[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 10 from proof inv-commit-rec (previous step)

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 = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(null, null, adr1, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), ((btree(null, r0, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * true))(zh), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), refsize-cons(istore0, alladrs0), refsize-cons(istore, as), adr1 ∈ istore, m ∈ istore, r0 ∈ zh, alladrs0 ⊆ dom nodes(istore0), alladrs0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as ⊆ dom nodes(istore), as ⊆ alladrs, istore0 ⊑ istore, ¬ dups(nx), ¬ adr1 ∈ as, height = 0 → as = ∅ ⊦ ⟪btree_commit_rec#(r0, adr1, 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 ∧ (err ≠ ESUCCESS → btree-inv(r0, adr0, istore, istore0, zh, ri, fi)) ∧ ( err = ESUCCESS → btree-inv(r0, adr1, istore, istore, zh, ri, ri) ∧ btree-inv (r0, adr0, istore[adr1.lnum, istore[adr1.lnum].ref-size:= (istore[adr1.lnum].ref-size - adr1.size)], istore0, zh[r0, zh[r0].dirty:= true], ri, fi) ∧ ¬ zh[r0].dirty))

User applied rule insert program lemma

And obtained the following new goals

Proof step 11:
adr0 = adr1, istore1 = istore, inv-journal(nx, gstore, gstore0, ginval), inv-index(istore, istore0), pstate.index?, inv-state(pstate, nx, gstore, istore, bv), ((btree(rp, r0, adr0, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh), r0 ∈ zh, zh[r0].dirty, alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore), as ∩ as0 = ∅, refsize-cons(istore, as0 ∪ as) ⊦ ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ inv-state(pstate, nx, gstore, istore, bv) ∧ inv-journal(nx, gstore, gstore0, ginval) ∧ inv-index(istore, istore0) ∧ 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), height, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh)))
Proof step 12:
dirty0 ↔ dirty, P0 = true, zh[r0] = mkznode(r, zbrar, boolvar, true, m0), adr0 = adr1, adr1 = adr1, adr2 = adr1, adr3 = adr1, alladrs1 = alladrs, as1 = as, as2 = ∅, ginval0 = ginval, gstore1 = gstore, gstore2 = gstore0, height1 = height, ino0 = ino, istore = istore, istore1 = istore, istore2 = istore, istore3 = istore0, ns0 = ns, nx0 = nx, pstate = index(m), r1 = r0, remove-refsize0(gstore0) = gstore0, remove-refsize0(istore0) = istore0, ri0 = ri, rp0 = null, elems nx ∩ ginval = ∅, as ∩ ∅ = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(null, r0, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * true))(zh), ((btree(null, null, adr1, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), ((btree(null, r0, adr1, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * true))(zh), zh[r0].dirty, pstate.index?, inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), inv-index(istore, istore0), inv-journal(nx, gstore, gstore0, ginval), inv-state(pstate, nx, gstore, istore, bv), refsize-cons(istore, as), refsize-cons(istore0, alladrs0), refsize-cons(istore, ∅ ∪ as), m ∈ istore, adr1 ∈ istore, r0 ∈ zh, as ⊆ alladrs, as ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore0), alladrs ⊆ dom nodes(istore), ∅ ⊆ dom nodes(istore), istore0 ⊑ istore, height = 0 → as = ∅ ⊦ [btree_commit_rec#(r0, adr1, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)] ( istore2 ⊑ istore ∧ inv-state(pstate, nx0, gstore1, istore, bv) ∧ inv-journal(nx0, gstore1, gstore2, ginval0) ∧ inv-index(istore, istore3) ∧ r1 ∈ zh ∧ (err = ESUCCESS → ¬ zh[r1].dirty ∧ pstate.index?) ∧ (∃ as3, alladrs0. refsize-cons(istore, as2 ∪ as3) ∧ as3 ∩ dom nodes(istore2) ⊆ as1 ∧ as3 ∩ as2 = ∅ ∧ alladrs0 ⊆ alladrs1 ∪ as3 ∧ ((btree(rp0, r1, (err = ESUCCESS ⊃ adr1;adr2), height1, err ≠ ESUCCESS, ri0, nodes(istore) | as3, as3, alladrs0, ok) * P0))(zh)) → istore1 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → bv) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ (err ≠ ESUCCESS → btree-inv(r0, adr0, istore, istore0, zh, ri, fi)) ∧ ( err = ESUCCESS → btree-inv(r0, adr1, istore, istore, zh, ri, ri) ∧ btree-inv (r0, adr0, istore[adr1.lnum, istore[adr1.lnum].ref-size:= (istore[adr1.lnum].ref-size - adr1.size)], istore0, zh[r0, zh[r0].dirty:= true], ri, fi) ∧ ¬ zh[r0].dirty)), dups(nx), adr1 ∈ as

Rule argument

spec: <toplevel> name: inv-commit-rec-ind seq: adr0 = adr1, istore1 = istore, inv-journal(nx, gstore, gstore0, ginval), inv-index(istore, istore0), pstate.index?, inv-state(pstate, nx, gstore, istore, bv), (( btree (rp, r0, adr0, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P) )(zh), r0 ∈ zh, zh[r0].dirty, alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore), as ∩ as0 = ∅, refsize-cons(istore, as0 ∪ as) ⊦ ⟪btree_commit_rec# (r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns ; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ inv-state(pstate, nx, gstore, istore, bv) ∧ inv-journal(nx, gstore, gstore0, ginval) ∧ inv-index(istore, istore0) ∧ 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), height, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P) )(zh))) subst: [ns, ino, nx, gstore0, gstore, ginval, istore0, adr, r0, bv, pstate, istore, zh, err, adr1, alladrs, adr0, rp, height, dirty, ri, as, P, as0, istore1] → [ns, ino, nx, gstore0, gstore, ginval, istore0, adr1, r0, bv, pstate, istore, zh, err, adr1, alladrs, adr1, null, height, dirty, ri, as, true, ∅, istore] flags: <all paths>

Additional information

Used simplifier rules: ⊦ inv-journal(nx, gstore, gstore0, ginval) ↔ inv(gstore \ ginval) ∧ inv(gstore0) ∧ elems nx ∩ ginval = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ remove-refsize0(gstore0) = gstore0; (s)%⊦ inv-index(istore, istore0) ↔ inv(istore) ∧ inv(istore0) ∧ istore0 ⊑ istore ∧ remove-refsize0(istore0) = istore0; (s)% ⊦ index(m).index?; (s)%⊦ inv-state(pstate, nx, gstore, istore, syn) ↔ ( pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn); (s)% ⊦ ¬ index(m).journal?; (s)% ⊦ index(m).lnum = m; (s)% ⊦ unbuffered ≠ index(m); (s)%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) → (( btree (rp, r0, adr, height, dirty0, ri, iast | as, as, alladrs, nstat) * P) )(zh0); (ss)% ⊦ ∅ ⊆ as; (s)% ⊦ as ∩ ∅ = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ as ∪ ∅ = as; (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]% ⊦ mkznode(r, zbrar, boolvar, boolvar0, m).dirty ↔ boolvar0; (s)