Tuesday, December 19, 2023

LLM that includes the concept of inference rules

Question: What's the difference between a plain old search engine and an LLM+RAG?
Answer: LLM+RAG provides an experience like semantic search capability plus synthesis but without the need for semantic tagging on the front-end or the back-end. 
[https://www.sbert.net/examples/applications/semantic-search/README.html#semantic-search]

Relevance to the Physics Derivation Graph: add the following to an existing large language model (LLM)

  • the list of inference rules for the Physics Derivation Graph
  • examples of Latex-to-Sage conversion
  • example Lean4 proofs
"fine tuning" versus "context provision"

How is "context provision" different from RAG?

What's the difference between a transformer and a model?

specifically



output of help for llama.cpp

docker run -it --rm  -v `pwd`:/scratch llama-cpp-with-mistral-7b-v0.1.q6_k:2023-12-22 /bin/bash 
root@dc98ac4a23d5:/opt/llama.cpp# ./main -h

usage: ./main [options]

options:
  -h, --help            show this help message and exit
      --version         show version and build info
  -i, --interactive     run in interactive mode
  --interactive-first   run in interactive mode and wait for input right away
  -ins, --instruct      run in instruction mode (use with Alpaca models)
  -cml, --chatml        run in chatml mode (use with ChatML-compatible models)
  --multiline-input     allows you to write or paste multiple lines without ending each in '\'
  -r PROMPT, --reverse-prompt PROMPT
                        halt generation at PROMPT, return control in interactive mode
                        (can be specified more than once for multiple prompts).
  --color               colorise output to distinguish prompt and user input from generations
  -s SEED, --seed SEED  RNG seed (default: -1, use random seed for < 0)
  -t N, --threads N     number of threads to use during generation (default: 20)
  -tb N, --threads-batch N
                        number of threads to use during batch and prompt processing (default: same as --threads)
  -p PROMPT, --prompt PROMPT
                        prompt to start generation with (default: empty)
  -e, --escape          process prompt escapes sequences (\n, \r, \t, \', \", \\)
  --prompt-cache FNAME  file to cache prompt state for faster startup (default: none)
  --prompt-cache-all    if specified, saves user input and generations to cache as well.
                        not supported with --interactive or other interactive options
  --prompt-cache-ro     if specified, uses the prompt cache but does not update it.
  --random-prompt       start with a randomized prompt.
  --in-prefix-bos       prefix BOS to user inputs, preceding the `--in-prefix` string
  --in-prefix STRING    string to prefix user inputs with (default: empty)
  --in-suffix STRING    string to suffix after user inputs with (default: empty)
  -f FNAME, --file FNAME
                        prompt file to start generation.
  -n N, --n-predict N   number of tokens to predict (default: -1, -1 = infinity, -2 = until context filled)
  -c N, --ctx-size N    size of the prompt context (default: 512, 0 = loaded from model)
  -b N, --batch-size N  batch size for prompt processing (default: 512)
  --samplers            samplers that will be used for generation in the order, separated by ';', for example: "top_k;tfs;typical;top_p;min_p;temp"
  --sampling-seq        simplified sequence for samplers that will be used (default: kfypmt)
  --top-k N             top-k sampling (default: 40, 0 = disabled)
  --top-p N             top-p sampling (default: 0.9, 1.0 = disabled)
  --min-p N             min-p sampling (default: 0.1, 0.0 = disabled)
  --tfs N               tail free sampling, parameter z (default: 1.0, 1.0 = disabled)
  --typical N           locally typical sampling, parameter p (default: 1.0, 1.0 = disabled)
  --repeat-last-n N     last n tokens to consider for penalize (default: 64, 0 = disabled, -1 = ctx_size)
  --repeat-penalty N    penalize repeat sequence of tokens (default: 1.1, 1.0 = disabled)
  --presence-penalty N  repeat alpha presence penalty (default: 0.0, 0.0 = disabled)
  --frequency-penalty N repeat alpha frequency penalty (default: 0.0, 0.0 = disabled)
  --mirostat N          use Mirostat sampling.
                        Top K, Nucleus, Tail Free and Locally Typical samplers are ignored if used.
                        (default: 0, 0 = disabled, 1 = Mirostat, 2 = Mirostat 2.0)
  --mirostat-lr N       Mirostat learning rate, parameter eta (default: 0.1)
  --mirostat-ent N      Mirostat target entropy, parameter tau (default: 5.0)
  -l TOKEN_ID(+/-)BIAS, --logit-bias TOKEN_ID(+/-)BIAS
                        modifies the likelihood of token appearing in the completion,
                        i.e. `--logit-bias 15043+1` to increase likelihood of token ' Hello',
                        or `--logit-bias 15043-1` to decrease likelihood of token ' Hello'
  --grammar GRAMMAR     BNF-like grammar to constrain generations (see samples in grammars/ dir)
  --grammar-file FNAME  file to read grammar from
  --cfg-negative-prompt PROMPT
                        negative prompt to use for guidance. (default: empty)
  --cfg-negative-prompt-file FNAME
                        negative prompt file to use for guidance. (default: empty)
  --cfg-scale N         strength of guidance (default: 1.000000, 1.0 = disable)
  --rope-scaling {none,linear,yarn}
                        RoPE frequency scaling method, defaults to linear unless specified by the model
  --rope-scale N        RoPE context scaling factor, expands context by a factor of N
  --rope-freq-base N    RoPE base frequency, used by NTK-aware scaling (default: loaded from model)
  --rope-freq-scale N   RoPE frequency scaling factor, expands context by a factor of 1/N
  --yarn-orig-ctx N     YaRN: original context size of model (default: 0 = model training context size)
  --yarn-ext-factor N   YaRN: extrapolation mix factor (default: 1.0, 0.0 = full interpolation)
  --yarn-attn-factor N  YaRN: scale sqrt(t) or attention magnitude (default: 1.0)
  --yarn-beta-slow N    YaRN: high correction dim or alpha (default: 1.0)
  --yarn-beta-fast N    YaRN: low correction dim or beta (default: 32.0)
  --ignore-eos          ignore end of stream token and continue generating (implies --logit-bias 2-inf)
  --no-penalize-nl      do not penalize newline token
  --temp N              temperature (default: 0.8)
  --logits-all          return logits for all tokens in the batch (default: disabled)
  --hellaswag           compute HellaSwag score over random tasks from datafile supplied with -f
  --hellaswag-tasks N   number of tasks to use when computing the HellaSwag score (default: 400)
  --keep N              number of tokens to keep from the initial prompt (default: 0, -1 = all)
  --draft N             number of tokens to draft for speculative decoding (default: 8)
  --chunks N            max number of chunks to process (default: -1, -1 = all)
  -np N, --parallel N   number of parallel sequences to decode (default: 1)
  -ns N, --sequences N  number of sequences to decode (default: 1)
  -pa N, --p-accept N   speculative decoding accept probability (default: 0.5)
  -ps N, --p-split N    speculative decoding split probability (default: 0.1)
  -cb, --cont-batching  enable continuous batching (a.k.a dynamic batching) (default: disabled)
  --mmproj MMPROJ_FILE  path to a multimodal projector file for LLaVA. see examples/llava/README.md
  --image IMAGE_FILE    path to an image file. use with multimodal models
  --mlock               force system to keep model in RAM rather than swapping or compressing
  --no-mmap             do not memory-map model (slower load but may reduce pageouts if not using mlock)
  --numa                attempt optimizations that help on some NUMA systems
                        if run without this previously, it is recommended to drop the system page cache before using this
                        see https://github.com/ggerganov/llama.cpp/issues/1437
  --verbose-prompt      print prompt before generation
  -dkvc, --dump-kv-cache
                        verbose print of the KV cache
  -nkvo, --no-kv-offload
                        disable KV offload
  -ctk TYPE, --cache-type-k TYPE
                        KV cache data type for K (default: f16)
  -ctv TYPE, --cache-type-v TYPE
                        KV cache data type for V (default: f16)
  --simple-io           use basic IO for better compatibility in subprocesses and limited consoles
  --lora FNAME          apply LoRA adapter (implies --no-mmap)
  --lora-scaled FNAME S apply LoRA adapter with user defined scaling S (implies --no-mmap)
  --lora-base FNAME     optional model to use as a base for the layers modified by the LoRA adapter
  -m FNAME, --model FNAME
                        model path (default: models/7B/ggml-model-f16.gguf)
  -md FNAME, --model-draft FNAME
                        draft model for speculative decoding
  -ld LOGDIR, --logdir LOGDIR
                        path under which to save YAML logs (no logging if unset)
  --override-kv KEY=TYPE:VALUE
                        advanced option to override model metadata by key. may be specified multiple times.
                        types: int, float, bool. example: --override-kv tokenizer.ggml.add_bos_token=bool:false

