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Proof step 3 from proof ls-add (previous step)

¬(∀ b0. b0 = a ∨ b0 ∈ s → b0 < b), ∀ b0. b0 ∈ s → b0 < b ⊦ ¬ a < b

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
b0 ∈ s, a < b, s.min ∈ s, s.max ∈ s, ∀ b0. b0 ∈ s → b0 < b ⊦ s = ∅, # s = 0, b0 < s.min, s.max < b0, b0 < b, s.max < b

Rule argument

Additional information

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