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Proof step 2 from proof min-put-ls (previous step)

st.min < a ⊦ st[a, d].min = st.min, st = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
st.keys.min < a, st.keys.min ∈ st ⊦ st[a, d].min = st.keys.min, st = ∅

Rule argument

Additional information

Used simplifier rules: min-def: ⊦ st ≠ ∅ → st.min = st.keys.min;%Min-in: ⊦ st ≠ ∅ → st.min ∈ st;%Min-min: ⊦ a ∈ st → ¬ a < st.min;%inf: ⊦ a ∈ st → a ∈ st.keys;%Keys: ⊦ a ∈ st.keys ↔ a ∈ st;% ⊦ ¬ a < a; (s)