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Proof step 9 from proof in-union-ext (previous step)

adr ∈ asar[n + m0] ∪ union(asar, n, m0), adr ∈ union(asar, n, m0), ∃ n0. n ≤ n0 ∧ n0 < n + m0 ∧ adr ∈ asar[n0] ⊦ #(asar[n + m0] ∪ union(asar, n, m0)) = 0, asar[n + m0] ∪ union(asar, n, m0) = ∅, ∃ n0. n ≤ n0 ∧ ¬ n + m0 < n0 ∧ adr ∈ asar[n0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
n0 < n + m0, adr ∈ asar[n + m0] ∪ union(asar, n, m0), adr ∈ union(asar, n, m0), adr ∈ asar[n0], n ≤ n0 ⊦ #(asar[n + m0] ∪ union(asar, n, m0)) = 0, # union(asar, n, m0) = 0, # asar[n0] = 0, asar[n0] = ∅, m0 = 0, union(asar, n, m0) = ∅, asar[n + m0] ∪ union(asar, n, m0) = ∅, ∃ n0. n ≤ n0 ∧ ¬ n + m0 < n0 ∧ adr ∈ asar[n0]

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[address-list-set]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < m + n ↔ n ≠ 0; (s)% ⊦ m < n → n ≠ 0; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[address-list-set]