### Building Lean expressions for LADR
> [! MathChat]
> Here's a first take on LEAN proofs for LADR Exercises
>
- These answers generated by GPT-4 on 2024-02-10
---
### 1A-1
Q:
Sure, here is how you can show that vector addition is commutative using Lean proof assistant. However, Lean does not have built-in vector types, so we will use natural numbers as a simple stand-in for demonstration purposes.
We can prove commutativity of addition on natural numbers which will give us an idea of how such a proof can be structured in Lean.
import data.nat.basic
open nat
theorem add_comm (a b : ℕ) : a + b = b + a :=
begin
induction a with d hd,
{
-- Base case:
-- We need to show that 0 + b = b + 0, which is true by definition of addition.
rw [add_zero, zero_add]
},
{
-- Inductive step:
-- We assume that d + b = b + d (this is our inductive hypothesis hd)
-- and need to show that (d + 1) + b = b + (d + 1).
rw [add_succ, hd, succ_add]
}
end
---
This code first imports the necessary library for natural numbers. Then it defines a theorem `add_comm` which states that for any two natural numbers `a` and `b`, `a + b` equals `b + a`. The proof is by induction on `a`. In the base case, `a` is 0 and the result follows directly from the definition of addition. In the inductive step, `a` is assumed to be `d + 1` for some natural number `d`. Assuming the result holds for `d` (this is the inductive hypothesis `hd`), the goal is to prove that it holds for `d + 1`. This is done by rewriting `(d + 1) + b` as `d + b + 1` and `b + (d + 1)` as `b + d + 1`, and then applying the inductive hypothesis to obtain the desired equality.
This is a simplistic example and actual vector spaces have a lot more structure, but the general idea would be the same: define the vector space and its operations, state the theorem to be proved, and then construct the proof, often by induction or other standard mathematical techniques.
---
Sequence of Learn code to define vectors:
In the Lean proof assistant, vectors are essentially lists of a fixed length. They are defined inductively in the standard library as follows:
```lean
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inductive vector (α : Type u) : ℕ → Type u
| nil {} : vector 0
| cons : Π {n}, α → vector n → vector (n + 1)
```
Here are some built-in operations for vectors:
1. Append: The append operation concatenates two vectors together. It is defined as follows:
```lean
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def append {α : Type u} : Π {n m : ℕ} , vector α n → vector α m → vector α (n + m)
| _ _ nil ys := ys
| _ _ (cons x xs) ys := cons x (append xs ys)
```
2. Head: The head operation returns the first element of a non-empty vector. It is defined as follows:
```lean
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def head {α : Type u} {n : ℕ} (v : vector α (n+1)) : α :=
vector.rec_on v (λ m a v, a)
```
3. Tail: The tail operation removes the first element of a non-empty vector. It is defined as follows:
```lean
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def tail {α : Type u} {n : ℕ} (v : vector α (n+1)) : vector α n :=
vector.rec_on v (λ m a v, v)
```
4. To_list: The to_list operation converts a vector to a list. It is defined as follows:
```lean
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def to_list {α : Type u} : Π {n : ℕ}, vector α n → list α
| _ nil := []
| _ (cons x xs) := x :: to_list xs
```
5. From_list: The from_list operation converts a list to a vector of a certain length. If the length of the list is less than the required length, the operation fails.
```lean
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def from_list {α : Type u} : Π (l : list α), Π {n : ℕ}, n = list.length l → vector α n
| [] _ rfl := nil
| (x :: xs) _ rfl := cons x (from_list xs rfl)
```
Remember that these are just the basic operations. More complex operations can be defined in terms of these, such as map, filter, reduce, etc.
---
Defining addition in the Lean proof assistant involves creating a function that takes two integers and returns their sum. Here is a simple example:
```lean
def add (a b : ℕ) : ℕ := a + b
```
In this function, `a` and `b` are the parameters, both of type `ℕ` (natural numbers). The function returns a `ℕ` which is the sum of `a` and `b`.
We could also define addition for other types, such as real numbers (`ℝ`). Here's how you do it:
```lean
def add_real (a b : ℝ) : ℝ := a + b
```
For the vector spaces, we can define addition as follows:
```lean
universe u
constant Vec : Type u → Type u
namespace Vec
constant add {d : Type u} : Vec d → Vec d → Vec d
end Vec
```
Here, `Vec` is a function that takes a type and returns a type. In this case, we're defining `Vec` to be the type of vectors over a given field `d`. Inside the `Vec` namespace, we define an `add` function that takes two vectors of the same type and returns a vector of the same type. The `{d : Type u}` syntax indicates that `d` is an implicit parameter to the `add` function, meaning that it can be inferred from the arguments you pass to `add`.
Please note that the above code is a formal way to define vector addition in Lean. In practice, Lean has built-in types and operations for vectors, and you would usually use those instead of defining your own.
### Next, generate working list of Exercises to create Linear Algebra World
[[Lean Proofs for Axler LADR Exercises]]