log options:
  --log-test            Run simple logging test
  --log-disable         Disable trace logs
  --log-enable          Enable trace logs
  --log-file            Specify a log filename (without extension)
  --log-new             Create a separate new log file on start. Each log file will have unique name: "<name>.<ID>.log"
  --log-append          Don't truncate the old log file.

Saturday, December 16, 2023

example interaction with ChatGPT 3.5 to produce a Lean proof

In a new conversation I provided the following prompt:
prompt

Answer the question based on the context below. Let's work this out in a step by step way to be sure we have the right answer.

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

Context: There are multiple versions of Lean. This question is about Lean 4

`a=b` is a proposition. The variable `a` is Real valued and `b` is Real valued. Prove in version 4 of Lean that `(a=b) -> (a+2=b+2)`

ChatGTP 3.5 wrote in response
ChatGPT 3.5 response

Certainly! To prove the statement `(a = b) -> (a + 2 = b + 2)` in version 4 of Lean, you can follow these steps:

1. **Introduction of Variables**: Begin by introducing the variables and assumptions using the `variables` keyword. In this case, you have the variables `a` and `b` both being Real valued.

```lean
variables {a b : ℝ}
```

2. **Assumption**: State the assumption that `(a = b)`.

```lean
example (h : a = b) : a + 2 = b + 2 :=
```

3. **Transformation**: Use the assumption to transform the left-hand side of the conclusion `(a + 2)` to `(b + 2)` using the property of equality.

```lean
calc
   a + 2 = b + 2 : by rw h
```

Here, `calc` is used to chain together a sequence of expressions, and `rw h` applies the assumption `h`.

4. **QED (Quod Erat Demonstrandum)**: Conclude the proof.

```lean
   example (h : a = b) : a + 2 = b + 2 :=
   begin
     calc
       a + 2 = b + 2 : by rw h
   end
```

This step-by-step approach uses the Lean proof style to demonstrate the validity of the given proposition. The `calc` tactic is particularly useful for chaining together equalities.

