| ⟨btree_commit_rec#(r1, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr2, err, zh0, istore, pstate, syn)⟩ (adr2 = adr4 ∧ err = err0 ∧ zh0 = zh2 ∧ istore = istore3 ∧ pstate = pstate0 ∧ (syn ↔ syn0)), ⟪btree_commit_rec#(r1, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr2, err, zh0, istore, pstate, syn)⟫ ( istore2 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore2) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r1 ∈ zh0 ∧ (err = ESUCCESS → ¬ zh0[r1].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ as1) ∧ as1 ∩ dom nodes(istore2) ⊆ asar[n0] ∧ as1 ∩ (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) = ∅ ∧ alladrs1 ⊆ asar0[n0] ∪ as1 ∧ (( btree (r0, r1, (err = ESUCCESS ⊃ adr2;adr3), height1, err ≠ ESUCCESS, key-range(zbrar0, n0, # asar, ri), nodes(istore) | as1, as1, alladrs1, ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore2) | 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(istore2) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (1 + n0, n1 ) * r0 => mkznode(rp, zbrar0, false, true, # asar) * P) )(zh0))), # asar0 = # asar, # zbrar0 = BRANCH_SIZE, # asar = (n1 + n0) + 1, zbrar0[n0] = mkzbranch(key, adr2, r1), istore1[adr] = istore[adr], zh0[r1] = mkznode(r3, zbrar1, boolvar2, true, m2), adr3 = adr2, err = ESUCCESS, istore2 = istore, pstate = index(m3), remove-refsize0(gstore1) = gstore1, remove-refsize0(istore0) = istore0, union(asar0, 0, # asar) = alladrs -- adr, union(asar, 0, # asar) ∩ as0 = ∅, as ∩ as0 = ∅, elems nx ∩ ginval1 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, (( 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), ((btree(rp, r0, adr, height1 + 1, true, ri, nodes(istore) | union(asar, 0, # asar), union(asar, 0, # asar), alladrs, ok) * P))(zh0), n0 + 1 < BRANCH_SIZE, # asar < BRANCH_SIZE, n0 < BRANCH_SIZE, btree-sub-clean(r0, n0, zh0), disj(asar, 0, # asar), disj(asar0, 0, # asar), inv(istore), inv(gstore2 \ ginval1), inv(gstore1), inv(istore1), inv(istore0), inv(istore3), refsize-cons(istore, as0 ∪ union(asar, 0, # asar)), refsize-cons(istore1, as0 ∪ as), sorted(zbrar0, 0, n1 + n0), r1 ∈ zh0, adr ∈ istore1, r0 ∈ zh1, m ∈ istore1, m3 ∈ istore, m ∈ istore, adr ∈ alladrs, adr ∈ istore, r1 ∈ zh2, union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ alladrs, union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ as, as ⊆ dom nodes(istore1), as ⊆ alladrs, alladrs ⊆ dom nodes(istore1), as0 ⊆ dom nodes(istore1), asar[n0] ⊆ asar0[n0], asar[n0] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), asar[n0] ⊆ dom nodes(istore), istore1 ⊑ istore, istore0 ⊑ istore1, istore0 ⊑ istore, istore2 ⊑ istore3, istore0 ⊑ istore3, height1 = 0 → asar[n0] = ∅, pstate0.journal? → nx ≠ [] ∧ pstate0.lnum = nx.last ∧ pstate0.lnum ∈ gstore2, pstate0.index? → pstate0.lnum ∈ istore3, pstate0 = unbuffered → syn0, err0 = ESUCCESS → ¬ zh2[r1].dirty ∧ pstate0.index?, ∀ n. n < # asar → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), ∀ n. n < # asar → asar[n] ⊆ alladrs, ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore1), ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore), ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ alladrs, ∀ n. ¬ n1 + n0 < n → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), ∃ as1, alladrs1. refsize-cons(istore3, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ as1) ∧ as1 ∩ dom nodes(istore2) ⊆ asar[n0] ∧ as1 ∩ (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) = ∅ ∧ alladrs1 ⊆ asar0[n0] ∪ as1 ∧ (( btree (r0, r1, (err0 = ESUCCESS ⊃ adr4;adr3), height1, err0 ≠ ESUCCESS, key-range(zbrar0, n0, # asar, ri), nodes(istore3) | as1, as1, alladrs1, ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore2) | 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(istore2) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (1 + n0, n1 ) * r0 => mkznode(rp, zbrar0, false, true, # asar) * P) )(zh2), ∀ 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))) ⊦ ⟪n := n0; if err0 = ESUCCESS then zh2[r0].zbranches[n].adr := adr4; n := n + 1⟫ ( commit-inv (n, pstate0, istore1, istore0, nx, gstore2, gstore1, ginval1, syn0, err0, rp, r0, adr, height1 + 1, true, ri, istore3, as, as0, alladrs, P, zh2) ∧ ¬ n1 < zh2[r0].usedsize - n), # alladrs = 0, alladrs = ∅, dups(nx), adr2 ∈ asar[n0], adr ∈ union(asar, 0, # asar), adr ∈ as |
