[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 77 from proof btree-clean-fwd-strong (previous step)

# zbrar0 = BRANCH_SIZE, # asar = # asar0, # asar2 = # asar0, # asar1 = # asar0, # zbrar = BRANCH_SIZE, # asar0 = (n0 + 1) + n, asar1[n] = asar2[n], asar[n] = asar0[n], zbrar0[n] = mkzbranch(key0, adr1, r4), zbrar[n] = mkzbranch(key0, adr1, r3), union(asar2, 0, # asar0) = alladrs1 -- adr, union(asar1, 0, # asar0) = alladrs1 -- adr, union(asar0, 0, # asar0) = alladrs0 -- adr, union(asar, 0, # asar0) = alladrs0 -- adr, ((btree(r2, r4, adr1, height, false, key-range(zbrar0, n, # asar0, ri0), iast0, asar0[n], asar0[n], ok) * P2))(zh), ((btree(r, r3, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast1, asar2[n], asar2[n], ok) * P))(zh3), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, # asar0)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, # asar0)))(zh5), disj(asar2, 0, # asar0), disj(asar1, 0, # asar0), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar0, 0, n0 + n), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar0), iast0), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar0), iast1), adr ∈ alladrs0, adr ∈ iast0, adr ∈ alladrs1, adr ∈ iast1, asar0[n] ⊆ dom iast0, asar2[n] ⊆ dom iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, zh4 ⊥ zh3, zh5 ⊥ zh, ((btree(r, r3, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast1, asar2[n], asar2[n], ok) * P))(zh3) ∧ ((btree(r2, r4, adr1, height, false, key-range(zbrar0, n, # asar0, ri0), iast0, asar0[n], asar0[n], ok) * P2))(zh) ∧ asar2[n] ⊆ dom iast1 ∧ asar0[n] ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height0 ∧ key-range(zbrar0, n, # asar0, ri0) = key-range(zbrar, n, # asar0, ri1) ∧ asar0[n] = asar2[n], (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0) ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar0, n, # asar0, ri0) = key-range(zbrar0, n, # asar0, ri1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 78:
# zbrar0 = BRANCH_SIZE, # asar = # asar0, # asar2 = # asar0, # asar1 = # asar0, # zbrar = BRANCH_SIZE, # asar0 = (n0 + 1) + n, asar0[n] = asar2[n], asar1[n] = asar2[n], asar[n] = asar2[n], zbrar0[n] = mkzbranch(key0, adr1, r4), zbrar[n] = mkzbranch(key0, adr1, r3), key-range(zbrar0, n, # asar0, ri0) = key-range(zbrar, n, # asar0, ri1), union(asar2, 0, # asar0) = alladrs1 -- adr, union(asar1, 0, # asar0) = alladrs1 -- adr, union(asar0, 0, # asar0) = alladrs0 -- adr, union(asar, 0, # asar0) = alladrs0 -- adr, ((btree(r2, r4, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast0, asar2[n], asar2[n], ok) * P2))(zh), ((btree(r, r3, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast1, asar2[n], asar2[n], ok) * P))(zh3), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, # asar0)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, # asar0)))(zh5), disj(asar2, 0, # asar0), disj(asar1, 0, # asar0), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar0, 0, n0 + n), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar0), iast0), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar0), iast1), adr ∈ alladrs0, adr ∈ iast0, adr ∈ alladrs1, adr ∈ iast1, asar2[n] ⊆ dom iast0, asar2[n] ⊆ dom iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, zh4 ⊥ zh3, zh5 ⊥ zh, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0) ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar, n, # asar0, ri1) = key-range(zbrar0, n, # asar0, ri1)

Rule argument

Additional information

Used simplifier rules: btree-adrs: ⊦ (( btree (rp, rp0, adr0, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ dom iast;%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]