Proof construction is the activity of assembling a sequence of justified inferential steps that establishes a target formula from premises or axioms within a specified formal system.

Principle

Each step must be justified by a permitted inference rule, a previously proven lemma, or an axiom; the sequence should maintain correctness and move toward the target while preserving context and scope.

Demonstration

Construct a natural-deduction proof that from A→B and A derives B by writing the assumption, applying modus ponens as a justified step, and discharging assumptions if needed to conclude an implication.

Misapplication

Assembling a chain of statements without explicit justification, using circular arguments where a step presupposes the target, or omitting necessary side conditions leads to invalid or non-reproducible proofs.

Consequence

A correctly constructed proof yields a verifiable certificate that the target follows from premises; it enables peer review, reuse of lemmas, and mechanized checking by proof assistants.

Reversal

The inverse is proof deconstruction or countermodel construction: instead of building a derivation, exhibit a counterexample or dismantle purported steps to show the target is not derivable.

Boundary

Proof construction refers to formal derivations in a chosen calculus or proof system; it excludes informal exposition that lacks stepwise justificatory structure and creative heuristics that do not result in formal derivations.

Semantic Tension

Tension exists between elegant, human-readable proofs and mechanically generated proofs that may be long or low-level; both establish correctness but differ in explanatory value and structure.

Synthesis

Proof construction is the disciplined process of chaining justified inferential moves under system rules to produce a checkable derivation that certifies a formula from premises or axioms.