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Proof step 18 from proof btree-write-node (previous step)

((r => mkznode(r0, zbrar, false, true, m0) * bnode(r, zbrar, 0, m0, m0, height0 + 1, true, ri, iast, as, alladrs -- adr0) * P))(zh), btree-sub-clean(r, m0, zh), nsize(m0, nstat), adr0 ∈ alladrs, as ⊆ dom iast, as ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ as, ∃ r1, zh1, zbrar0, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh[r, mkznode(r0, zbrar, false, false, m0)]);r1 = r ∧ zh1 = zh[r, mkznode(r0, zbrar, false, false, m0)]) ∧ (dirty → false) ∧ (adr1 ∈ as ++ adr1 ↔ ¬ dirty) ∧ adr1 ∈ (alladrs -- adr0) ++ adr1 ∧ nsize(nused, nstat) ∧ unmodified(adr1, mkznode(r0, zbrar0, height0 + 1 = 0, dirty, nused), iast[adr1, save(mkznode(r0, zbrar, false, true, m0))]) ∧ (( r1 => mkznode(r0, zbrar0, height0 + 1 = 0, dirty, nused) * bnode (r1, zbrar0, 0, nused, nused, height0 + 1, dirty, ri, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], (as ++ adr1) -- adr1, ((alladrs -- adr0) ++ adr1) -- adr1) * P) )(zh1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
((r => mkznode(r0, zbrar, false, true, m0) * bnode(r, zbrar, 0, m0, m0, height0 + 1, true, ri, iast, as, alladrs -- adr0) * P))(zh), btree-sub-clean(r, m0, zh), nsize(m0, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ as, adr1 ∈ alladrs, adr1 ∈ iast, ∃ zbrar0, nused. nsize(nused, nstat) ∧ unmodified(adr1, mkznode(r0, zbrar0, false, false, nused), iast[adr1, save(mkznode(r0, zbrar, false, true, m0))]) ∧ (( r => mkznode(r0, zbrar0, false, false, nused) * bnode (r, zbrar0, 0, nused, nused, height0 + 1, false, ri, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], as -- adr1, alladrs -- adr0 -- adr1) * P) )(zh[r, mkznode(r0, zbrar, false, false, m0)])

Rule argument

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;% ⊦ (as ++ adr) -- adr = as -- adr; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ ¬ adr ∈ as → (adr ∈ (as -- adr0) ++ adr1 ↔ adr = adr1); (s)% ⊦ adr ∈ as ++ adr0 ↔ adr = adr0 ∨ adr ∈ as; (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]