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Unspooling the Equality

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Question

Now please do the same thing, but for payment:

```python

pay = ixpayment(wallet, source, quantity + conciliation(wallet, source, target, quantity))

```

Answer

Sure, let's go through the derivation process with the new definition of `pay`:

We start with the equation:

```python your_value_feign == your_value_honest ```

This can be transformed to:

```python value(ix_feigned, honest_valuation) == value(ix_honest, honest_valuation) ```

Now, we substitute the `pay` function:

```python value(pay(wallet, feigned_valuation, supplier_valuation, quantity), honest_valuation) == value(pay(wallet, honest_valuation, supplier_valuation, quantity), honest_valuation) ```

We're checking this particular payment function:

```python pay = ixpayment(wallet, source, quantity + conciliation(wallet, source, target, quantity)) ```

So, we substitute the `pay` function:

```python value(ixpayment(wallet, feigned_valuation, quantity + conciliation(wallet, feigned_valuation, supplier_valuation, quantity)), honest_valuation) == value(ixpayment(wallet, honest_valuation, quantity + conciliation(wallet, honest_valuation, supplier_valuation, quantity)), honest_valuation) ```

In this context we're using this definition of `conciliation`:

```python def conciliation(wallet, source, target, quantity): return (quantity*wallet).dot(source - target)/wallet.dot(target) ```

So, we substitute the `conciliation` function:

```python value(ixpayment(wallet, feigned_valuation, quantity + (quantity*wallet).dot(feigned_valuation - supplier_valuation)/wallet.dot(supplier_valuation)), honest_valuation) == value(ixpayment(wallet, honest_valuation, quantity + (quantity*wallet).dot(honest_valuation - supplier_valuation)/wallet.dot(supplier_valuation)), honest_valuation) ```

Now we might as well make some substitutions for `ixpayment`:

```python def ixpayment(wallet, valuation, quantity): return wallet * (quantity / value(wallet, valuation)) ```

So, we substitute the `ixpayment` function:

```python value(wallet * ((quantity + (quantity*wallet).dot(feigned_valuation - supplier_valuation)/wallet.dot(supplier_valuation)) / wallet.dot(feigned_valuation)), honest_valuation) == value(wallet * ((quantity + (quantity*wallet).dot(honest_valuation - supplier_valuation)/wallet.dot(supplier_valuation)) / wallet.dot(honest_valuation)), honest_valuation) ```

And now finally we replace `value`:

```python def value(wallet, valuation): return wallet.dot(valuation) ```

So, we substitute the `value` function:

```python (wallet * ((quantity + (quantity*wallet).dot(feigned_valuation - supplier_valuation)/wallet.dot(supplier_valuation)) / wallet.dot(feigned_valuation))).dot(honest_valuation) == (wallet * ((quantity + (quantity*wallet).dot(honest_valuation - supplier_valuation)/wallet.dot(supplier_valuation)) / wallet.dot(honest_valuation))).dot(honest_valuation) ```

This is the final derived equation.

⁠

Then we convert to symbols:

(w * ((q + (q*w).(f -s)/w.s)/w.f)).h == (w * ((q + (q*w).(h - s)/w.s)/w.h)).h

And simplify using wolfram alpha:

(w ((q w).(h - s) + q w.s))/(w.h w.s).h = (w ((q w).(f - s) + q w.s))/(w.f w.s).h

⁠

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