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Proof step 21 from proof btree-clean-fwd (previous step)

((bnode(r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast, alladrs1 -- adr, alladrs1 -- adr) * P1))(zh3), (( r2 => mkznode(rp0, zbrar0, false, false, nused) * bnode(r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast, alladrs0 -- adr, alladrs0 -- adr) * P0) )(zh2), ((r => mkznode(rp1, zbrar, false, false, nused)))(zh4), nsize(nused, nstat), unmodified(adr, mkznode(rp1, zbrar, false, false, nused), iast), unmodified(adr, mkznode(rp0, zbrar0, false, false, nused), iast), adr ∈ iast, adr ∈ alladrs1, adr ∈ alladrs0, alladrs1 ⊆ dom iast, alladrs0 ⊆ dom iast, zh4 ⊥ zh3, (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh2 = zh0), (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ ((P * P2))(zh) ↔ (∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh = zh0 ∪ zh1 ∧ P(zh0) ∧ P2(zh1))
Proof step 23:
((bnode(r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast, alladrs1 -- adr, alladrs1 -- adr) * P1))(zh3), ((r => mkznode(rp1, zbrar, false, false, nused)))(zh4), nsize(nused, nstat), unmodified(adr, mkznode(rp0, zbrar0, false, false, nused), iast), unmodified(adr, mkznode(rp1, zbrar, false, false, nused), iast), adr ∈ alladrs0, adr ∈ alladrs1, adr ∈ iast, alladrs0 ⊆ dom iast, alladrs1 ⊆ dom iast, zh4 ⊥ zh3, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh2 = zh0), ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1, ∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh2 = zh0 ∪ zh1 ∧ ((r2 => mkznode(rp0, zbrar0, false, false, nused)))(zh0) ∧ ((bnode(r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast, alladrs0 -- adr, alladrs0 -- adr) * P0))(zh1) ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

Rule argument

spec: heap-sep, inst: zheap name: sep seq: ⊦ ((P * P2))(zh) ↔ (∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh = zh0 ∪ zh1 ∧ P(zh0) ∧ P2(zh1)) subst: [zh, P2, P] → [zh2, bnode (r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast, alladrs0 -- adr, alladrs0 -- adr) * P0, r2 => mkznode(rp0, zbrar0, false, false, nused)] flags: <all paths>

Additional information

Used terms: zh2, bnode (r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast, alladrs0 -- adr, alladrs0 -- adr) * P0, r2 => mkznode(rp0, zbrar0, false, false, nused)