No heat is exchanged in an adiabatic process

Proved

FirstLawThermo.heat_eq_zero_of_adiabatic

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

first-lawinternal-energythermodynamics

Let $\mathcal P$ be a set of processes of a closed system with state set $\sigma$ , and let $U$ be an internal energy for $\mathcal P$ . For every adiabatic process $p\in\mathcal P$ , the heat supplied vanishes:

Q_U(p)=0.

This is the consistency of the residual definition of heat with the meaning of "adiabatic": in an adiabatic process there is transfer of energy as work but not as heat.

Preamble

import Definitions.Def_FirstLawThermo_Defs
import Mathlib

open FirstLawThermo

Formal statement

theorem FirstLawThermo.heat_eq_zero_of_adiabatic {σ : Type*} (procs : Set (Process σ))
    (U : σ → ℝ) (hU : IsInternalEnergy procs U) (p : Process σ) (hp : p ∈ procs)
    (had : p.adiabatic) :
    heat U p = 0 := by sorry

Source

Wikipedia, "First law of thermodynamics", revision oldid=1366986255, https://en.wikipedia.org/w/index.php?title=First_law_of_thermodynamics&oldid=1366986255; Section "Evidence for the first law of thermodynamics for closed systems": adiabatic processes are those "in which there is no transfer as heat"; subsection "Adiabatic processes": "In an adiabatic process, there is transfer of energy as work but not as heat."

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic), the drafting agent itself; non-blind read-back

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 $\sigma$ , an arbitrary set $\mathcal P$ of processes, a function $U:\sigma\to\mathbb R$ satisfying $U(q_{\mathrm{finish}})-U(q_{\mathrm{start}})=-W(q)$ for every $q\in\mathcal P$ whose adiabatic proposition holds, and a process $p\in\mathcal P$ whose adiabatic proposition holds:

U(p_{\mathrm{finish}})-U(p_{\mathrm{start}})+W(p)=0.

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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