feat: initial properties to test squiggle validity

packages/squiggle-lang/__tests__/docs/README.md

@@ -1,15 +1,17 @@
 # Properties
 
 Using `nix`. Where `o` is `open` on OSX and `xdg-open` on linux, 
+
 ```sh
 nix-build
-o result/properties.pdf
+o result/property-tests.pdf
 ```
+
 Without `nix`, you can install `pandoc` yourself and run
 ```sh
-pandoc --from markdown --to latex --out properties.pdf --pdf-engine xelatex properties.md
+pandoc -s property-tests.md -o property-tests.pdf
 ```
 
 ## _Details_
 
-The `properties.pdf` document is _normative and aspirational_. It does not document tests as they exist in the codebase, but somewhat represents how we think squiggle ought to be tested. 
+The `properties-tests.pdf` document is _normative and aspirational_. It does not document tests as they exist in the codebase, but somewhat represents how we think squiggle ought to be tested. 

packages/squiggle-lang/__tests__/docs/make.sh

@@ -0,0 +1 @@
+ pandoc -s property-tests.md -o property-tests.pdf

packages/squiggle-lang/__tests__/docs/property-tests.md

@@ -0,0 +1,38 @@
+# Property tests for squiggle
+
+Here are some property tests for squiggle. I am testing mostly for the mean and the standard deviation. I know that squiggle doesn't yet have functions for the standard deviation, but they could be added.
+
+The keywords to search for are "[algebra of random variables](https://wikiless.org/wiki/Algebra_of_random_variables?lang=en)". 
+
+## Sums
+
+$$ mean(f+g) = mean(f) + mean(g) $$
+
+$$ Std(f+g) = sqrt(std(f)^2 + std(g)^2) $$
+
+In the case of normal distributions,
+
+$$ normal(a,b) + normal(c,d) = normal(a+c, sqrt(b^2 + d^2) $$
+
+## Substractions
+
+$$ mean(f-g) = mean(f) - mean(g) $$
+
+$$ std(f-g) = sqrt(std(f)^2 + std(g)^2) $$
+
+## Multiplications
+
+$$ mean(f \cdot g) =  mean(f) \cdot mean(g) $$
+
+$$ std(f \cdot g) = sqrt( (std(f)^2 + mean(f)) \cdot (std(g)^2 + mean(g)) - (mean(f) \cdot mean(y))^2) $$
+
+## Divisions
+
+Divisions are tricky, and in general we don't have good expressions to characterize properties of ratios. In particular, the ratio of two normals is a Cauchy distribution, which doesn't have to have a mean.
+
+## To do:
+
+- Provide sources or derivations, useful as this document becomes more complicated
+- Provide definitions for the probability density function, exponential, inverse, log, etc.
+- Provide at least some tests for division
+- See if playing around with characteristic functions turns out anything useful