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Mathematics · unit version 1

Domains, invariants and structural counting

Find the assumptions that make a composition, accumulated reward, reachability argument or structural count valid, and repair proofs when those assumptions change.

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Readiness and preparation

Check your starting point

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When does equality of outputs imply equality of inputs?

A composite can discard inputs outside its domain, hiding collisions.

Compare reasoning and find a refresher

Injectivity applies only to two inputs in the actual domain. State that domain before using the implication.

Review function composition and injectivity

Does a preserved quantity prove that a target is reachable?

A necessary obstruction and a constructive sufficiency proof are separate.

Compare reasoning and find a refresher

No. Matching the invariant only passes a necessary check. Prove a legal construction or another sufficient criterion.

Review state invariants and legal transitions

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Audit the domain and certify the conclusion

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Lesson sequence

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  1. Partial functions: keep the domain in the proof

    Lesson details and objectives

    Track defined inputs through composition and repair injectivity, surjectivity and inverse arguments when an operation is partial.

    • Derive the exact domain of a composition and distinguish undefined inputs from ordinary outputs.
    • Construct a nonsingleton collision-hiding example and prove the image-coverage condition restoring injectivity.
    • Separate totality from surjectivity and verify partial inverses and algebraic domain exclusions.

    6 practice tasks.

  2. Recover accumulated rewards from endpoint potentials

    Lesson details and objectives

    Prove a local reward identity, telescope it between arbitrary configurations and separate stopping choices from terminal states.

    • Derive a merge reward from the change of a sum-of-squares potential.
    • Prove endpoint and stopping-state conditions for path-independent rewards.
    • Separate potential bookkeeping, termination and changes to the permitted moves.

    6 practice tasks.

  3. Turn a modular obstruction into a reachability theorem

    Lesson details and objectives

    Discover an invariant on an integer grid, then solve for move counts to distinguish reversible and forward-only reachability.

    • Use displacement from an arbitrary start to prove a modular necessary condition.
    • Prove sufficiency with explicit integer move coefficients when reverse moves are allowed.
    • Derive and test the extra nonnegative-coefficient restrictions for forward-only motion.

    6 practice tasks.

  4. Count distinct labels by checking constructor assumptions

    Lesson details and objectives

    Separate vertex occurrences from label sets, locate overlap losses, and repair structural induction using disjointness and freshness.

    • Compute occurrence counts and distinct-label sets separately for finite labeled trees.
    • Prove the label-count formula under disjoint-child and fresh-root constructor conditions.
    • Diagnose failed induction steps and derive an exact overlap correction without uniqueness assumptions.

    6 practice tasks.