Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Minimal Super Set induction on inductive

SuperSymmetry.SU5.ChargeSpectrum.minimalSuperSet_induction_on_inductive

physics quantum field theory relativity minimal super set

Project documentation

Given a collection of charges x in ofFinset S5 S10, the minimal charges y in ofFinset S5 S10 which are a super sets of x. -/ def minimalSuperSet (S5 S10 : Finset 𝓩) (x : ChargeSpectrum 𝓩) : Finset (ChargeSpectrum 𝓩) := let SqHd := if x.qHd.isSome then ∅ else S5.map ⟨fun y => ⟨some y, x.qHu, x.Q5, x.Q10⟩, by intro y1 y2; simp⟩ let SqHu := if x...

Exact Lean statement

lemma minimalSuperSet_induction_on_inductive {S5 S10 : Finset 𝓩}
    (p : ChargeSpectrum 𝓩 → Prop)
    (hp : (x : ChargeSpectrum 𝓩) → p x → ∀ y ∈ minimalSuperSet S5 S10 x, p y)
    (x : ChargeSpectrum 𝓩) (hbase : p x)
    (y : ChargeSpectrum 𝓩) (hy : y ∈ ofFinset S5 S10) (hsubset : x ⊆ y) :
    (n : ℕ) → (hn : n = y.card - x.card) → p y
  | 0, hn => by
    have hxy : x = y

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Lean source

Canonical source

Full Lean sourceLean 4

lemma minimalSuperSet_induction_on_inductive {S5 S10 : Finset 𝓩}    (p : ChargeSpectrum 𝓩 → Prop)    (hp : (x : ChargeSpectrum 𝓩) → p x → ∀ y ∈ minimalSuperSet S5 S10 x, p y)    (x : ChargeSpectrum 𝓩) (hbase : p x)    (y : ChargeSpectrum 𝓩) (hy : y ∈ ofFinset S5 S10) (hsubset : x ⊆ y) :    (n : ℕ) → (hn : n = y.card - x.card) → p y  | 0, hn => by    have hxy : x = y := by      refine eq_of_subset_card hsubset ?_      have hl : card x ≤ card y := card_mono hsubset      omega    subst hxy    simp_all  | Nat.succ n, hn => by    have hxy : x ≠ y := by      intro h      subst h      simp at hn    obtain ⟨z, hz, hsubsetz⟩ := exists_minimalSuperSet S5 S10 hy hsubset hxy    refine minimalSuperSet_induction_on_inductive p hp z ?_ y hy ?_ n ?_    · exact hp x hbase z hz    · exact hsubsetz    · rw [card_of_mem_minimalSuperSet z hz]      omega

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimalSuperSet.lean:326-349

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