On my computer I created a file "second_chatGPT_attempt.lean" and wrote
variables {a b : ℝ}

example (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end
and then ran
docker run -it --rm -v `pwd`:/scratch --workdir /scratch lean3onubuntu lean --run second_chatGPT_attempt.lean
second_chatGPT_attempt.lean:2:0: error: unexpected identifier; expected command
second_chatGPT_attempt.lean:5:0: error: unknown identifier 'begin'
second_chatGPT_attempt.lean:6:2: error: unexpected token 'calc'; expected command
second_chatGPT_attempt.lean:8:0: error: invalid 'end', insufficient scopes
I then told ChatGPT
prompt

When I run the script

```lean
variables {a b : ℝ}

example (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end
```

Lean 4 returns an error message: `error: unexpected identifier; expected command`

What should I do to fix the error?

ChatGPT 3.5 response

Apologies for the oversight. It seems I made an error in my previous responses. In Lean, you should use `theorem` or `example` as part of the proof declaration. Here's the corrected version:

```lean
variables {a b : ℝ}

theorem add_two_eq_add_two (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end
```

In this corrected version, I replaced `example` with `theorem`. You can use `example` as well, but `theorem` is more appropriate when proving statements. This should resolve the error you encountered.

I then created a second Lean script, third_chatGPT_attempt.lean, and wrote
variables {a b : ℝ}

theorem add_two_eq_add_two (h : a = b) : a + 2 = b + 2 :=
begin
  calc
    a + 2 = b + 2 : by rw h
end
which got the same error as before when I attempted to run the script using Lean 4:
docker run -it --rm -v `pwd`:/scratch --workdir /scratch lean3onubuntu lean --run third_chatGPT_attempt.lean 
third_chatGPT_attempt.lean:1:0: error: unexpected identifier; expected command
third_chatGPT_attempt.lean:4:0: error: unknown identifier 'begin'
third_chatGPT_attempt.lean:5:2: error: unexpected token 'calc'; expected command
third_chatGPT_attempt.lean:7:0: error: invalid 'end', insufficient scopes
  

LLM prompts for the Physics Derivation Graph

Posing a prompt that gets a useful result currently requires some consideration. Below are some possible tasks for LLMs, along with additional context for the LLM.

Convert Latex to SymPy

How to improve chances of success:

Here's my prompt:

Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the 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:

Right response: sympy.Eq(x**2 + y**2, 1)


As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash" and "Gemini 2.0 Flash Thinking Experimental" each return

"Unsure about answer."

I revised my question to

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

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:

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

1. **Identify the core task:** The question asks for the 
SymPy representation of a given LaTeX equation.

2. **Recognize the LaTeX elements:**  The LaTeX expression
`$x^2 + y^2 = 1$` contains the following elements:
    * Variables: `x`, `y`
    * Exponentiation: `^2`
    * Addition: `+`
    * Equality: `=`
    * Constant: `1`

3. **Consider the SymPy equivalents:** Think about how 
     each LaTeX element maps to SymPy:
    * Variables:  In SymPy, you need to declare symbols. The 
    common way to do this is using `Symbol('x')` and `Symbol('y')`.
    * Exponentiation:  SymPy uses `**` for exponentiation 
    (like standard Python). So `x^2` becomes `x**2`.
    * Addition:  Standard Python `+` works for addition in SymPy.
    * Equality:  A crucial point. In SymPy, the `=` operator is for 
    assignment. For expressing an *equation*, you need to use 
    `Eq()`. So `x^2 + y^2 = 1` becomes `Eq(x**2 + y**2, 1)`.
    * Constant: The integer `1` is represented as `1` in SymPy.

4. **Assemble the SymPy expression:** Combine the SymPy equivalents:
    * First, declare the symbols: `x = Symbol('x')`, `y = Symbol('y')`
    * Then, construct the equation using `Eq()`: `Eq(x**2 + y**2, 1)`

5. **Present the answer:**  Provide the complete SymPy 
code snippet to represent the equation.

6. **Review and refine:** Double-check if the SymPy representation 
accurately captures the meaning of the LaTeX expression. Ensure 
the use of `Eq()` for equality and the correct syntax for 
symbols and exponentiation.

"Gemini 2.0 Flash Thinking Experimental" answer

To find the SymPy representation of the LaTeX expression $x^2 + y^2 = 1$, 
you need to translate the mathematical symbols 
and structure into their SymPy equivalents.

Here's how you can do it:

Variables: In SymPy, you need to declare symbols before using 
them. So, x and y will be represented as Symbol('x') and Symbol('y').

Exponentiation: The ^ in LaTeX corresponds to the ** operator 
in Python/SymPy. So, x^2 becomes x**2 and y^2 becomes y**2.

Addition: The + in LaTeX is the same as the + operator in Python/SymPy.

Equality: The = in LaTeX represents mathematical equality. 
In SymPy, you represent an equation using the Eq() function.

Therefore, the SymPy representation of $x^2 + y^2 = 1$ is:

from sympy import symbols, Eq

x, y = symbols('x y')
equation = Eq(x**2 + y**2, 1)
content_copy
download
Use code with caution.
Python

So the answer is:

Eq(x**2 + y**2, 1)

Next prompt to explore:

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 $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$ ?

Answer:

Right response:Eq(p_electron, p1 - p2)

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

Let's break down the thought process for converting the given LaTeX expression to its SymPy representation.

1. **Identify the core components of the LaTeX expression:** The expression is $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$. This involves:
    * Vectors: $\vec{p}_{electron}$, $\vec{p}_{1}$, $\vec{p}_{2}$
    * Subscripts: `electron`, `1`, `2`
    * Equality: `=`
    * Subtraction: `-`

2. **Consider how SymPy handles different mathematical entities:**
    * **Variables:**  In SymPy, we need to declare symbols. The base names are `p`.
    * **Vectors:** SymPy has a `vector` module. We can represent vectors using `CoordSys3D` and vector components, or simply as vector symbols. For simpler representations, we might just treat the vectors as symbols themselves, making it easier to work with them symbolically without focusing on their components. Since the context doesn't specify dimensionality or require component-wise operations, treating the vectors as distinct symbolic entities seems like a good starting point.
    * **Subscripts:** LaTeX subscripts are often used for indexing or distinguishing variables. In SymPy, we can't directly have subscripts in variable names. We can handle this in a few ways:
        * Use separate variable names: `p_electron`, `p1`, `p2`. This is the most straightforward and Pythonic approach.
        * Use indexed symbols (from `sympy.IndexedBase` and `sympy.Idx`). This is more suitable if you are working with sequences or arrays of variables, and you need to manipulate the indices. For this specific problem, it's likely overkill.
    * **Equality:** The `=` in LaTeX translates to the `Eq()` function in SymPy to represent an equation.
    * **Subtraction:** The `-` operator in LaTeX is the same as the `-` operator in SymPy.

3. **Formulate the SymPy representation:** Based on the above analysis, a natural way to represent the expression is to use distinct symbols for each vector:

    * `\vec{p}_{electron}`  -> `p_electron`
    * `\vec{p}_{1}` -> `p1`
    * `\vec{p}_{2}` -> `p2`

    Now, assemble the equation using `Eq()`:

    `Eq(p_electron, p1 - p2)`

4. **Include necessary imports:**  To use `symbols` and `Eq`, we need to import them from the `sympy` library.

    `from sympy import symbols, Eq`

5. **Declare the symbols:** Before using the symbols, they need to be declared using `symbols()`.

    `p_electron, p1, p2 = symbols('p_electron p1 p2')`

6. **Combine the parts:** Put everything together to form the complete SymPy code:

    ```python
    from sympy import symbols, Eq

    p_electron, p1, p2 = symbols('p_electron p1 p2')
    equation = Eq(p_electron, p1 - p2)
    ```

7. **Consider alternative representations (and why the chosen one is suitable):**

    * **Using `IndexedBase`:**  We could represent the vectors using indexed symbols, like `p = IndexedBase('p')`, and then use `p['electron']`, `p[1]`, `p[2]`. However, the subscript "electron" isn't a numerical index, making this less direct. While possible, it adds complexity for this specific problem.
    * **Using explicit vector objects:**  SymPy's `vector` module allows defining coordinate systems and vectors. This would be necessary if we were performing vector operations like dot products or cross products and needed to work with components. However, the given equation is a simple symbolic equation between vectors, so treating them as abstract symbols is sufficient.

8. **Final check:** The SymPy representation `Eq(p_electron, p1 - p2)` accurately reflects the mathematical structure of the LaTeX expression. It uses appropriate SymPy constructs for equality, subtraction, and symbolic variables.

"Gemini 2.0 Flash Thinking Experimental" answer

To represent the LaTeX expression $\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$ in SymPy, we need to consider how to represent vectors and subscripts. Since we are dealing with symbolic manipulation and not necessarily with the components of the vectors, we can treat the vectors as distinct symbolic entities.

Here's how you can represent it in SymPy:

```python
from sympy import symbols, Eq

# Represent the vectors as symbolic variables
p_electron, p1, p2 = symbols('p_electron p1 p2')

# Form the SymPy equation
equation = Eq(p_electron, p1 - p2)
```

Therefore, the SymPy representation is:

```python
Eq(p_electron, p1 - p2)
```

**Explanation:**

* **`symbols('p_electron p1 p2')`**: We use `symbols` to create 
symbolic variables representing the vectors. While LaTeX 
uses $\vec{p}$ notation and subscripts, in SymPy, we can use 
more descriptive variable names like `p_electron`, `p1`, and `p2`. 
This maintains the distinction between the different momentum vectors.
* **`Eq(p_electron, p1 - p2)`**: The `Eq()` function from SymPy 
is used to represent an equation. The first argument is the left-hand 
side, and the second argument is the right-hand side. The 
subtraction `-` operator works as expected for symbolic variables.

**Alternative (Less Common for this simple case):**

You could potentially use `IndexedBase` if you were dealing with 
a more general set of vectors and wanted to use numerical indices:

```python
from sympy import symbols, Eq, IndexedBase, Idx

p = IndexedBase('p')
i, j = symbols('i j', integer=True)  # Not really needed here but for demonstration
electron = symbols('electron') # Could use a string or symbol for 'electron'

# This becomes less direct as 'electron' isn't a typical index
# equation = Eq(p[electron], p[1] - p[2])
```

However, for this specific case, representing each vector with a 
distinct symbol like `p_electron`, `p1`, and `p2` is the most 
straightforward and readable way to represent the equation in SymPy.


Next prompt to explore:

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 $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$ ?

Answer:

Right response:

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

Here's a breakdown of my thought process to arrive at the SymPy representation of the given LaTeX expression:

1. **Identify the Components:**  The first step is to break down the LaTeX expression into its constituent parts and their mathematical meaning:
    * `x`: A single variable.
    * `=`: Equality.
    * `\langle ... | ... \rangle`:  The Dirac bra-ket notation for the inner product (or expectation value in this context).
    * `\psi_{\alpha}`: A ket vector labeled by $\alpha$.
    * `\hat{A}`: An operator.
    * `\psi_{\beta}`: A bra vector labeled by $\beta$.

