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Proof step 9 from proof btree-clean-alladrs (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
((r0 => mkznode(rp, zbrar, false, false, nused) * bnode(r0, zbrar, 0, nused, nused, height0 + 1, false, ri, iast, as -- adr, alladrs -- adr) * P) )(zh0), nsize(nused, nstat), unmodified(adr, mkznode(rp, zbrar, false, false, nused), iast), adr ∈ as, adr ∈ alladrs, adr ∈ iast, as ⊆ alladrs, as ⊆ dom iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs ⊦ # as = 0, # alladrs = 0, alladrs = ∅, as = ∅, as = alladrs

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]% ⊦ adr ∈ dom iast ↔ adr ∈ iast; (s)%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]%sub-in: ⊦ adr ∈ as ∧ as ⊆ as0 → adr ∈ as0; (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)