Cost-zero type offsets and their difference space

For the type set $\mathcal{G}$ and cost $F$ of Decorated finite-chain costs with catalysts, define the relation $\mathrm{\Gamma }{\approx }_0{\mathrm{\Gamma }}^{\mathrm{\prime }}$ by $F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})=F({\mathrm{\Gamma }}^{\mathrm{\prime }},\mathrm{\Gamma })=0$. Let $\mathcal{P}:=\mathcal{G}\times \mathcal{G}$ be the set of raw type pairs. Define a relation on raw pairs by

$(A,B)\equiv (A^{\mathrm{\prime }},B^{\mathrm{\prime }})⟺\mathrm{\exists }C\text{ such that }A\times B^{\mathrm{\prime }}\times C{\approx }_0{A}^{\mathrm{\prime }}\times B\times C\mathrm{.}$

The separate well-definedness lemma proves that ${\approx }_0$ is an equivalence compatible with composition and positive scaling, that $\equiv$ is an equivalence on $\mathcal{P}$, and that the following operations are well-defined on its classes. After that proof, write $V_F:=\mathcal{P}\mathrm{/}\equiv$. Define addition by $[(A,B)]+[(D,E)]:=[(A\times D,B\times E)]$, negation by $-[(A,B)]:=[(B,A)]$, and positive scaling by $t[(A,B)]:=[(tA,tB)]$. Set zero to $[(1,1)]$, where $1$ is the empty unit type. Complete scalar multiplication for all $t\in \mathbb{R}$ by $0[(A,B)]:=[(1,1)]$ and, for $t<0$, $t[(A,B)]:=[((-t)B,(-t)A)]$; here $-t>0$, so only the already specified positive scaling of types is used. The same lemma establishes that these operations make $V_F$ a real vector space; no vector-space law is assumed by this Definition.