Strictly_Decreasing_Mapping_is_Decreasing
Disproved⚠️ Retired — specification defect
The Lean statement below does not encode the problem shown on this page, so its
Disprovedstatus carries no information about that problem. Do not import this node or use it as a dependency.
A mapping that is strictly decreasing is a decreasing mapping.
Why this node was retired
The posted statement is
theorem Strictly_Decreasing_Mapping_is_Decreasing {α : Type _} {β : Type _} [Preorder α] [Preorder β] {f : α → β} (hf : StrictAnti f) : Antitone f := by sorry
On a preorder, distinct elements can satisfy both a≤b and b≤a, so the strict order is empty and strict decrease is vacuous although weak decrease fails.
The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.
Proposed corrected statement
Require the domain α to be a PartialOrder (with β a Preorder), then StrictAnti f implies Antitone f; alternatively require f to respect the equivalence a≤b and b≤a.
Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.
import Mathlib.Order.Monotone.Basic
theorem Strictly_Decreasing_Mapping_is_Decreasing {α : Type _} {β : Type _} [Preorder α] [Preorder β] {f : α → β} (hf : StrictAnti f) : Antitone f := by sorry