Evaluation of LLM prompts

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.

Question: What is the SymPy representation for the Latex expression $x^2 + y^2 = 1$ ?

Answer:

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.

Question: What is the SymPy representation for the Latex expression $x^2 + y^2 = 1$ ?

Answer:

from sympy import symbols, Eq

x, y = symbols('x y')
equation = Eq(x**2 + y**2, 1)

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: SymPy is an open-source Python library for symbolic computation. SymPy provides a computer algebra system. SymPy can convert Latex math to SymPy, and SymPy can render a mathematical expression as Latex.

Question: What is the SymPy representation for the Latex expression $x + y = 1$ ?

Answer:

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: Period (symbol T) is the interval of time between events. Frequency (symbol f) is the number of occurrences of a repeating event per unit of time

Specify the mathematical relation between period and frequency as an equation in Latex.

\[ f = \frac{1}{T} \]

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: Angular frequency (symbol \omega) is rate at which the object rotates. Period (symbol T) is the interval of time between events.

Specify the mathematical relation between period and angular frequency as an equation in Latex.

\[ \omega = \frac{2\pi}{T} \]

Context: arxiv is an open-access repository of electronic preprints

Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.

Cite three papers from arxiv that contain mathematical derivations with more than four steps.

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.

Question: What mathematical steps relate the Latex math expression $i x = log(y)$ and $\exp(i x) = y$ ?

Answer:

1. Take the exponential of both sides of the first equation: \(\exp(ix) = \exp(\log(y))\).
2. Simplify the right side using the property \(\exp(\log(a)) = a\).

So, the final relation is \(\exp(ix) = y\).

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.

Question: What is the derivative of $y = \cos(x) + i \sin(x)$ with respect to $x$ ?

Answer:

\[\frac{dy}{dx} = -\sin(x) + i \cos(x)\]

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.

Question: derive the wave function for a quantum particle in a 1D box

Answer:

\[-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\]

where \( \psi(x) \) is the wave function, \( m \) is the mass of the particle, \( V(x) \) is the potential energy, \( E \) is the total energy, and \( \hbar \) is the reduced Planck constant.

In the case of a particle in a 1D box with infinite potential walls, \( V(x) = 0 \) within the box, and \( V(x) = \infty \) outside the box. The solution involves finding the eigenfunctions and eigenvalues of this system, resulting in the quantized energy levels and corresponding wave functions:

\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \]

where \( n \) is the quantum number, \( L \) is the length of the box.

Context: Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in Lean that `(a=b) -> (a+2=b+2)`

Context: Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in Lean that `(a=b) -> (a+2=b+2)`

In Lean, you would typically use tactics to prove a theorem. The goal is to show that if `a = b`, then `a + 2 = b + 2`.

Here's a possible proof in Lean:

```lean
import data.real.basic

-- Assume a, b are real numbers
variables (a b : ℝ)

-- Assume the proposition a = b
variable h : a = b

-- To prove a + 2 = b + 2, add
-- 2 to both sides of the
-- equation a = b
example : a + 2 = b + 2 :=
begin
  -- Use the tactic rw (rewrite)
  -- to substitute a = b in
  -- the goal
  rw h,

  -- Use the tactic ring to
  -- simplify the goal using
  -- the properties of
  -- the real numbers
  ring,
end
```

Let me explain the steps:

1. `import data.real.basic`: This imports the basic properties of real numbers in Lean.
2. `variables (a b : ℝ)`: This declares `a` and `b` as real numbers.
3. `variable h : a = b`: This declares `h` as the hypothesis that `a = b`.
4. `example : a + 2 = b + 2 :=`: This sets up the goal to prove `a + 2 = b + 2`.
5. `rw h`: This uses the `rw` tactic to rewrite the goal using the hypothesis `a = b`.
6. `ring`: This uses the `ring` tactic to simplify the goal using the properties of the real numbers.

So, the proof in Lean is essentially showing that if `a = b`, then you can add 2 to both sides to get `a + 2 = b + 2`.

Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in Lean that `(a=b) -> (a+2=b+2)`

Here is the proof in **Lean 4**.

To work with Real numbers in Lean, you typically need **Mathlib**. The proof relies on the substitution property of equality (rewriting).

```lean
import Mathlib.Data.Real.Basic

-- We define the example with variables a and b as Reals (ℝ)
example (a b : ℝ) : (a = b) → (a + 2 = b + 2) := by
  -- 1. Introduce the hypothesis `a = b` into the context
  intro h
  -- 2. Rewrite `a` to `b` in the goal using hypothesis `h`
  rw [h]
  -- Lean automatically closes the goal because `b + 2 = b + 2` is reflexive
```

### Explanation of the Steps

1.  **`import Mathlib.Data.Real.Basic`**: This imports the definition of real numbers.
2.  **`example (a b : ℝ)`**: We declare variables `a` and `b` of type `ℝ`.
3.  **`intro h`**: Since the goal is an implication ($P \to Q$), we assume the premise $P$ (that $a=b$) and name it `h`.
4.  **`rw [h]`**: The `rw` (rewrite) tactic looks at `h` (which proves $a=b$) and replaces occurrences of $a$ with $b$ in the goal.
    *   The goal changes from `a + 2 = b + 2` to `b + 2 = b + 2`.
    *   Since equality is reflexive ($x = x$), Lean closes the proof automatically.

