def definition def or abbrev

`ChainInfClosed`

No prose has been written for this declaration yet. The Lean source and graph data below render without it.

formal statement (Lean)

 153def ChainInfClosed (P : Set (Setoid C)) : Prop :=

proof body

Definition body.

 154  ∀ (chain : Set (Setoid C)), chain ⊆ P → chain.Nonempty → IsChain (· ≤ ·) chain →
 155    sInf chain ∈ P
 156
 157/-- **Theorem (Zorn for Recognizers)**: Let P be a nonempty family of setoids
 158    on C that is closed under chain infima. Then P contains a minimal element—
 159    a finest recognizer-induced equivalence relation.
 160
 161    Formally: ∃ m ∈ P such that no s ∈ P is strictly finer than m.
 162
 163    This is the dual of Zorn's Lemma (existence of minimal elements):
 164    every chain has a lower bound (the infimum) ⟹ minimal element exists. -/

depends on (14)

Lean names referenced from this declaration's body.

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of in IndisputableMonolith.Foundation.DAlembert.LedgerFactorization decl_use
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is in IndisputableMonolith.Foundation.SimplicialLedger.EdgeLengthFromPsi decl_use
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for in IndisputableMonolith.Foundation.UniversalForcingSelfReference decl_use
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of in IndisputableMonolith.Information.PhysicsComplexityStructure decl_use
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that in IndisputableMonolith.NumberTheory.PhiLadderLattice decl_use
contains in IndisputableMonolith.Numerics.Interval.Basic decl_use