MATH.18:4 - Solution

Local mantra: choose the consequence; interpret the ingredients; derive what transfers; compare the round trips; use the agreement or its failure.

MATH.18:4.1 - Set the scope of the comparison

Name the source account, target account and intended use. Identify the objects involved, their operations and relations, and what counts as an answer. If transformations between objects matter, include those transformations in the comparison.

For each ingredient that the intended consequence uses, ask how the target will express it. Keep optional additional structure separate. For example, an additive operation, an order and a distance support different questions even when carried by the same underlying set.

Decide the strength of the needed conclusion. Transferring one equation, constructing a solution, comparing all compositions in a selected family, and establishing equivalence of two presentations are different claims. Inspect the ingredients and reasoning needed for that claim. Expand the comparison when the next use expands it.

MATH.18:4.2 - Construct the interpretation

Specify the source objects’ target counterparts and the operations or relations that interpret each required primitive. Give a construction rather than a matching name. If an operation is to be recovered through a defining condition, establish that a result exists and is determined to the degree required by the source.

Respect the domains and arities. A binary operation needs two admitted inputs; a relation needs the participants on which it is asserted. When the source uses equivalence classes, establish that different representatives give equivalent interpreted results. MATH.2 supplies the detailed quotient test.

Extend the interpretation through constructed expressions. Interpret each input, then interpret the operation applied to those inputs. A composite expression is handled by repeating this rule through its construction. MATH.5 develops the extension from generators when the source is presented that way.

An assertion also has a construction. Translate equality, conditions and quantifier ranges. For there exists x in A with P(x), the target must express both the interpreted domain A and the interpreted condition P. Enlarging that range can admit a solution that the original problem excludes; :5.2 works this failure.

When an interpretation uses a basis, representative or another choice, supply it or a means of obtaining it. Determine whether changing the choice changes the result, or only changes a representation with a known comparison.

MATH.18:4.3 - Establish what the interpretation preserves and reflects

For the consequence being transferred, follow its construction or argument through the interpretation. Establish that the required operations remain applicable and that each used equality or relation remains valid.

Distinguish the two directions. Preservation takes a source consequence to a target consequence. Reflection takes an interpreted target consequence back to the source. To use a target calculation as the source answer, obtain the required recovery and reflection argument.

For operations composed as maps, an object assignment F also assigns F(f):F(A) -> F(B) to each source map f:A -> B. To translate a sequence stage by stage, establish:

F(g∘f)=F(g)∘F(f) and F(id_A)=id_F(A).

Such an assignment is a functor. Fix source objects A and B. To recover maps, determine which target maps F(A) -> F(B) have counterparts A -> B. To recover an equality, take two source maps f,g:A -> B and determine whether F(f)=F(g) implies f=g. The equality test concerns maps with these same endpoints. Recovery of the objects themselves is a further question, handled in :4.4.

Formal-theory branch. When the intended result is transport of theorems, give the translation of the relevant syntax and logic. Establish the interpreted axioms and justify the inference rules used by the theorem. Comparing collections of models through maps supplies a different mathematical result until its connection to this theorem-translation claim is established.

A failed preservation or reflection equation is useful. Work its inputs far enough to show what answer or construction changes. Then restrict the claim, enrich the interpretation or keep the accounts distinct for that purpose.

MATH.18:4.4 - Construct the return and examine the composites

When two-way use is needed, construct a return interpretation G. Apply G∘F to source ingredients and F∘G to target ingredients. Inspect what each round trip returns.

Sometimes the ingredients are recovered literally, as in the order-and-operation case below. Sometimes a specified isomorphism supplies recovery: an invertible map preserving the structure used by the comparison.

When whole systems of objects and maps are compared, pointwise isomorphisms need compatibility with the maps. If eta_A:A -> G(F(A)) provides source recovery, require for every source map f:A -> B:

G(F(f))∘eta_A = eta_B∘f.

Both routes start in A and finish in G(F(B)). This equation means that translating and then applying the map agrees with applying the map and then translating. Require the corresponding compatibility on the target side as well. In categorical language, functors with these natural isomorphisms give an equivalence of categories.

Use that equivalence for properties supported by the chosen structure and invariant under its isomorphisms. If a later question uses additional structure, include it in the comparison. The length question in :5.3 shows why this return matters.

If only one direction is constructed or needed, retain that useful interpretation at its established scope. If two-way recovery fails, name the unrecovered operation, assertion or distinction that matters to the work.

MATH.18:4.5 - Transfer the result and reopen only the changed demand

Carry the intended calculation, construction or argument into the receiving account and recover its answer where required. State the conditions that make the transfer usable. A receiver needs the interpretation and relevant consequence; an equivalence label alone leaves the work to be reconstructed.

A changed primitive, quantifier range or allowed-map class can change the comparison. Revisit the affected construction and its argument. Keep conclusions whose ingredients and conditions are unchanged.

For mathematical work, the result may be an equivalence at the selected scope, a useful one-way interpretation or a separating consequence. For a computational or working-method application, C.29 and C.29.2 supply the further correspondence and execution questions. Mathematical agreement can then inform the application through those explicit connections.