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Proof step 24 from proof btree-mark-dirty (previous step)

zh[r] = mkznode(rp0, zbrar, false, false, height0), ((bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), r ∈ zh, adr ∈ iast, adr ∈ alladrs, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((r =>_ * bnode(r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ zh ≡ zh0 -- r → (((r =>_ * P0))(zh0) ↔ P0(zh) ∧ r ∈ zh0)
Proof step 26:
zh[r] = mkznode(rp0, zbrar, false, false, height0), ((bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ alladrs, adr ∈ iast, r ∈ zh, zh -- r ≡ zh -- r, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((bnode(r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r) ∧ r ∈ zh

Rule argument

spec: separation, inst: zheap name: free-unknown seq: ⊦ zh ≡ zh0 -- r → (((r =>_ * P0))(zh0) ↔ P0(zh) ∧ r ∈ zh0) subst: [zh, zh0, P0, r] → [zh -- r, zh, bnode (r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P, r] flags: <all paths>

Additional information

Used simplifier rules: ⊦ zh ≡ zh; (s) Used terms: zh -- r, zh, bnode (r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P, r