ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Barotropic Kelvin and Bernoulli with exact hypotheses — formal worked solution

Example

Let $I = R$ , $Ω = R^{3}$ , $P (q) = q$ for $q > 0$ , $ρ_{r e f} = 1$ , $ρ (t, x) = 1$ , $V = 0$ , and $u (t, x) = (1, 0, 0)$ . These fields solve the barotropic Euler system. The enthalpy is $h (q) = \log q$ . Define the Bernoulli quantity by $B = ∣ u ∣^{2} / 2 + h (ρ) + V$ ; here it is the constant $1 / 2$ . The closed loop $X (t, s) = (\cos (2 π s) + t, \sin (2 π s), 0)$ is material and has circulation $0$ for every $t$ .

Example references

Barotropic Kelvin and Bernoulli with exact hypotheses

Facts & Assumptions

Given: Use the classical fields, positive density, equation of state, potential and closed material loop in [A1].

[A1]

$I = R$ , $Ω = R^{3}$ , $P (q) = q$ , $ρ_{r e f} = 1$ , $ρ = 1$ , $V = 0$ , $u = (1, 0, 0)$ , and $X (t, s) = (\cos (2 π s) + t, \sin (2 π s), 0)$ for $0 \leq s \leq 1$ .

[F1]

Barotropic Kelvin and Bernoulli with exact hypotheses: for classical positive-density barotropic Euler with conservative $V$ , $h (q) = \int_{ρ_{r e f}}^{q} P^{'} (r) / r d r$ ; steady Bernoulli is constant on streamlines, and a closed $C^{1}$ material loop has constant circulation under the stated mixed-derivative condition.

Verification

1.1givenA1F1algebra

The fields have the required smoothness and positive density. Since $ρ$ is constant, $ρ_{t} + div (ρ u) = 0$ ; also $u_{t} = 0$ , $(u \cdot \nabla) u = 0$ , $\nabla (P (ρ)) = 0$ , and $\nabla V = 0$ . Thus both barotropic Euler equations hold pointwise. The enthalpy formula in [F1] gives $h (q) = \int_{1}^{q} d r / r = \log q$ , so $h (1) = 0$ and $∣ u ∣^{2} / 2 + h (1) + V = 1 / 2$ .

2.1givenA1F1algebra∎

The loop is closed because its endpoint values at $s = 0, 1$ agree, and $X_{t} = (1, 0, 0) = u (t, X)$ ; its spatial and mixed derivatives are continuous. Its tangent is $X_{s} = (- 2 π \sin (2 π s), 2 π \cos (2 π s), 0)$ , so $Γ (t) = \int_{0}^{1} u \cdot X_{s} d s = - 2 π \int_{0}^{1} \sin (2 π s) d s = 0$ . This verifies the theorem's circulation conclusion directly.

Depends on

Barotropic Kelvin and Bernoulli with exact hypotheses