### Term-Mode Proof (Functional approach)

Since you mentioned Lean is based on the **Calculus of Constructions**, you might appreciate the "Term Mode" proof. This treats proofs as functions. Here, we use `congrArg`, which applies a function to both sides of an equality.

```lean
import Mathlib.Data.Real.Basic

example (a b : ℝ) : (a = b) → (a + 2 = b + 2) :=
  fun h => congrArg (fun x => x + 2) h
```

**Logic:**
*   `h` has type `a = b`.
*   We define a function $f(x) = x + 2$.
*   `congrArg` creates a proof that $f(a) = f(b)$ given $a = b$.

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Context: Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

Context: a derivation in mathematical Physics consists of a sequence of steps. Each step relates mathematical expressions to an inference rule. An expression is comprised of symbols and operators. An inference rule typically transforms input expressions into output expressions.

$i x = log(y)$ is a proposition. The variable `x` is Real valued and `y` is Real valued. The constant $i$ is the square root of one. Prove in Lean that `(i x = log(y)) -> (\exp(i x) = y)`

However, if the intent is to show a relationship between complex numbers and exponentiation, a correct proposition might be \(\exp(ix) = \cos(x) + i \sin(x)\). If this is what you intended, please let me know, and I can help you with a Lean proof for that.

Context: Lean is a proof assistant and programming language. Lean is based on the calculus of constructions with inductive types.

`i x = log(y)` is a proposition. The variable `x` is Real valued and `y` is Real valued. The constant $i$ is the square root of one. Prove in Lean that `(i x = log(y)) -> (\exp(i x) = y)`

```lean
import data.real.basic
import data.complex.basic

-- Assume Real numbers x and y
variables (x y : ℝ)

-- Assume the proposition
-- i x = log(y)
variable h : complex.I * x = log y

-- Prove that exp(i x) = y
example : complex.exp (complex.I * x) = y :=
begin
  -- Use the property
  -- exp(i x) = cos(x) + i * sin(x)
  rw complex.exp_eq_cos_add_sin,

  -- Substitute the given
  -- proposition i x = log(y)
  rw h,

  -- Simplify using trigonometric
  -- identities
  simp [real.cos, real.sin],

  -- The goal is now simplified
  -- to y = y, which is true
  refl,
end
```

This proof uses the fact that \(\exp(i x) = \cos(x) + i \sin(x)\) and then substitutes the given proposition \(i x = \log(y)\). The `rw` tactic is used for rewriting, and the `simp` tactic is used for simplifying trigonometric identities. The final goal is to show that \(y = y\), which is true, completing the proof.

**Derivation of Newton's Law of Universal Gravitation:**

1. **Start with the Law of Force between two masses:**
\[ F = ma \]
where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.

2. **Consider the gravitational force between two masses:**
\[ F = G \frac{m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the separation between their centers.

3. **Equate the gravitational force to the mass times gravitational acceleration:**
\[ ma = G \frac{m_1 m_2}{r^2} \]

4. **Cancel the mass \( m \) from both sides:**
\[ a = G \frac{m_1}{r^2} \]

5. **Recognize that \( a \) is the gravitational acceleration \( g \) near the Earth's surface:**
\[ g = G \frac{m_1}{r^2} \]

6. **For a mass \( m_2 \) near the Earth's surface, \( g \) becomes:**
\[ g = G \frac{m_{\text{Earth}}}{r^2} \]

7. **Express the gravitational acceleration in terms of mass and radius of the Earth:**
\[ g = \frac{GM_{\text{Earth}}}{r^2} \]

8. **Newton's Law of Universal Gravitation:**
\[ F = G \frac{m_1 m_2}{r^2} \]

This derivation shows how Newton's Law of Universal Gravitation is derived from the basic principles of force and gravitational acceleration.