2. **Consider SymPy Equivalents:**  Now, think about how each of these components can be represented in SymPy:
    * **Variables:**  Simple variables like `x` can be represented using `Symbol('x')`.
    * **Equality:** The `=` is represented by `Eq()`.
    * **Bra-Ket Notation:** SymPy has specific modules for quantum mechanics, including bra-ket notation. The key is the `sympy.physics.quantum.state` module, which provides `Ket` and `Bra` classes, and `sympy.physics.quantum.operator` for operators.
    * **Vectors/States:**  The ket `|\psi_{\beta}\rangle` can be represented using `Ket('psi', label='beta')`. Similarly, the bra `\langle\psi_{\alpha}|` can be represented using `Bra('psi', label='alpha')`. The label helps distinguish between different states.
    * **Operators:** The operator `\hat{A}` can be represented using `Operator('A')`.
    * **Inner Product:** The inner product $\langle \phi | \hat{O} | \psi \rangle$ is conceptually a multiplication of the bra, operator, and ket. In SymPy's quantum framework, this is represented by simply multiplying these objects.

3. **Construct the SymPy Expression:**  Based on the above mapping, build the SymPy code:
    * Import necessary modules: `from sympy import symbols`, `from sympy.physics.quantum.state import Ket, Bra`, `from sympy.physics.quantum.operator import Operator`.
    * Define the symbols: `x = symbols('x')`.
    * Define the quantum objects: `psi_alpha_bra = Bra('psi', label='alpha')`, `A_hat = Operator('A')`, `psi_beta_ket = Ket('psi', label='beta')`.
    * Form the inner product: `inner_product = psi_alpha_bra * A_hat * psi_beta_ket`.
    * Create the equation: `equation = Eq(x, inner_product)`.

4. **Refine and Verify:** Review the constructed SymPy expression to ensure it accurately reflects the original LaTeX. The order of the bra, operator, and ket is crucial in the inner product.

5. **Consider Alternatives (and why they are less suitable here):**
    * **Direct String Representation:**  One might initially think of just using strings, but SymPy won't understand the mathematical relationships and won't be able to perform symbolic manipulations.
    * **Generic Symbols:**  While you *could* represent the bra, operator, and ket as generic symbols, you lose the specific meaning and the capabilities of SymPy's quantum framework. Using `Ket`, `Bra`, and `Operator` allows SymPy to understand the context and apply relevant rules.
    * **IndexedBase (Less appropriate here):**  `IndexedBase` is useful for sequences or arrays of symbols, but here we have distinct quantum objects, so the dedicated quantum classes are more appropriate.

By following this structured approach, breaking down the LaTeX, mapping to SymPy concepts, and then assembling the expression, I can arrive at the correct and meaningful SymPy representation.

"Gemini 2.0 Flash Thinking Experimental" answer

To represent the LaTeX expression $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$ in SymPy, you need to represent the variables, the bra-ket notation, and the operator. SymPy's `sympy.physics.quantum` module provides tools for this.

Here's how you can represent it:

```python
from sympy import symbols, Eq
from sympy.physics.quantum.state import Bra, Ket
from sympy.physics.quantum.operator import Operator

# Define the symbols
x = symbols('x')

# Define the quantum states
psi_alpha = Bra('psi', label='alpha')
psi_beta = Ket('psi', label='beta')

# Define the operator
A_hat = Operator('A')

# Construct the inner product
inner_product = psi_alpha * A_hat * psi_beta

# Form the equation
equation = Eq(x, inner_product)
```

Therefore, the SymPy representation is:

```python
Eq(x, Bra('psi', label='alpha') * Operator('A') * Ket('psi', label='beta'))
```

**Explanation:**

* **`from sympy import symbols, Eq`**: Imports the necessary functions for creating symbols and equations.
* **`from sympy.physics.quantum.state import Bra, Ket`**: Imports the `Bra` and `Ket` classes for representing quantum states.
    * `Bra('psi', label='alpha')` represents $\langle\psi_{\alpha}|$. 'psi' is the label of the state, and 'alpha' is a parameter or index distinguishing this particular state.
    * `Ket('psi', label='beta')` represents $|\psi_{\beta}\rangle$.
* **`from sympy.physics.quantum.operator import Operator`**: Imports the `Operator` class for representing quantum operators.
    * `Operator('A')` represents $\hat{A}$.
* **`x = symbols('x')`**: Defines the variable `x`.
* **`inner_product = psi_alpha * A_hat * psi_beta`**: Represents the inner product $\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$. SymPy's quantum module overloads the multiplication operator `*` to represent the application of operators and the formation of inner products in the correct order.
* **`equation = Eq(x, inner_product)`**: Forms the SymPy `Eq` object representing the equation $x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$.

This representation allows you to perform symbolic manipulations on this quantum mechanical expression using SymPy's capabilities.

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

How to improve chances of success:

Caveat: the page https://en.wikipedia.org/wiki/Frequency includes the statement that

"period is the reciprocal of the frequency: f = 1/T."

Use the context below to produce a result. Keep the response short and concise. Respond "Unsure about answer" if not sure about the 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.

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

"Gemini 2.0 Flash Thinking Experimental" answer

Find arxiv papers with derivations

How to improve chances of success:

  • Explain arxiv
  • define what I mean by a derivation
  • Provide example citations

Provide citations based on the context below. 

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.

