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Proof step 7 from proof btree-repl-dirty-address (previous step)

zh[r] = mkznode(r0, zbrar, boolvar, true, m), r ∈ zh, as ⊆ alladrs, as ⊆ dom iast, height = 0 → as = ∅, ∃ r1, zh1, zbrar, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty → true) ∧ (adr0 ∈ as ↔ ¬ dirty) ∧ adr0 ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr0, 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 -- adr0, alladrs -- adr0) * P) )(zh1) ⊦ ((btree(rp, r, adr1, height, true, ri, iast, as, (alladrs -- adr0) ++ adr1, nstat) * P))(zh), adr0 ∈ as, adr1 ∈ alladrs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
((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 = ∅, ((btree(rp, r, adr1, height, true, ri, iast, as, (alladrs -- adr0) ++ adr1, nstat) * P))(zh), adr1 ∈ alladrs, adr0 ∈ as

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]% ⊦ ((r => znd * P0))(zh) → r ∈ zh; (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)%⊦ mkznode(r, zbrar, boolvar, boolvar0, m) = mkznode(r0, zbrar0, boolvar1, boolvar2, m0) ↔ r = r0 ∧ zbrar = zbrar0 ∧ (boolvar ↔ boolvar1) ∧ (boolvar0 ↔ boolvar2) ∧ m = m0; (ws)% ⊦ ((r => znd * P0))(zh) → (zh[r] = znd0 ↔ znd = znd0); (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]%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]% ⊦ r ∈ zh → r ≠ null; (s)% ⊦ ¬ adr ∈ as → as -- adr = as; (s)