[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 31 from proof refine-add2 (previous step)

1 = # ax0, abs-log(nx0, gstore1 ↓) = abs-log(nx, gstore ↓) + adr3 ' + ax0, ax0 = adr1 + ax, nodes(gstore1 ↓, jro) = nodes(gstore ↓, jro0)[adr3, nd2][ax0, nd1 '], remove-refsize0(gstore0) = gstore0, elems nx ∩ jro0 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore = ∅, elems nx0 ∩ jro = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ jro0, dom gstore0 ∪ elems nx0 = dom gstore1 ∪ jro, jro1.keys.inodes?, p_os.keys.inodes?, jro1 < jmaxino1, p_os < pmaxino, ri.inos < jmaxino1, fi1.inos < pmaxino, inv(gstore \ jro0), inv(gstore0), inv(gstore1 \ jro), log-cons(abs-log(nx, gstore ↓), nodes(gstore ↓, jro0)), new(ax0, nodes(gstore ↓, jro0)), node-cons(nd2.key, nd2), node-cons(nd1.key, nd1), nodes-cons(adr3 ' + ax0, nd2 ' + nd1 ', gstore1, jro), refsize-cons(gstore, range ri), refsize-cons(gstore0, range fi1), refsize-cons(gstore1, range ri), ri-cons(ri, nodes(gstore ↓, jro0)), ri-inj(ri), ri-inj(fi1), pmaxino ≤ jmaxino1, ri.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore, fi1.lebs ⊆ dom gstore0, dom gstore ⊆ dom gstore1, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore1, jro), jvalid5 → jvalid4, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ∃ adr4, adr5. ¬ adr4 ∈ nodes(gstore ↓, jro0) ∧ ¬ adr5 ∈ nodes(gstore ↓, jro0) ∧ adr4 ≠ adr5 ∧ ⟨adr0 := adr4, adr := adr5; fs := nodes(gstore ↓, jro0)[adr0, nd2][adr, nd1], log := (abs-log(nx, gstore ↓) + adr0 ') + adr ', err := ESUCCESS; jvalid := ?⟩ (log = abs-log(nx0, gstore1 ↓) ∧ fs = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ adr3 = adr0 ∧ ax0[0] = adr ∧ ESUCCESS = err), jmaxino1 = 0, pmaxino = 0, dups(ax0), dups(nx0), dups(nx), adr3 ∈ ax0, adr3 ∈ nodes(gstore ↓, jro0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 32:
abs-log(nx0, gstore1 ↓) = abs-log(nx, gstore ↓) + adr3 ' + adr1 ', nodes(gstore1 ↓, jro) = nodes(gstore ↓, jro0)[adr3, nd2][adr1, nd1], remove-refsize0(gstore0) = gstore0, elems nx ∩ jro0 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore = ∅, elems nx0 ∩ jro = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ jro0, dom gstore0 ∪ elems nx0 = dom gstore1 ∪ jro, adr3 ≠ adr1, jmaxino1 ≠ 0, pmaxino ≠ 0, jro1.keys.inodes?, p_os.keys.inodes?, jro1 < jmaxino1, p_os < pmaxino, ri.inos < jmaxino1, fi1.inos < pmaxino, inv(gstore \ jro0), inv(gstore0), inv(gstore1 \ jro), log-cons(abs-log(nx, gstore ↓), nodes(gstore ↓, jro0)), node-cons(nd2.key, nd2), node-cons(nd1.key, nd1), nodes-cons(adr3 ' + adr1 ', nd2 ' + nd1 ', gstore1, jro), refsize-cons(gstore, range ri), refsize-cons(gstore0, range fi1), refsize-cons(gstore1, range ri), ri-cons(ri, nodes(gstore ↓, jro0)), ri-inj(ri), ri-inj(fi1), adr1 ∈ abs-log(nx0, gstore1 ↓), pmaxino ≤ jmaxino1, ri.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore, fi1.lebs ⊆ dom gstore0, dom gstore ⊆ dom gstore1, ¬ dups(nx0), ¬ dups(nx), ¬ adr3 ∈ nodes(gstore ↓, jro0), ¬ adr1 ∈ nodes(gstore ↓, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore1, jro), jvalid5 → jvalid4, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ∃ adr2, adr4. ¬ adr2 ∈ nodes(gstore ↓, jro0) ∧ ¬ adr4 ∈ nodes(gstore ↓, jro0) ∧ adr2 ≠ adr4 ∧ ⟨adr0 := adr2, adr := adr4; fs := nodes(gstore ↓, jro0)[adr0, nd2][adr, nd1], log := (abs-log(nx, gstore ↓) + adr0 ') + adr ', err := ESUCCESS; jvalid := ?⟩ (log = abs-log(nx0, gstore1 ↓) ∧ fs = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ adr3 = adr0 ∧ adr1 = adr ∧ ESUCCESS = err)

Rule argument

Additional information

Used simplifier rules: in-15: ⊦ ax + adr ' = ax0 → adr ∈ ax0; (forward) from specification list[address-list-set]% ⊦ ¬ dups([]); (s)% ⊦ ¬ adr ∈ []; (s)% ⊦ new([], fs); (s)% ⊦ fs[adr ', nd '] = fs[adr, nd]; (s)% ⊦ ax + [] = ax; (s)% ⊦ (adr ' + ax)[0] = adr; (s)% ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)% ⊦ adr ∈ adr0 ' + ax ↔ ¬(adr ≠ adr0 ∧ ¬ adr ∈ ax); (ss)% ⊦ new(adr ' + log, fs) ↔ ¬ adr ∈ fs ∧ new(log, fs); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ # ax = 0 ↔ ax = []; (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ adr + ax = adr ' + ax; (s)