MATH.19:1 - Problem frame

Use this pattern when you have a mathematical claim to establish and some relevant definitions or results, but no argument connecting them. A calculation may suggest the answer while leaving its generality unexplained. An induction step may need information that its hypothesis does not provide. A familiar theorem may almost fit, with one missing premise.

Construct intermediate claims that connect what is available to what is required. A lemma is an auxiliary proved claim used in another argument. Finding a useful lemma is part of the work: its conclusion must help the next step, and its premises must be obtainable where it is used.

First useful move: unfold the wanted conclusion once. Ask what would suffice to establish it, and which available statement or construction could supply that condition. Try to complete this connection on an arbitrary admitted object.

The reader needs elementary mathematical statements, functions and the idea of a proof from assumptions. The worked cases explain their additional notation. The method supports arguments about numbers, functions, structures and mathematical models; each branch retains its own assumptions.

Use a supplied theorem directly when its premises are met and its conclusion answers the question. B.5.RA helps recover an existing unfamiliar argument. MATH.4 develops the inductive construction when induction is the required step; MATH.12 recovers an obtaining procedure from a proof. Here the missing contribution is the proof’s construction and organization.