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Proof step 1 from proof min-delmin-01 (conclusion)

⊦ s ≠ ∅ ∧ s ≠ ∅ ++ s.min → (s -- s.min).min ∈ s

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 2:
s.min ∈ s, s.max ∈ s ⊦ s = ∅ ++ s.min, s = ∅, # s = 0, (s -- s.min).min ∈ s

Rule argument

Additional information

Used simplifier rules: Max-in: ⊦ s ≠ ∅ → s.max ∈ s;%Min-in: ⊦ s ≠ ∅ → s.min ∈ s;%Max-max: ⊦ a ∈ s → ¬ s.max < a;%Min-min: ⊦ a ∈ s → ¬ a < s.min;%minmax-04: ⊦ s ≠ ∅ → ¬ s.max < s.min;%irreflexivity: ⊦ ¬ a < a; from specification oelem%ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic%size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic