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Proof step 8 from proof btree-refed-subseteq-alladrs (previous step)

∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs, ∃ r1, zh1, zbrar, dirty1, nused. (r = null ⊃ ¬ dirty1 ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr ∈ as ↔ ¬ dirty1) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused), iast) ∧ (( r1 => mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused) * bnode(r1, zbrar, 0, nused, nused, height0 + 1, dirty1, fi, iast, as -- adr, alladrs -- adr) * P) )(zh1) ⊦ as ⊆ alladrs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
adr ∈ as ↔ ¬ dirty0, ((r0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(r0, zbrar, 0, nused, nused, height0 + 1, dirty0, fi, iast, as -- adr, alladrs -- adr) * P) )(zh0), nsize(nused, nstat), unmodified(adr, mkznode(rp, zbrar, false, dirty0, nused), iast), adr ∈ alladrs, dirty0 → dirty, (r = null ⊃ adr ∈ as ∧ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, as ⊆ alladrs

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)