[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 41 from proof inv-commit-rec-ind (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 42:
# asar = # asar0, # zbrar0 = BRANCH_SIZE, # asar0 = m1 + 1, zh0[zbrar0[n0].child] = mkznode(r3, zbrar1, boolvar2, true, m2), istore2[adr3] = istore[adr3], adr = zbrar0[n0].adr, err = ESUCCESS, pstate = index(m3), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, union(asar0, 0, # asar0) = alladrs -- adr3, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, union(asar, 0, # asar0) ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree (r0, zbrar0[n].child, zbrar0[n].adr, height1, true, key-range(zbrar0, n, # asar0, ri), nodes(istore) | union(asar, 0, # asar0), asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r0 => mkznode(rp, zbrar0, false, true, # asar0)) * P) )(zh0), ((btree(rp, r0, adr3, height1 + 1, true, ri, nodes(istore) | union(asar, 0, # asar0), union(asar, 0, # asar0), alladrs, ok) * P))(zh0), n0 + 1 < BRANCH_SIZE, n0 < # asar0, # asar0 < BRANCH_SIZE, n0 < BRANCH_SIZE, btree-sub-clean(r0, n0, zh0), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), inv(istore0), inv(istore2), inv(gstore1), inv(gstore2 \ ginval1), inv(istore), refsize-cons(istore2, as0 ∪ as), refsize-cons(istore, as0 ∪ union(asar, 0, # asar0)), sorted(zbrar0, 0, m1), adr3 ∈ istore, adr3 ∈ alladrs, m ∈ istore, m3 ∈ istore, m ∈ istore2, r0 ∈ zh1, adr3 ∈ istore2, as0 ⊆ dom nodes(istore2), alladrs ⊆ dom nodes(istore2), as ⊆ alladrs, as ⊆ dom nodes(istore2), union(asar, 0, # asar0) ∩ dom nodes(istore2) ⊆ as, union(asar, 0, # asar0) ∩ dom nodes(istore2) ⊆ alladrs, istore0 ⊑ istore, istore0 ⊑ istore2, istore2 ⊑ istore, ∀ n. n < # asar0 → asar[n] ⊆ dom nodes(istore) ∩ union(asar, 0, # asar0), ∀ n. n < # asar0 → asar[n] ⊆ alladrs, ∀ n. n < # asar0 → asar[n] ⊆ dom nodes(istore), ∀ n. n < # asar0 → asar[n] ⊆ dom nodes(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 < 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#(zbrar0[n0].child, adr1, istore0, ginval1, gstore2, gstore1, nx, maxino, ginval; adr, err, zh0, istore, pstate, syn); n := n0; if err = ESUCCESS then zh0[r0].zbranches[n].adr := adr; 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) ∧ (zh0[r0].usedsize - n) + n0 < # asar0), # asar0 = 0, # alladrs = 0, zbrar0[n0].child = null, alladrs = ∅, dups(nx), adr3 ∈ as, adr3 ∈ union(asar, 0, # asar0)

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# asar0, m1]

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # asar0, m1