Proof step 51 from proof inv-commit-rec-ind (previous step)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 52:

  # asar = (n1 + n0) + 1, # zbrar0 = BRANCH_SIZE, # asar0 = # asar, zh0[r1] = mkznode(r3, zbrar1, boolvar2, true, m2), istore2[adr3] = istore[adr3], zbrar0[n0] = mkzbranch(key, adr2, r1), err = ESUCCESS, pstate = index(m3), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, union(asar0, 0, # asar) = alladrs -- adr3, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, union(asar, 0, # asar) ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, # alladrs ≠ 0, alladrs ≠ ∅, ((btree(rp, r0, adr3, height1 + 1, true, ri, nodes(istore) | union(asar, 0, # asar), union(asar, 0, # asar), alladrs, ok) * P))(zh0), (( btree(r0, r1, adr2, height1, true, key-range(zbrar0, n0, # asar, ri), nodes(istore) | asar[n0], asar[n0], asar0[n0], ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (0, n0 ) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (1 + n0, n1 ) * r0 => mkznode(rp, zbrar0, false, true, # asar) * P) )(zh0), n0 < BRANCH_SIZE, # asar < BRANCH_SIZE, n0 + 1 < BRANCH_SIZE, btree-sub-clean(r0, n0, zh0), disj(asar0, 0, # asar), disj(asar, 0, # asar), inv(istore0), inv(istore2), inv(gstore1), inv(gstore2 \ ginval1), inv(istore), refsize-cons(istore2, as0 ∪ as), refsize-cons(istore, as0 ∪ union(asar, 0, # asar)), sorted(zbrar0, 0, n1 + n0), adr3 ∈ istore, adr3 ∈ alladrs, m ∈ istore, m3 ∈ istore, m ∈ istore2, r0 ∈ zh1, adr3 ∈ istore2, r1 ∈ zh0, asar[n0] ⊆ dom nodes(istore), asar[n0] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), asar[n0] ⊆ asar0[n0], as0 ⊆ dom nodes(istore2), alladrs ⊆ dom nodes(istore2), as ⊆ alladrs, as ⊆ dom nodes(istore2), union(asar, 0, # asar) ∩ dom nodes(istore2) ⊆ as, union(asar, 0, # asar) ∩ dom nodes(istore2) ⊆ alladrs, istore0 ⊑ istore, istore0 ⊑ istore2, istore2 ⊑ istore, ¬ dups(nx), ¬ adr3 ∈ as, ¬ adr3 ∈ union(asar, 0, # asar), ¬ adr2 ∈ asar[n0], height1 = 0 → asar[n0] = ∅, ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ alladrs, ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore), ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore2), ∀ n. n < # asar → asar[n] ⊆ alladrs, ∀ n. n < # asar → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), ∀ 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 < height1 +1 → 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))) ⊦ ⟪btree_commit_rec#(r1, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr2, err, zh0, istore, pstate, syn); n := n0; if err = ESUCCESS then zh0[r0].zbranches[n].adr := adr2; n := n + 1⟫ ( commit-inv (n, pstate, istore2, istore0, nx, gstore2, gstore1, ginval1, syn, err, rp, r0, adr3, height1 + 1, true, ri, istore, as, as0, alladrs, P, zh0) ∧ ¬ n1 < zh0[r0].usedsize - n), as0 ∪ (union(asar, 0, # asar) \ asar[n0]) ⊆ dom nodes(istore) ∧ asar0[n0] ⊆ dom nodes(istore)

Rule argument

Additional information

Used simplifier rules: n < m, m ≤ # asar ⊦ asar[n] ∪ union(asar, 0, m) = union(asar, 0, m); (s)%⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → (( btree(rp, r, adr, height, true, fi, iast, as, alladrs, nstat) * P) )(zh); (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%n < m, m ≤ # asar ⊦ as ∩ union(asar, 0, m) = ∅ → as ∩ asar[n] = ∅; (s)% ⊦ m + 1 ≠ 0; (s)% ⊦ n = n0 + n1 → (n ≤ n0 ↔ n1 = 0 ∧ n = n0); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ as0 ∪ (as1 \ as0) = as0 ∪ as1; (s)%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ 0 ≤ m; (s)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ as1 ⊆ as2 → (as0 \ as1 ⊆ as2 ↔ as0 ⊆ as2); (s)% ⊦ as ∪ as0 ⊆ as1 ↔ as ⊆ as1 ∧ as0 ⊆ as1; (ws)% ⊦ m ≤ m; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ (as0 ∪ (as1 \ as)) ∩ as = as0 ∩ as; (s)