Bitcoin’s mathematical definition (1)

We want to formulate the data behind Bitcoin more scientifically since it got also attention from the science area. We start with some simple and basic defintions:

• $$T$$: Set of all transactions
• $$t_{x} \in T$$: Transaction $$t$$ with id $$X$$
• $$A$$: Set of all addresses
• $$(S_{t_{x}}^{in}, <)$$: Bounded lattice-ordered set of all inputs of $$t_{x}$$
• $$(S_{t_{x}}^{out}, <)$$: Bounded lattice-ordered set of all outputs of $$t_{x}$$
• $$(r_{i}, a_{i}, v_{i}) \in S_{t_{x}}^{in}: r_{i} \in \mathbb{N}_0, a_{i} \in A, v_{i} \in \mathbb{N}_0^{+}$$: $$i$$th element of the inputs where $$r_{i}$$ defines the order starting at 0
• $$(r_{j}, a_{j}, v_{j}) \in S_{t_{x}}^{out}: r_{j} \in \mathbb{N}_0, a_{j} \in A, v_{j} \in \mathbb{N}_0^{+}$$: $$j$$th element of the outputs where $$r_{j}$$ defines the order starting at 0 with the condition: $$(r_{i}, a_{i}, v_{i}) \in S_{t_{x}}^{in} . \exists! (r_{j}, a_{j}, v_{j}) \in S_{t_{y}}^{out} \land x \neq y \land a_{i} = a_{j} \land v_{i} = v_{j}$$