Research proof index

Lower bound property of the minimal associated prime. Given an intermediate submodule N'' in an interval I, any associated prime of I.val.2 / N'' is ≥ the minimal element of _μ R M I. This uses the admitted equivalence between minimal associated primes and minimal support, plus the existence of minimal primes in the support.

Singleton lower bound for μA: the chosen minimal prime is ≤ every tail _μ. Specializing the previous lemma to the minimal element of a smaller interval, we obtain the order relation needed to show that the singleton {min} is the infimum in the definition of μA.

Proposition 3.4: nonemptiness of the set of stable breakpoints StI μ I. Under well-foundedness and the DCC hypothesis, and assuming convexity on I, the selection predicates S₁I/S₂I can be satisfied by a canonical choice produced by the recursion prop3d4₀func. API note: this provides the key existential input for later uniqueness/maximality argum...

Another key property of the recursion: step i+1 is chosen to be “maximal among those with at least its μA-value”, in the sense that no z strictly between step i+1 and step i can have μA (I.left, z) greater-or-equal to μA (I.left, step(i+1)). This is a tie-breaking/optimality condition derived from minimality in the well-founded has_min cho...

Optimality at the last pre-termination step. Let len be the first index such that step len equals I.left. Then at index len-1, no intermediate point y between I.left and (func (len-1)).val yields a strictly larger value of μA (I.left, y). This is used to show that the final candidate satisfies the selection predicate S₁I.

Proposition 3.8 (part 1): totality on StI μ I under comparability/attainment hypotheses. Under convexity and well-foundedness, if either: - the target S is totally ordered, or - all relevant μA infima are attained, then the order on the set of stable breakpoints becomes total. API note: this produces an instance of Std.Total for the subtype `StI μ...