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Can't give a full review yet, but I think some of this should be separated into other lemmas. For example, define V' and p' in general and then pretty much each of your have statements can be its own lemma.
@morrison-daniel I did something similar to this. I have defined a more general notion of an EffectivePotential and then define properties of that, the proof is slightly different from what I had before, but hopefully it is now slightly more generalizable.
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Adding the theorem which says that the Higgs potential is the most general potential up to order 4 which is invariant under the SM action.