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Proof step 5 from proof butmin-ins (previous step)

s.min < a, s.min ∈ s, s.max ∈ s ⊦ s = ∅, (s ++ a) -- (s ++ a).min = (s -- s.min) ++ a, # s = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ a ∈ s ∧ ¬ b < a → (s ++ b).min = s.min
Proof step 7:
s.min ∈ s ∧ ¬ a < s.min, s.min < a, s.min ∈ s, s.max ∈ s ⊦ s = ∅, (s ++ a) -- s.min = (s -- s.min) ++ a, # s = 0

Rule argument

spec: oset inst: name: min-ins-01 seq: ⊦ a ∈ s ∧ ¬ b < a → (s ++ b).min = s.min subst: [a, s, b] → [s.min, s, a] flags: rewrite, current

Additional information

Used simplifier rules: min-insert: ⊦ a < s.min ∧ s ≠ ∅ → (s ++ a).min = a;%del-addeq-05: ⊦ (s ++ a) -- a = s -- a; from specification set-union%del-neq-02: ⊦ s -- a ≠ s0 ++ a; from specification set-union%transitivity: ⊦ a < b ∧ b < c → a < c; (forward) from specification oelem%irreflexivity: ⊦ ¬ a < a; from specification oelem Used terms: s.min, s, a