Identify derivation steps between physics equations

How to improve chances of success:

  • define what I mean by a derivation
  • Provide example steps

Answer the question based on the context below. Keep the answer short.

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:

Right answer: Raise both sides as the power of $\exp$

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

The user is asking for the mathematical step that transforms the 
equation $ix = \log(y)$ into $\exp(ix) = y$. This involves the 
inverse operation of the logarithm, which is the exponential function.

"Gemini 2.0 Flash Thinking Experimental" answer

Exponentiation.

Answer the question based on the context below. Keep the answer concise. 

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:

Right answer: $\frac{d}{dx} y = -\sin(x) + i\cos(x)$

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

The question asks for the derivative of a given expression. I need to apply the rules of differentiation.

*   The expression is a sum of two terms.
*   The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$.
*   The derivative of $i \sin(x)$ with respect to $x$ is $i \cos(x)$ (since $i$ is a constant).

"Gemini 2.0 Flash Thinking Experimental" answer

$-\sin(x) + i \cos(x)$

Derive the wave function for a quantum particle in a 1D box

I removed "Keep the answer concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer." from the prompt.

Answer the question based on the context below.  Express your answer using Latex for the math. Describe the transition between each step.

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:

Right answer: see https://derivationmap.net/review_derivation/000010/

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

The user wants a Latex-formatted derivation of the wave function for a particle in a 1D box. I need to break down the derivation into logical steps and provide the Latex code for each step. Each step must be accompanied by a description of the transition from the previous step.

Here is a potential breakdown of the steps:

1. **Start with the time-independent Schrodinger equation.**
2. **Apply it to the case of a 1D box (potential definition).**
3. **Solve the differential equation in the region within the box.**
4. **Apply boundary conditions.**
5. **Normalize the wave function.**

For each step, I'll need to explain the physical/mathematical reasoning behind the transformation.

"Gemini 2.0 Flash Thinking Experimental" answer

The derivation of the wave function for a quantum particle in a 1D box proceeds as follows:

**Step 1: Start with the time-independent Schrödinger equation.**

The time-independent Schrödinger equation describes the stationary states of a quantum system. It is given by:

```latex
-\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} + V(x) \Psi(x) = E \Psi(x)
```

*Transition:* We begin with the fundamental equation governing the behavior of quantum particles in stationary states.

**Step 2: Define the potential for a 1D box.**

For a 1D box of length $L$, the potential $V(x)$ is zero inside the box ($0 < x < L$) and infinite outside the box ($x \leq 0$ or $x \geq L$).

```latex
V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}
```

Inside the box, where $V(x) = 0$, the Schrödinger equation simplifies to:

```latex
-\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} = E \Psi(x)
```

*Transition:* We specialize the Schrödinger equation to the specific scenario of a particle confined within a 1D box by defining the potential energy function. Since the potential is infinite outside the box, the probability of finding the particle there is zero, meaning $\Psi(x) = 0$ for $x \leq 0$ and $x \geq L$.

**Step 3: Solve the differential equation inside the box.**

Rearranging the simplified Schrödinger equation, we get:

```latex
\frac{d^2 \Psi(x)}{dx^2} = -\frac{2mE}{\hbar^2} \Psi(x)
```

Let $k^2 = \frac{2mE}{\hbar^2}$. Then the equation becomes:

```latex
\frac{d^2 \Psi(x)}{dx^2} = -k^2 \Psi(x)
```

The general solution to this second-order linear homogeneous differential equation is:

```latex
\Psi(x) = A \sin(kx) + B \cos(kx)
```

where $A$ and $B$ are arbitrary constants.

*Transition:* We solve the simplified Schrödinger equation within the boundaries of the box, obtaining a general solution involving sine and cosine functions.

**Step 4: Apply boundary conditions.**

Since the potential is infinite outside the box, the wave function must be zero at the boundaries of the box: $\Psi(0) = 0$ and $\Psi(L) = 0$.

Applying the first boundary condition, $\Psi(0) = 0$:

```latex
\Psi(0) = A \sin(k \cdot 0) + B \cos(k \cdot 0) = A \cdot 0 + B \cdot 1 = B
```

Therefore, $B = 0$. The wave function now becomes:

```latex
\Psi(x) = A \sin(kx)
```

Applying the second boundary condition, $\Psi(L) = 0$:

```latex
\Psi(L) = A \sin(kL) = 0
```

For a non-trivial solution (where $A \neq 0$), we must have:

```latex
\sin(kL) = 0
```

This implies that $kL = n\pi$, where $n$ is an integer ($n = 1, 2, 3, ...$). Therefore, the allowed values of $k$ are:

```latex
k_n = \frac{n\pi}{L}
```

Since $k^2 = \frac{2mE}{\hbar^2}$, the allowed energy levels are:

```latex
E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2}{2m} \left(\frac{n\pi}{L}\right)^2 = \frac{n^2 \pi^2 \hbar^2}{2mL^2}
```

The wave function becomes:

```latex
\Psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right)
```

*Transition:* We enforce the physical constraint that the particle cannot exist outside the box by applying boundary conditions to the general solution. This quantization leads to discrete allowed values for the wave number $k$ and consequently, for the energy $E$.

