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Proof step 10 from proof btree-repl-dirty-address (previous step)

((r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast, height = 0 → as = ∅ ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ as, adr1 ∈ alladrs, ∃ r1, zh1, zbrar, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty → true) ∧ (adr1 ∈ as ↔ ¬ dirty) ∧ adr1 ∈ (alladrs -- adr0) ++ adr1 ∧ nsize(nused, nstat) ∧ unmodified(adr1, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r1 => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode(r1, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr1, ((alladrs -- adr0) ++ adr1) -- adr1) * P) )(zh1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
((r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ dom iast, as ⊆ alladrs, height = 0 → as = ∅ ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ alladrs, adr0 ∈ as, ∃ zbrar, dirty, nused0. (adr1 ∈ as ↔ ¬ dirty) ∧ nsize(nused0, nstat) ∧ unmodified(adr1, mkznode(rp, zbrar, height = 0, dirty, nused0), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused0) * bnode(r, zbrar, 0, nused0, nused0, height, dirty, ri, iast, as -- adr1, alladrs -- adr0 -- adr1) * P) )(zh)

Rule argument

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;% ⊦ (as ++ adr) -- adr = as -- adr; (s)% ⊦ ¬ adr ∈ as → (adr ∈ (as -- adr0) ++ adr1 ↔ adr = adr1); (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]