Reflexivity and Planck's form of the zeroth law give an equivalence relation

Proved

ZerothLawThermo.equivalence_of_refl_zerothLawPlanck

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

equivalence-relationthermodynamicszeroth-law

Let $S$ be a type of thermodynamic systems and $\mathrm{TE}$ a binary relation on $S$ . Assume

$\mathrm{TE}(A,A)$ for all $A\in S$ ;
Planck's form of the zeroth law: if a body $C$ is in thermal equilibrium with two bodies $A$ and $B$ , i.e. $\mathrm{TE}(C,A)$ and $\mathrm{TE}(C,B)$ , then $\mathrm{TE}(A,B)$ .

Then $\mathrm{TE}$ is an equivalence relation on $S$ .

This shows that Planck's (left-Euclidean) formulation of the law serves the same purpose as the usual one.

Preamble

import Definitions.Def_ZerothLawThermo_Defs
import Mathlib

open ZerothLawThermo

Formal statement

theorem ZerothLawThermo.equivalence_of_refl_zerothLawPlanck {S : Type*} (TE : S → S → Prop)
    (hrefl : ∀ A : S, TE A A) (hzero : ZerothLawPlanck TE) :
    Equivalence TE := by sorry

Source

Wikipedia, "Zeroth law of thermodynamics", revision oldid=1328026962, https://en.wikipedia.org/w/index.php?title=Zeroth_law_of_thermodynamics&oldid=1328026962; Section "Equivalence relation": Planck's statement "If a body C, be in thermal equilibrium with two other bodies, A and B, then A and B are in thermal equilibrium with one another" (ref. 8), followed by "This statement asserts that thermal equilibrium is a left-Euclidean relation ... If we also define that every thermodynamic system is in thermal equilibrium with itself, then thermal equilibrium is also a reflexive relation. Binary relations that are both reflexive and Euclidean are equivalence relations."

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Disclosure — this read-back is NOT blind. It was written by the same agent (Aristotle, by Harmonic) that drafted these Lean statements, with full knowledge of the source text and of the intended meaning. It is therefore not independent testimony and must not be mistaken for an independent audit; reviewers should check the Lean code against the source themselves.

For an arbitrary type $S$ (any universe, possibly empty) and an arbitrary binary relation $\mathrm{TE}$ on $S$ , assume:

$\mathrm{TE}(A,A)$ for every $A\in S$ ;
ZerothLawPlanck TE, i.e. for all $A,B,C\in S$ , $\mathrm{TE}(C,A)$ and $\mathrm{TE}(C,B)$ together imply $\mathrm{TE}(A,B)$ .

Conclusion: $\mathrm{TE}$ is an equivalence relation (reflexive, symmetric and transitive, in the same sense as for the right-Euclidean milestone). The hypotheses are jointly satisfiable, e.g. by equality.

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

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