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Proof step 8 from proof del-lists-unions (previous step)

n = n0 + # x, ((ls(r, x) * del-list(rx0, n0)))(H0), # x ≤ n, H0 ⊥ H1, ∀ n, H0, H1, m, P. del-list(rx0, n)(H0) ∧ H0 ⊥ H1 → (((del-list(rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)) ⊦ ((del-list(r ' + rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1), r = null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
((ls(r, x) * del-list(rx0, n0)))(H0), H0 ⊥ H1, ∀ n, H0, H1, m, P. del-list(rx0, n)(H0) ∧ H0 ⊥ H1 → (((del-list(rx0, m) * P))(H0 ∪ H1) ↔ n = m ∧ P(H1)) ⊦ ((del-list(r ' + rx0, m) * P))(H0 ∪ H1) ↔ n0 + # x = m ∧ P(H1), r = null

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ m + n; (s)