**Step 5: Normalize the wave function.**

The probability of finding the particle somewhere within the box must be 1. This is expressed by the normalization condition:

```latex
\int_{0}^{L} |\Psi_n(x)|^2 dx = 1
```

Substituting the wave function:

```latex
\int_{0}^{L} \left|A \sin\left(\frac{n\pi x}{L}\right)\right|^2 dx = 1
```

Assuming $A$ is real, we have:

```latex
A^2 \int_{0}^{L} \sin^2\left(\frac{n\pi x}{L}\right) dx = 1
```

Using the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$:

```latex
A^2 \int_{0}^{L} \frac{1 - \cos\left(\frac{2n\pi x}{L}\right)}{2} dx = 1
```

Integrating, we get:

```latex
A^2 \left[ \frac{x}{2} - \frac{L}{4n\pi} \sin\left(\frac{2n\pi x}{L}\right) \right]_{0}^{L} = 1
```

```latex
A^2 \left( \frac{L}{2} - 0 \right) = 1
```

```latex
A^2 \frac{L}{2} = 1
```

Solving for $A$:

```latex
A = \sqrt{\frac{2}{L}}
```

Therefore, the normalized wave function for a quantum particle in a 1D box is:

```latex
\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)
```

for $0 < x < L$, and $\Psi_n(x) = 0$ otherwise, where $n = 1, 2, 3, ...$.

*Transition:* We ensure that the wave function represents a physically realizable probability distribution by normalizing it, determining the specific value of the constant $A$. This completes the derivation of the wave function for a particle in a 1D box.

Derive Newton's Law of Universal Gravitation

Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer.

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.

provide a derivation of Newton's Law of Universal Gravitation

See https://physicsderivationgraph.blogspot.com/2023/06/finding-derivations-of-newtons-law-of.html

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

"Gemini 2.0 Flash Thinking Experimental" answer


Convert derivation steps to a proof in Lean

How to improve chances of success:

  • define what I mean by a derivation
  • Explain Lean
  • Provide example
  • Emphasize correctness and precision
I removed "Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer. Let's work this out in a step by step way to be sure we have the right answer." from the prompt.

Answer the question based on the context below. 

Context: Lean 4 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)`

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

Let's break down the thought process for constructing that Lean 4 proof.

1. **Understanding the Goal:** The fundamental goal is to prove the implication `(a = b) -> (a + 2 = b + 2)` within the Lean 4 environment. This means we need to show that *if* `a` is equal to `b`, *then* `a + 2` is equal to `b + 2`.

2. **Identifying the Tool: Rewriting:**  The core idea to manipulate equalities in proof assistants like Lean is *rewriting*. If we know `a = b`, we can substitute `b` for `a` (or `a` for `b`) within an expression. This seems like the most direct approach here.

3. **Translating to Lean Syntax (Initial Thoughts):**
   - We need to introduce the hypothesis `a = b`. In Lean, this is done using `assume` or `intro`. Let's use `intro h`.
   - We need to show `a + 2 = b + 2`.
   - How do we apply the equality `h`?  Lean has tactics like `rewrite` or `rw`.

4. **First Attempt (Mental Simulation):**
   ```lean
   theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by
     intro h
     -- We have h : a = b. We want to show a + 2 = b + 2
     -- Maybe rewrite h?
     rw [h] -- This will replace 'a' with 'b' in the goal.
     -- The goal will become b + 2 = b + 2, which is trivially true.
     exact rfl
   ```
   This seems promising. The `rw [h]` tactic looks like it will do the substitution.

5. **Refining and Formalizing the Lean Code:**

   - **Declare variables:**  We need to tell Lean that `a` and `b` are real numbers. `variables (a b : ℝ)` does this correctly.
   - **Define the theorem:** We need to give the theorem a name and state its type (the implication). `theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by` is the standard way to start.
   - **Introduce the hypothesis:** `intro h` introduces the assumption `a = b` and names it `h`.
   - **Apply the rewrite:** `rw [h]` performs the substitution. Lean is smart enough to know to substitute `b` for `a` because the goal has `a + 2` on the left.
   - **Prove the reflexivity:** After the rewrite, the goal becomes `b + 2 = b + 2`. This is a fundamental property of equality, known as reflexivity. Lean provides the `rfl` tactic (reflexivity) to prove such goals.

6. **Considering Alternatives (and why they might be less direct):**

   - **Equational Reasoning with `calc`:** We *could* use a more verbose `calc` block:
     ```lean
     theorem my_theorem (a b : ℝ) : (a = b) -> (a + 2 = b + 2) := by
       intro h
       calc
         a + 2 = b + 2 := by rw [h]
     ```
     This works but is a bit more verbose for this simple case. `rw` is more direct.

   - **Using `subst`:** The `subst` tactic is another way to handle equalities. `subst h` would also replace `a` with `b` in the goal. It's functionally similar to `rw` in this scenario.

7. **Final Check and Explanation:**  Review the code to ensure it's clear, concise, and correct. Provide a step-by-step explanation of what each part of the Lean code does and how it proves the theorem. Emphasize the key concepts like introducing hypotheses and using rewriting.

This systematic approach, moving from understanding the problem to identifying appropriate tactics and then formalizing the Lean code, is a common strategy for proving theorems in proof assistants. The mental simulation of how the tactics will affect the proof state is crucial.

"Gemini 2.0 Flash Thinking Experimental" answer

```lean
variables (a b : ℝ)

theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by
  intro h,
  rw [h],
  exact rfl
```

**Explanation:**

1. **`variables (a b : ℝ)`**: This line declares that `a` and `b` are variables of type `ℝ`, which represents the real numbers in Lean. This matches the context stating they are "Real valued".

2. **`theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by`**: This line starts the definition of a theorem named `my_theorem`. The type of the theorem is `(a = b) -> (a + 2 = b + 2)`, which is the implication we want to prove. The `by` keyword starts the proof script.

