Research proof index

Proposition 2.6 (c): a case split yielding either equality or a strict inequality chain. The hypothesis allows either comparability of the two adjacent μA values, or attainment of the infimum defining μA (x,z). The conclusion then provides a dichotomy between equality and a strict improvement.

Proposition 2.8 (auxiliary step): a disjunction bounding one of two μA values by a μmax value. This is an interval-local statement used to derive the “meet” inequality in Proposition 2.8.

Proposition 2.8 (a): μA (u, x ⊔ y) dominates the meet μA (u,x) ⊓ μA (u,y). This is obtained by taking an infimum and using prop2d8₀I to select the relevant branch.

Proposition 2.8 (b): under comparability or attainment, one of the two μA values is dominated by μA (u, x ⊔ y). This is a “one-sided dominance” conclusion that matches the alternative in the paper statement.

Proposition 3.12 (internal): explicit computation of μA (μ R M). For any strict interval I : N₁ < N₂, the auxiliary function μA evaluates to the singleton finset containing the minimal element of _μ R M I (in the S₀ R order). Proof idea: * Show that the intermediate submodule ker_of_quot_comp_localization I realizes an element of the defining...

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.