systems
realization process.

The hidden properties of the mathematical physics equations make it possible
to describe the process of Hamiltonian systems realization from differential
equations. This is described in the article [9]. However, the article [9]
describes not the properties of the Hamiltonian system and the Hamiltonian
function (to which many papers are devoted), but shows how Hamiltonian systems
are realized from differential equations, and describes the implementation
process.

In classical mechanics, the Hamiltonian system is realized from the Euler-
Lagrange equations as the Euler-Lagrange equation integrability [9]. It should
be noted here that the Hamiltonian system can be obtained from the Euler-
Lagrange equation if no additional conditions are imposed on it. In this case,
the Euler-Lagrange equation turns out to be non-integrable on the tangent
space. However, when some degrees of freedom are realized (when the
constraints locally become holonomic), the integrability condition (one of the
Hamiltonian system relations) is satisfied, and the Hamiltonian system for the
Hamiltonian function is realized from the Euler-Lagrange equation.

The Hamiltonian system is realized discretely. In this case, the transition
from a tangent manifold with the Euler-Lagrange equation to sections of a
cotangent manifold with a Hamiltonian system realized by the Legendre
transformation, which is a degenerate transformation (a transformation that
does not preserve the differential).

### The emergence of the physical structures and the observed formations.

The mathematical physics equations reveal the process of the emergence of
physical structures and the emergence of observed formations [10].

At a degenerate transformation from the evolutionary form, the closed inexact
exterior form and relevant dual form are realized, which form the differential
geometrical structure, namely, the pseudostructure with conserved quantity.
The differential-geometrical structure describes the physical structure, i.e.,
the structure on which the exact conservation law is fulfilled. The physical
fields and relevant manifolds are formed by such physical structures [10].
Massless particles, structures made up by eikonal surfaces and wave fronts,
and so on are examples of physical structures.

The emergence of the physical structures is accompanied by the emergence of
observable formations such as waves, vortices, fluctuations, turbulent
pulsations, and so on in the described media [10].

The description of process the physical structures emergence has features that
are not present in any of the existing mathematical formalisms. The physical
structures emergence is described by the transition from a tangent, non-
integrable, space to a cotangent, integrable manifold (i.e., from one spatial
object to another not equivalent spatial object). This can only be described
by skew-symmetric forms using the degenerate transform.

It can be noted here that there is a connection between a degenerate
transformation and a non-degenerate one. This can be shown on the Lagrangian
formalism example. The degenerate transformation is a transition from
coordinate space ($q_{j},\,\dot{q}_{j})$) to integrable structures cotangent
manifold ($q_{j},\,p_{j}$). And the nondegenerate transformation is a
transition in cotangent manifold from some structure ($q_{j},\,p_{j}$) to
another structure ($Q_{j},\,P_{j}$).

The degenerate transformations describe the process of emergence of physical
structures, whereas the nondegenerate transformations describe transitions
from one structure to another.

### Discrete transitions and quantum leaps.

The mathematical physics equations can describe discrete quantum transitions
[11].

It should be emphasized that discrete processes can only describe the
mathematical physics equations on which the integrability conditions are not
imposed. It is the realization of the integrability conditions, which occurs
discretely in the equations solving process, that reveals the hidden
properties of the mathematical physics equations, which make it possible to
describe discrete transitions, the emergence of various structures and
observed formations.

This also reveals the mechanism of occurrence of quantum leaps. In transition
from one structure to another, the conserved quantity described by a closed
exterior form and the pseudo structure described by the dual form, changes
discretely. The discrete change in the conserved quantity and pseudo structure
produces a quantum which is emerged in transition from one structure to
another. Evolutionary form commutator, formed at the time of the structure
emergence, determines this quantum characteristics.

It can be emphasized once again that discrete transitions are described by the
degenerate transformation, in which there is a transition from the original
coordinate space to integrable structures.

##  6 Conclusion

The unique hidden possibilities of the mathematical physics differential
equations were discovered when studying the mathematical physics equations
integrability. In this case, it was taken into account that the condition for
integrability is the consistency of the desired function derivatives (i.e.,
the presence of a differential) and the consistency of equations if the
mathematical physics equations are the system of equations. (These conditions
are usually not taken into account in solving the mathematical physics
equations.)

It was found that the mathematical physics equations integrability is realized
discretely in the equations solving process (and is not a constant property of
the mathematical physics equations). At the same time, hidden properties of
mathematical physics equations are revealed, which turn out to be
unconventional. Firstly, there is a transition from the tangent, non-
integrable space, on which the mathematical physics equations are defined, to
the integrable structures of the cotangent manifolds . Secondly, on integrable
structures, the solution of mathematical physics equations becomes a discrete
function. This is the so-called generalized solution. In this case, the
solution in the original coordinate space remains non-integrable, i.e., it is
not a function. It turns out that the mathematical physics equations have
double solutions.

Such properties of the mathematical physics equations, as shown, allow us to
describe discrete transitions, quantum leaps, the emergence of various
structures and observed formations, such as waves, vortices, turbulent
pulsations, and so on. Due to such possibilities, the mathematical physics
equations reveal the mechanism of many ongoing processes and phenomena. This
is demonstrated in the published works of the author, which are briefly
presented in this paper.

Such results were obtained by skew-symmetric differential forms, which are
differentials and differential expressions, which allows solving integrability
problems. At the same time, in addition to closed exterior forms that have
invariant properties, the skew-symmetric forms obtained by the author were
used, which are defined on non-integrable manifolds and have evolutionary
properties. The evolutionary forms theory contains elements such as non-
identical relations and degenerate transformations that are practically not
explicitly applied in the existing mathematical formalisms.

## References

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2010, http://arxiv.org/pdf/1007.4757.pdf

[2] Bott R., Tu L. W., Differential Forms in Algebraic Topology, Springer, NY,
1982.

[3] Tonnelat M.-A., Les principles de la theorie electromagnetique et la
relativite, Masson, Paris, 1959.

[4] Petrova L. Connection between functionals of the field-theory equations
and state functionals of the mathematical physics equations. Journal of
Physics: Conference Series, IOP Publishing ([Bristol, UK], England), 2018, Vol
1051, No. 012025, 1- 8.

[5] Petrova L.I. Double Solutions of the Euler and Navier-Stokes Equations.
Process of Origination the Vorticity and Turbulences. Fluid Mechanics, Science
Publishing Group, 2017, Mar. 21, 6-12.

[6] Clark J F., Machesney M. The Dynamics of Real Gases, Butterworths, London,
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[7] Petrova L. Evolutionary Relation of Mathematical Physics Equations
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1-10; https://doi.org/10.3390/axioms10020046

[8] Petrova L.I. Duality of conservation laws and their role in the processes
of emergence of physical structures and formations, Mathematics for
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No.9, 55-70.

[9] Petrova L, Qualitative Investigation of Hamiltonian Systems by Application
of Skew-Symmetric Differential Forms, Symmetry, MDPI (Basel, Switzerland),
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[10] Petrova L.I. Spontaneous emergence of physical structures and observable
formations: fluctuations, waves, turbulent pulsations and so on, Journal of
Applied Mathematics and Physics, Scientific Research Publishing (United
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[11] Petrova L. Discrete Quantum Transitions, Duality: Emergence of Physical
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