Let me see if I get what you're saying about the difference between a Petri net and a wiring diagram!

Let's stick with the example where our resources are

* \\(H\\)

* \\(O\\)

* \\(H_2O\\)

The Petri net makes \\(\mathbb N[S]\\) into a commutative monoidal preorder by specifying the relation \\[ (a + 2n)[H] + (b + n)[O] + c [H_2O] \leq a [H] + b[O] + (c + n)[H_2O] \\] for all \\(a,b,c,n \in \mathbb N\\).

The wiring diagram with 2 hydrogen wires and 1 oxygen wires going into a box and 1 \\(H_2O\\) wire exiting the box would be a proof of the inequality \\[ 2[H] + [O] \leq [H_2O].\\] Two of these wiring diagrams set up in parallel would be a proof of the inequality \\[4[H] + 2[O] \leq 2 [H_2O].\\] And the wiring diagram with 2 hydrogen wires and 1 oxygen wires going into a box, 1 \\(H_2O\\) wire exiting the box, and 1 oxygen wire running uninterrupted in parallel would be a proof of the inequality \\[2[H] + 2[O] \leq [O] + [H_2O]\\]

Let's stick with the example where our resources are

* \\(H\\)

* \\(O\\)

* \\(H_2O\\)

The Petri net

The wiring diagram with 2 hydrogen wires and 1 oxygen wires going into a box and 1 \\(H_2O\\) wire exiting the box would be a proof of the inequality \\[ 2[H] + [O] \leq [H_2O].\\] Two of these wiring diagrams set up in parallel would be a proof of the inequality \\[4[H] + 2[O] \leq 2 [H_2O].\\] And the wiring diagram with 2 hydrogen wires and 1 oxygen wires going into a box, 1 \\(H_2O\\) wire exiting the box, and 1 oxygen wire running uninterrupted in parallel would be a proof of the inequality \\[2[H] + 2[O] \leq [O] + [H_2O]\\]