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Proof step 11 from proof min-put (previous step)

⊦ # st.keys = 0, st.keys = ∅, (st.keys ++ a).min = st.min, a = (st.keys ++ a).min, a < st.keys.min

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ a ∈ s ∧ ¬ b < a → (s ++ b).min = s.min
Proof step 13:
st.keys.min ∈ st.keys ⊦ # st.keys = 0, st.keys = ∅, st.keys.min = st.min, a = st.keys.min, a < st.keys.min

Rule argument

spec: oset name: min-ins-01 seq: ⊦ a ∈ s ∧ ¬ b < a → (s ++ b).min = s.min subst: [a, s, b] → [st.keys.min, st.keys, a] flags: <all paths>

Additional information

Used simplifier rules: ⊦ s.min ∈ s ↔ s ≠ ∅; (ss) Used terms: st.keys.min, st.keys, a