# Steps to Proving a Theorem

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1 Steps to Proving a Theorem Step 1: Understand Goal What am I looking for? What should the last sentence in my proof say? What are some equivalent ways to state what I want? Are there any complicated definitions that need to be unwound? Can I see any assumptons that would make this true? Step 2: Understand Assumptions What am I allowed to assume? What are some equivalent ways to state what I know? Are there any complicated definitions that need to be unwound? What problems do I face when trying to find a counterexample? Can I find any counterexamples to my conclusion when some of these assumptions are eliminated or weakened? What evidence do I have that these assumptions yield that conclusion? Step 3: Understand Relevant Information Can I think of any similar problems? Are there any theorems with similar conclusions? Is there a way to connect the assumptions I have to similar conclusions? What information can I obtain from the assumptions? Are there any theorems with similar assumptions? Step 4: Form a Plan Can I use any of the relevant information that I found? Can any of the proofs that have similar conclusions or conclusions be adapted for this circumstance? Are there any auxiliary theorems I can prove first? Can some assumptions are strengthened or added can I reach my conclusion? If so, can that proof be adapted for the assumptions I do have? Can some of my conclusions be weakened or eliminated to reach that modified conclusion? If so, can that proof by adapted for the conclusions I do have? Step 5: Execute Plan Is this step justified? Is this step relevant? Can I prove this fact? Am I using all the assumptions I have at my disposal? Step 6: Reflection Are all my steps justified? Can I prove all the claims that I made? Do all steps seem clear to my readers? Is there a way to verify the result? Did I actually achieve my goal? Did I use all my assumptions? Step 7: Expand Did I use all of my assumptions? Did I use the full strength of all of my assumptions? Can I weaken any of my assumptions? Did I prove more than I intended? Can I strengthen my conclusion?

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