⟨btree_commit_rec#(r1, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr2, err, zh0, istore, pstate, syn)⟩ (adr2 = adr4 ∧ err = err0 ∧ zh0 = zh2 ∧ istore = istore3 ∧ pstate = pstate0 ∧ (syn ↔ syn0)), ⟪btree_commit_rec#(r1, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr2, err, zh0, istore, pstate, syn)⟫ ( istore2 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore2) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r1 ∈ zh0 ∧ (err = ESUCCESS → ¬ zh0[r1].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ as1) ∧ as1 ∩ dom nodes(istore2) ⊆ asar[n0] ∧ as1 ∩ (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) = ∅ ∧ alladrs1 ⊆ asar0[n0] ∪ as1 ∧ (( btree (r0, r1, (err = ESUCCESS ⊃ adr2;adr3), height1, err ≠ ESUCCESS, key-range(zbrar0, n0, # asar, ri), nodes(istore) | as1, as1, alladrs1, ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore2) | 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(istore2) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (1 + n0, n1 ) * r0 => mkznode(rp, zbrar0, false, true, # asar) * P) )(zh0))), # asar = (n1 + n0) + 1, # zbrar0 = BRANCH_SIZE, # asar0 = # asar, zh0[r1] = mkznode(r3, zbrar1, boolvar2, true, m2), istore1[adr] = istore[adr], zbrar0[n0] = mkzbranch(key, adr2, r1), adr3 = adr2, err = ESUCCESS, istore2 = istore, n = n0, pstate = index(m3), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, union(asar0, 0, # asar) = alladrs -- adr, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, union(asar, 0, # asar) ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, ((btree(rp, r0, adr, 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(istore3), inv(istore0), inv(istore1), inv(gstore1), inv(gstore2 \ ginval1), inv(istore), refsize-cons(istore1, as0 ∪ as), refsize-cons(istore, as0 ∪ union(asar, 0, # asar)), sorted(zbrar0, 0, n1 + n0), r1 ∈ zh2, adr ∈ istore, adr ∈ alladrs, m ∈ istore, m3 ∈ istore, m ∈ istore1, r0 ∈ zh1, adr ∈ istore1, r1 ∈ zh0, asar[n0] ⊆ dom nodes(istore), asar[n0] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), asar[n0] ⊆ asar0[n0], as0 ⊆ dom nodes(istore1), alladrs ⊆ dom nodes(istore1), as ⊆ alladrs, as ⊆ dom nodes(istore1), union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ as, union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ alladrs, istore0 ⊑ istore3, istore2 ⊑ istore3, istore0 ⊑ istore, istore0 ⊑ istore1, istore1 ⊑ istore, err0 = ESUCCESS → ¬ zh2[r1].dirty ∧ pstate0.index?, pstate0 = unbuffered → syn0, pstate0.index? → pstate0.lnum ∈ istore3, pstate0.journal? → nx ≠ [] ∧ pstate0.lnum = nx.last ∧ pstate0.lnum ∈ gstore2, 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(istore1), ∀ n. n < # asar → asar[n] ⊆ alladrs, ∀ n. n < # asar → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), ∃ as1, alladrs1. refsize-cons(istore3, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ as1) ∧ as1 ∩ dom nodes(istore2) ⊆ asar[n0] ∧ as1 ∩ (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) = ∅ ∧ alladrs1 ⊆ asar0[n0] ∪ as1 ∧ (( btree (r0, r1, (err0 = ESUCCESS ⊃ adr4;adr3), height1, err0 ≠ ESUCCESS, key-range(zbrar0, n0, # asar, ri), nodes(istore3) | as1, as1, alladrs1, ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar, ri), nodes(istore2) | 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(istore2) | union(asar, 0, # asar), asar[n], asar0[n], ok) )⊗_ (1 + n0, n1 ) * r0 => mkznode(rp, zbrar0, false, true, # asar) * P) )(zh2), ∀ 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))) ⊦ ⟪if err0 = ESUCCESS then zh2[r0].zbranches[n].adr := adr4; n := n + 1⟫ ( commit-inv (n, pstate0, istore1, istore0, nx, gstore2, gstore1, ginval1, syn0, err0, rp, r0, adr, height1 + 1, true, ri, istore3, as, as0, alladrs, P, zh2) ∧ ¬ n1 < zh2[r0].usedsize - n), # alladrs = 0, alladrs = ∅, dups(nx), adr ∈ as, adr ∈ union(asar, 0, # asar), adr2 ∈ asar[n0]