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Proof step 47 from proof btree-write-other (previous step)

∀ rp, r, adr0, dirty, ri, iast, adr, ind, as, alladrs, nstat, P, zh. ( ((btree(rp, r, adr0, height0, dirty, ri, iast[adr, ind], as, alladrs, nstat) * P))(zh) ↔ ((btree(rp, r, adr0, height0, dirty, ri, iast, as, alladrs, nstat) * P))(zh)) ∨ adr ∈ as, ∃ r0, zh0, zbrar, dirty0, nused. (r = null ⊃ ¬ dirty0 ∧ P(zh);r0 = r ∧ zh0 = zh) ∧ (dirty0 → dirty) ∧ (adr0 ∈ as ↔ ¬ dirty0) ∧ adr0 ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar, false, dirty0, nused), iast) ∧ (( r0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(r0, zbrar, 0, nused, nused, height0 + 1, dirty0, ri, iast, as -- adr0, alladrs -- adr0) * P) )(zh0) ⊦ adr ∈ as, ∃ r0, zh0, zbrar, dirty0, nused. (r = null ⊃ ¬ dirty0 ∧ P(zh);r0 = r ∧ zh0 = zh) ∧ (dirty0 → dirty) ∧ (adr0 ∈ as ↔ ¬ dirty0) ∧ adr0 ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar, false, dirty0, nused), iast[adr, ind]) ∧ (( r0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(r0, zbrar, 0, nused, nused, height0 + 1, dirty0, ri, iast[adr, ind], as -- adr0, alladrs -- adr0) * P) )(zh0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 48:
adr0 ∈ as ↔ ¬ dirty0, (( r0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(r0, zbrar, 0, nused, nused, height0 + 1, dirty0, ri, iast, as -- adr0, alladrs -- adr0) * P) )(zh0), nsize(nused, nstat), unmodified(adr0, mkznode(rp, zbrar, false, dirty0, nused), iast), adr0 ∈ alladrs, dirty0 → dirty, (r = null ⊃ adr0 ∈ as ∧ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr0, dirty, ri, iast, adr, ind, as, alladrs, nstat, P, zh. ( ((btree(rp, r, adr0, height0, dirty, ri, iast[adr, ind], as, alladrs, nstat) * P))(zh) ↔ ((btree(rp, r, adr0, height0, dirty, ri, iast, as, alladrs, nstat) * P))(zh)) ∨ adr ∈ as ⊦ # alladrs = 0, alladrs = ∅, adr ∈ as, ∃ r1, zh1, zbrar0, dirty1, nused0. (r = null ⊃ ¬ dirty1 ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr0 ∈ as ↔ ¬ dirty1) ∧ nsize(nused0, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar0, false, dirty1, nused0), iast[adr, ind]) ∧ (( r1 => mkznode(rp, zbrar0, false, dirty1, nused0) * bnode(r1, zbrar0, 0, nused0, nused0, height0 + 1, dirty1, ri, iast[adr, ind], as -- adr0, alladrs -- adr0) * P) )(zh1)

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]