Set Shatters subset
Cslib.MachineLearning.PACLearning.SetShatters.subset
Plain-language statement
Shattering is anti-monotone in the shattered set: if C shatters W and V ⊆ W, then C shatters V.
Exact Lean statement
theorem SetShatters.subset {C : ConceptClass α Bool} {W V : Set α}
(hW : SetShatters C W) (hVW : V ⊆ W) : SetShatters C VFormal artifact
Lean source
theorem SetShatters.subset {C : ConceptClass α Bool} {W V : Set α} (hW : SetShatters C W) (hVW : V ⊆ W) : SetShatters C V := by intro V' hV'V obtain ⟨c, hc, hc_eq⟩ := hW (V' ∪ (W \ V)) (union_subset (hV'V.trans hVW) sdiff_subset) refine ⟨c, hc, ?_⟩ rw [show V = W ∩ V from (inter_eq_self_of_subset_right hVW).symm, ← inter_assoc, hc_eq] ext x simp only [mem_inter_iff, mem_union, mem_sdiff] refine ⟨?_, fun h => ⟨Or.inl h, hV'V h⟩⟩ rintro ⟨h1 | ⟨_, h2⟩, h3⟩ · exact h1 · exact absurd h3 h2- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/MachineLearning/PACLearning/VCDimension.lean:58-71
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Person-level attribution pending.
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Cslib.Automata.NA.Buchi.buchiFamily_cover
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Source project: Lean Computer Science Library
Person-level attribution pending.
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Cslib.Automata.NA.Buchi.buchiFamily_saturation
Plain-language statement
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Source project: Lean Computer Science Library
Person-level attribution pending.