The point is that software does not implement an invention. You're making a type error. No software does. It's like saying "a green tastes good." Green doesn't taste anything, and you can't have "a green." It's a color. When you write those words it's just a nonsense string. Software doesn't implement any patent just like 5 or pi don't implement any patent. It's a formal mathematical object. It also can't violate a patent because you can't patent mathematical objects, only inventions. An implementation of a patent may contain a computer that runs software.

It's like saying a circuit diagram could infringe on a patent. No, only a circuit can. Software is the diagram. A thing running software can infringe, but software isn't the right type of thing to be able to infringe. You can print books full of detailed diagrams and instructions all day long. The patent is already supposed to describe the thing! It's not secret! A description of every last detail is not protected!

This is also why you can prove things about software. You cannot prove things about inventions; only models of the inventions. You can actually prove statements about software itself because it is already part of formal mathematics.

Again, you are conflating "what can be patented" with "what can infringe"

If you make, use, produce, supply, induce someone to use anything that does a thing that is listed in a claim of a patent -- you have violated the patent.

The argument that "software is just abstract math and so there's no way it can infringe" is bologna. Any actual articulable piece of written software is not some abstract mathematical concept, it is the configuration for a machine. Courts simply do not treat software in the way you are asserting, they treat it like a component in a system.

If what you are asserting was true, it would be great for the software industry -- we wouldn't have to worry about patents anymore!