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Proof step 1 from proof butmin-ins (conclusion)

⊦ s.min < a ∧ s ≠ ∅ → (s ++ a).butmin = s.butmin ++ a

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: Butmin: ⊦ s ≠ ∅ → s.butmin = s -- s.min;%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