Research proof index

A collection of charges x : Charges is said to minimally allow the potential term T if it allows T and no strict subset of it allows T. -/ def MinimallyAllowsTerm (x : ChargeSpectrum 𝓩) (T : PotentialTerm) : Prop := ∀ y ∈ x.powerset, y = x ↔ y.AllowsTerm T /-! ### A.1. Decidability of MinimallyAllowsTerm We show that MinimallyAllowsTerm is de...

For charges x : Charges, the proposition which states that the singlets needed to regenerate the Yukawa couplings regenerate a dangerous coupling (in the superpotential) with up-to n insertions of the scalars. Note: If defined as (x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset ≠ ∅ the execution time is greatly increased. -/ de...

The support projection of A acts as identity on B when A.ker ≤ B.ker. Since A.supportProj projects onto ker(A)⊥ and B is zero on ker(A), the projection preserves B.

The SupRegularized value is also lower bounded.

For Monotone functions, the SupRegularized is the supremum of values.

A more precise version of sfHomotopy_in. -/ theorem sfHomotopy_in' {ι} (h₀ : SurroundingFamily g b γ₀ U) (h₁ : SurroundingFamily g b γ₁ U) (τ : ι → ℝ) (x : ι → E) (i : ι) {V : Set E} (hx : x i ∈ V) {t : ℝ} (ht : t ∈ I) {s : ℝ} (h_in₀ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠ 1 → (x i, γ₀ (x i) t s) ∈ Ω) (h_in₁ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠...