| zh0[r0] = mkznode(r2, zbrar0, boolvar0, false, m1), istore[adr3] = istore2[adr3], zh[r0] = mkznode(r1, zbrar, true, true, m0), dom nodes(istore2) = dom nodes(istore) ++ adr, remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, elems nx ∩ ginval1 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, ((btree(rp, r0, adr3, 0, dirty, ri, nodes(istore), ∅, ∅ ++ adr3, ok) * P))(zh), ((btree(rp, r0, adr3, 0, dirty, ri, ∅, ∅, ∅ ++ adr3, ok) * P))(zh), ((btree(rp, r0, adr, 0, false, ri, nodes(istore2), ∅ ++ adr, ∅ ++ adr, ok) * P))(zh0), btree-sub-clean(r0, m0, zh), inv(istore0), inv(istore), inv(gstore1), inv(gstore2 \ ginval1), inv(istore2), refsize-cons(istore, as0), refsize-cons(istore2, as0 ++ adr), adr ∈ istore2, adr3 ∈ istore2, m ∈ istore2, adr3 ∈ istore, r0 ∈ zh, m ∈ istore, r0 ∈ zh0, m2 ∈ istore2, as0 ⊆ dom nodes(istore), istore0 ⊑ istore, istore ⊑ istore2, istore0 ⊑ istore2, ∀ height0, zh, r0, r, zbrar, boolvar, m0, adr0, adr1, istore1, istore, pstate, m, istore0, gstore0, nx, ginval, as, as0, gstore, rp, dirty, ri, alladrs, P, adr, ino, ns, err, bv. height0 < 0 → zh[r0] = mkznode(r, zbrar, boolvar, true, m0) ∧ adr0 = adr1 ∧ istore1 = istore ∧ pstate = index(m) ∧ remove-refsize0(istore0) = istore0 ∧ remove-refsize0(gstore0) = gstore0 ∧ elems nx ∩ ginval = ∅ ∧ as ∩ as0 = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ ((btree(rp, r0, adr1, height0, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh) ∧ inv(istore0) ∧ inv(istore) ∧ inv(gstore0) ∧ inv(gstore \ ginval) ∧ refsize-cons(istore, as0 ∪ as) ∧ m ∈ istore ∧ r0 ∈ zh ∧ as0 ⊆ dom nodes(istore) ∧ alladrs ⊆ dom nodes(istore) ∧ as ⊆ alladrs ∧ as ⊆ dom nodes(istore) ∧ istore0 ⊑ istore ∧ ¬ dups(nx) ∧ ¬ adr1 ∈ as ∧ (height0 = 0 → as = ∅) → ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → bv) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r0 ∈ zh ∧ (err = ESUCCESS → ¬ zh[r0].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, as0 ∪ as1) ∧ as1 ∩ dom nodes(istore1) ⊆ as ∧ as1 ∩ as0 = ∅ ∧ alladrs1 ⊆ alladrs ∪ as1 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr1;adr0), height0, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh))) ⊦ dups(nx), adr ∈ istore, adr ∈ istore0, ∃ as, alladrs. refsize-cons(istore2, as0 ∪ as) ∧ as ∩ dom nodes(istore) = ∅ ∧ alladrs ⊆ as ++ adr3 ∧ ((btree(rp, r0, adr, 0, false, ri, nodes(istore2) | as, as, alladrs, ok) * P))(zh0) |
Used simplifier rules: ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ as ∪ ∅ = as; (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%⊦ ¬ adr ∈ istore → (as ++ adr) ∩ dom nodes(istore) = as ∩ dom nodes(istore); (s)% ⊦ as ∩ ∅ = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ ∅ ++ adr ⊆ as ↔ adr ∈ as; (s)% ⊦ adr ∈ as ++ adr0 ↔ adr = adr0 ∨ adr ∈ as; (s)% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) → (( btree (rp, r0, adr, height, dirty0, ri, iast | as, as, alladrs, nstat) * P) )(zh0); (ss) Used terms: ∅ ++ adr, ∅ ++ adr