[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 73 from proof inv-commit-rec-ind (previous step)

# asar0 = # asar, # zbrar0 = BRANCH_SIZE, # asar = (n1 + n0) + 1, zbrar0[n0] = mkzbranch(key, adr2, r1), istore1[adr] = istore3[adr], zh0[r1] = mkznode(r3, zbrar1, boolvar2, true, m2), istore[adr] = istore3[adr], pstate0 = index(m1), remove-refsize0(gstore1) = gstore1, remove-refsize0(istore0) = istore0, union(asar0, 0, # asar) = alladrs -- adr, union(asar, 0, # asar) ∩ as0 = ∅, as ∩ as0 = ∅, elems nx ∩ ginval1 = ∅, alladrs0 ∩ as0 = ∅, alladrs0 ∩ (union(asar, 0, # asar) \ asar[n0]) = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, (( btree(r0, r1, adr4, height1, false, key-range(zbrar0, n0, # asar, ri), nodes(istore3) | alladrs0, alladrs0, alladrs0, 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) )(zh2), pstate0.index?, n0 + 1 < BRANCH_SIZE, # asar < BRANCH_SIZE, n0 < BRANCH_SIZE, n0 < # zh2[r0].zbranches, 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(istore3, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ alladrs0), refsize-cons(istore, as0 ∪ union(asar, 0, # asar)), refsize-cons(istore1, as0 ∪ as), sorted(zbrar0, 0, n1 + n0), m1 ∈ istore3, m3 ∈ istore3, m ∈ istore3, adr ∈ istore, adr ∈ istore3, r1 ∈ zh0, adr ∈ istore1, r0 ∈ zh1, m ∈ istore1, m3 ∈ istore, m ∈ istore, adr ∈ alladrs, r1 ∈ zh2, alladrs0 ⊆ dom nodes(istore3), union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ alladrs, alladrs0 ∩ dom nodes(istore) ⊆ asar[n0], 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), alladrs0 ∩ dom nodes(istore) ⊆ asar0[n0], alladrs0 ∩ dom nodes(istore) ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), istore1 ⊑ istore3, istore1 ⊑ istore, istore0 ⊑ istore1, istore0 ⊑ istore, istore ⊑ istore3, istore0 ⊑ istore3, height1 = 0 → asar[n0] = ∅, pstate0.journal? → nx ≠ [] ∧ m1 = nx.last ∧ m1 ∈ gstore2, ∀ 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), ∀ 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))) ⊦ # alladrs = 0, alladrs = ∅, pstate0 = unbuffered, zh2[r1].dirty, dups(nx), adr ∈ as, adr ∈ union(asar, 0, # asar), adr2 ∈ asar[n0], commit-inv (n0 + 1, pstate0, istore1, istore0, nx, gstore2, gstore1, ginval1, syn0, ESUCCESS, rp, r0, adr, height1 + 1, true, ri, istore3, as, as0, alladrs, P, zh2[r0, zh2[r0].zbranches:= zh2[r0].zbranches[n0, zh2[r0].zbranches[n0].adr:= adr4]]) ∧ ¬ n1 < zh2[r0].usedsize - n0 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 74:
# asar = (n1 + n0) + 1, # zbrar0 = BRANCH_SIZE, # asar0 = # asar, istore[adr] = istore3[adr], zh0[r1] = mkznode(r3, zbrar1, boolvar2, true, m2), istore1[adr] = istore3[adr], zbrar0[n0] = mkzbranch(key, adr2, r1), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, union(asar0, 0, # asar) = alladrs -- adr, alladrs0 ∩ (union(asar, 0, # asar) \ asar[n0]) = ∅, alladrs0 ∩ as0 = ∅, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, union(asar, 0, # asar) ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, (( btree(r0, r1, adr4, height1, false, key-range(zbrar0, n0, # asar, ri), nodes(istore3) | alladrs0, alladrs0, alladrs0, 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) )(zh2), n0 < # zh2[r0].zbranches, 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)), refsize-cons(istore3, (as0 ∪ (union(asar, 0, # asar) \ asar[n0])) ∪ alladrs0), sorted(zbrar0, 0, n1 + n0), r1 ∈ zh2, adr ∈ alladrs, m ∈ istore, m3 ∈ istore, m ∈ istore1, r0 ∈ zh1, adr ∈ istore1, r1 ∈ zh0, adr ∈ istore3, adr ∈ istore, m ∈ istore3, m3 ∈ istore3, m1 ∈ istore3, alladrs0 ∩ dom nodes(istore) ⊆ dom nodes(istore) ∩ union(asar, 0, # asar), alladrs0 ∩ dom nodes(istore) ⊆ asar0[n0], 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, alladrs0 ∩ dom nodes(istore) ⊆ asar[n0], union(asar, 0, # asar) ∩ dom nodes(istore1) ⊆ alladrs, alladrs0 ⊆ dom nodes(istore3), istore0 ⊑ istore3, istore ⊑ istore3, istore0 ⊑ istore, istore0 ⊑ istore1, istore1 ⊑ istore, istore1 ⊑ istore3, 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), ∀ 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))) ⊦ # alladrs = 0, alladrs = ∅, zh2[r1].dirty, dups(nx), adr2 ∈ asar[n0], adr ∈ union(asar, 0, # asar), adr ∈ as, commit-inv (n0 + 1, index(m1), istore1, istore0, nx, gstore2, gstore1, ginval1, syn0, ESUCCESS, rp, r0, adr, height1 + 1, true, ri, istore3, as, as0, alladrs, P, zh2[r0, zh2[r0].zbranches:= zh2[r0].zbranches[n0, zh2[r0].zbranches[n0].adr:= adr4]]) ∧ ¬ n1 < zh2[r0].usedsize - n0 + 1

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ index(m).journal?; (s)% ⊦ index(m).index?; (s)% ⊦ unbuffered ≠ index(m); (s)