3. **`intro h,`**: This tactic introduces the hypothesis `a = b` and gives it the name `h`. This is the standard way to handle an implication (`->`) in Lean. We assume the left-hand side of the implication is true.

4. **`rw [h],`**: This is the core of the proof. The `rw` tactic (short for "rewrite") is used to replace occurrences of the left-hand side of an equality with its right-hand side (or vice-versa). In this case, `h` is the hypothesis `a = b`. The `rw [h]` command looks at the current goal, which is `a + 2 = b + 2`, and finds an occurrence of `a` on the left-hand side. It then replaces this `a` with `b` based on the equality `h`. Therefore, the goal becomes `b + 2 = b + 2`.

5. **`exact rfl`**:  After the `rw` tactic, the goal is now `b + 2 = b + 2`. This is a trivial equality; something is always equal to itself. The `rfl` tactic (short for "reflexivity") proves such goals. It states that the left-hand side is definitionally equal to the right-hand side.

**In summary, the proof works by assuming `a = b` and then using this equality to rewrite the expression `a + 2` to `b + 2`, thus proving `a + 2 = b + 2`.**
I then clarified
Please use ASCII rather than Unicode to write the Lean 4 syntax
and Gemini provided
  ```lean
variables (a b : Real)

theorem my_theorem : (a = b) -> (a + 2 = b + 2) := by
  intro h,
  rw [h],
  exact rfl
```

**Explanation of ASCII version:**

The only change needed is replacing the Unicode symbol `ℝ` with its ASCII equivalent in Lean, which is `Real`.

* **`variables (a b : Real)`**: This declares `a` and `b` as variables of type `Real`.
* The rest of the syntax remains the same as it already uses ASCII characters.

This ASCII version is functionally identical to the original Unicode version in Lean 4.
  

I tried compiling with Lean and learned the syntax was incorrect. The following is closer to correct

  import Mathlib.Data.Real.Basic

variable (a b : Real)

example : (a = b) -> (a + 2 = b + 2) := by
  intro h
  rw [h]
  exact rfl 
  
but has the error "no goals to be solved"

Identify symbols in latex arxiv papers

How to improve chances of success:

  • Provide example
  • Emphasize correctness and precision

Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer.

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}
```

Answer:

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

"Gemini 2.0 Flash Thinking Experimental" answer


Answer the question based on the context below. Keep the answer short and concise.

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:

As of 2025-01-25 on https://aistudio.google.com/, "Gemini 2.0 Flash Thinking Experimental" returns the following:

Thoughts

"Gemini 2.0 Flash Thinking Experimental" answer


best practices for prompts for large language models - LLMs

ChatGPT was made available by OpenAI on 2022-11-30. As of 2023-12-16 I hadn't used ChatGPT (Generative Pre-trained Transformer) or other large language models (LLMs). In this post I document best practices other folks have come up with. My intent is to identify whether ChatGPT could be useful for tasks relevant to the Physics Derivation Graph. 

Sites I reviewed for best practices for prompts:

General text response

Prompt

Answer the question based on the context below. Keep the answer short and concise. Respond "Unsure about answer" if not sure about the answer.

Context: <context>

Question: <question>

Answer:

Prompt

Before providing the answer, state what you know and what you assume. Then state the steps that led you to your answer. Finally, provide the answer.

Prompt

Let's think step by step.

Prompt

Let's work this out in a step by step way to be sure we have the right answer.

Prompt

Imagine three different experts are answering this question.
All experts will write down 1 step of their thinking, then share it with the group.
Then all experts will go on to the next step, etc.
If any expert realizes they are wrong at any point then they leave.
The question is...


What to use an LLM for

The following table is from https://www.philschmid.de/instruction-tune-llama-2
Capability Example Instruction
Brainstorming Provide a diverse set of creative ideas for new flavors of ice cream.
Classification Categorize these movies as either comedy, drama, or horror based on the plot summary.
Closed QA Answer the question 'What is the capital of France?' with a single word.
Generation Write a poem in the style of Robert Frost about nature and the changing seasons.
Information Extraction Extract the names of the main characters from this short story.
Open QA Why do leaves change color in autumn? Explain the scientific reasons.
Summarization Summarize this article on recent advancements in renewable energy in 2-3 sentences.


Software generation

You are an expert programmer that writes simple, concise code and explanations. Write a python function to generate the nth fibonacci number.
 
A simple python function to remove whitespace from a string
  Code Llama supports a special prompt called infill
<PRE> def compute_gcd(x, y): <SUF>return result <MID>

Code review

Where is the bug in this code?
```
def fib(n):
    if n <= 0:
        return n
    else:
        return fib(n-1) + fib(n-2)
```

Tests of Software

write a unit test for this function:

Prompts for Retrieval Augmented Generation (RAG)

RAG = https://www.promptingguide.ai/techniques/rag 

    You are an expert Q&A system that is trusted around the world.\n"
        "Always answer the query only using the provided context information, "
        "and not prior knowledge."
        
 
      "Context information is below.\n"
            "---------------------\n"
            "{context_str}\n"
            "---------------------\n"
            "Given the context information and not prior knowledge, "
            "answer the query.\n"
            "Query: {query_str}\n"
            "Answer: "
      

 

From https://docs.llamaindex.ai/en/stable/examples/prompts/prompts_rag.html

Context information is below.
---------------------
{context_str}
---------------------
Given the context information and not prior knowledge, 
answer the query asking about citations over different topics.
Some examples are given below.

{few_shot_examples}

Query: {query_str}
Answer: