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Proof step 3 from proof btree-root-in-heap (previous step)

((btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P))(zh), as ⊆ dom iast, as ⊆ alladrs ⊦ r = null, r ∈ zh

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: btree seq: ⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified (adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode (r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty0, height, adr, r0, rp] → [zh, P, nstat, alladrs, as, iast, fi, dirty, height, adr, r, rp] flags: <all paths>

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]% ⊦ ((r => znd * P0))(zh) → r ∈ zh; (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s) Used terms: zh, P, nstat, alladrs, as, iast, fi, dirty, height, adr, r, rp