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Question: What mathematical expressions are present in the following Latex?

```
\begin{equation}
a = b + c
\end{equation}
Assuming you know $b$, then
\begin{equation}
b*c = f
\end{equation}
```

1. \(a = b + c\) (Equation 1)
2. \(b \cdot c = f\) (Equation 2)

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Question: What mathematical expressions are present in the following Latex?

```
\begin{equation}
a = b + c
\end{equation}
Assuming you know $b$, then
\begin{equation}
b*c = f
\end{equation}
```

1. \(a = b + c\) (Equation 1)
2. \(b \cdot c = f\) (Equation 2)

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Question: What variables are present in the following Latex?

```
\begin{equation}
a = b + c
\end{equation}
Assuming you know $b$, then
\begin{equation}
b*c = f
\end{equation}
```

Answer:

1. \(a\)
2. \(b\)
3. \(c\)
4. \(f\)

Context: Latex is software for document preparation to generate PDF files. Mathematical equations can be expressed in Latex using markup syntax.

Question: What mathematical expressions are present in the following Latex?

```
Several notable properties of condensed
 matter phases are defined by fundamental
 physical constants. The Bohr radius gives
 a characteristic scale of interatomic
 distance on the order of the Angstrom,
 in terms of electron mass $m_e$, charge
 $e$, and Planck constant $\hbar$. These
 same fundamental constants enter the
 Rydberg energy, setting the scale of a
 characteristic bonding energy in condensed
 phases and chemical compounds
 \cite{ashcroft}.

Among the fundamental constants, those
 that are {\it dimensionless} and do
 not depend on the choice of units, play
  a special role in physics \cite{barrow}.
  Two important dimensionless constants
  are the fine structure constant $\alpha$
  and the proton-to-electron mass ratio,
  $\frac{m_p}{m_e}$. The finely-tuned
  values of $\alpha$ and $\frac{m_p}{m_e}$,
  and the balance between them, governs
  nuclear reactions such as proton decay
  and nuclear synthesis in stars, leading
  to the creation of the essential
  biochemical elements, including carbon.
  This balance provides a narrow
  ``habitable zone'' in the
  ($\alpha$,$\frac{m_p}{m_e}$) space
  where stars and planets can form and
  life-supporting molecular structures
  can emerge \cite{barrow}.

We show that a simple combination of
 $\alpha$ and $\frac{m_p}{m_e}$
 results in another dimensionless
 quantity which has an unexpected
 and specific implication for a key
 property of condensed phases, the
 speed at which waves travel in solids
 and liquids, or the speed of sound,
 $v$. We find that this combination
 provides an upper bound for $v$, $v_u$, as
\begin{equation}
\frac{v_u}{c}=\alpha\left(\frac{m_e}{2m_p}
\right)^{\frac{1}{2}},
\label{v0}
\end{equation}
\noindent where $c$ is the speed
of light in vacuum.

We support this result with a large
set of experimental data for different
systems, and the first principles
modelling of atomic hydrogen.

Identifying and understanding bounds
 on physical properties is important
 from the point of view of fundamental
 physics, predictions for theory and
 experiment, as well as searching for
 and rationalizing universal behavior
 (see, e.g., \cite{kss,zaanen3,hartnoll,
zaanen2,spin,behnia,zaanen1,behnia1,
hartnoll1}). Properties for which
bounds were recently discussed include
viscosity and diffusivity. The
proposed {\it lower} bounds for
these two properties feature in a
range of areas including, for example,
strongly-interacting field theories,
quark-gluon plasmas, holographic
duality, electron diffusion, transport
properties in metals and superconductors,
and spin transport in Fermi gases
\cite{kss,zaanen3,hartnoll,zaanen2,
  spin,behnia,zaanen1,behnia1,hartnoll1}.
  Recently, two of us found a lower
  bound for the kinematic viscosity of
  liquids set by fundamental physical
  constants \cite{sciadv}. Here, we
  propose a new, {\it upper}, bound
  for the speed of sound in condensed
  matter phases in terms of
  fundamental constants.

Apart from setting the speed of elastic
 interactions in solids, $v$ is related
 to elasticity, hardness and affects
 important low-temperature
 thermodynamic properties such as
 energy, entropy and heat capacity
 \cite{landau}. As discussed below,
 the upper bound of $v$ sets the smallest
possible entropy and heat capacity
at a given temperature.

In solids, $v$ depends on elastic
properties and density. These strongly
 depend on the bonding type and
 structure which are inter-dependent
 \cite{phillips}. As a result, it
 was not thought that $v$ can be
 predicted analytically without
 simulations, contrary to other
 properties such as energy or heat
 capacity which are universal in the
 classical harmonic approximation
 \cite{landau}. In view of this,
 representing the upper bound of
 $v$ in terms of fundamental constants
 is notable.
```

1. \(\alpha\)
2. \(\frac{m_p}{m_e}\)
3. \(\frac{v_u}{c} = \alpha \left(\frac{m_e}{2m_p}\right)^{\frac{1}{2}}\) (Equation \(v_u\))
4. \(c\) (speed of light in vacuum)