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Proof step 24 from proof refine-add2 (previous step)

# ax = 2, abs-log(nx0, gstore1 ↓) = abs-log(nx, gstore ↓) + ax, fs = nodes(gstore ↓, jro0), log = abs-log(nx, gstore ↓), nodes(gstore1 ↓, jro) = nodes(gstore ↓, jro0)[ax, nd2 ' + nd1 '], remove-refsize0(gstore0) = gstore0, elems nx0 ∩ jro = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx0 = dom gstore1 ∪ jro, dom gstore0 ∪ elems nx = dom gstore ∪ jro0, ax ≠ [], jmaxino1 ≠ 0, pmaxino ≠ 0, p_os.keys.inodes?, jro1.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino1, p_os < pmaxino, jro1 < jmaxino1, inv(gstore1 \ jro), inv(gstore0), inv(gstore \ jro0), log-cons(abs-log(nx, gstore ↓), nodes(gstore ↓, jro0)), new(ax, nodes(gstore ↓, jro0)), node-cons(nd1.key, nd1), node-cons(nd2.key, nd2), nodes-cons(ax, nd2 ' + nd1 ', gstore1, jro), refsize-cons(gstore1, range ri), refsize-cons(gstore0, range fi1), refsize-cons(gstore, range ri), ri-cons(ri, nodes(gstore ↓, jro0)), ri-inj(fi1), ri-inj(ri), pmaxino ≤ jmaxino1, dom gstore ⊆ dom gstore1, fi1.lebs ⊆ dom gstore0, ri.lebs ⊆ dom gstore, ri.lebs ⊆ dom gstore1, ¬ dups(nx0), ¬ dups(nx), ¬ dups(ax), jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), jvalid5 → jvalid4, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore1, jro) ⊦ ∃ err0. ⌊err0⌋ ∧ ⟨err := err0; adr0 := ?, adr := ?; jvalid := ?⟩ ( abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) ∧ nodes(gstore ↓, jro0) = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ ax.head = adr0 ∧ ax[1] = adr ∧ ESUCCESS = err), ⟨choose adr1, adr2 with new(adr1 ' + adr2 ', fs) ∧ ¬ dups(adr1 ' + adr2 ') in {adr0 := adr1, adr := adr2; fs := fs[adr0, nd2][adr, nd1], log := (log + adr0) + adr, err := ESUCCESS} ifnone err := ENOSPC; jvalid := ?⟩ (log = abs-log(nx0, gstore1 ↓) ∧ fs = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ ax.head = adr0 ∧ ax[1] = adr ∧ ESUCCESS = err)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 25:
# ax = 2, abs-log(nx0, gstore1 ↓) = abs-log(nx, gstore ↓) + ax, ax = adr3 + ax0, fs = nodes(gstore ↓, jro0), log = abs-log(nx, gstore ↓), nodes(gstore1 ↓, jro) = nodes(gstore ↓, jro0)[ax, nd2 ' + 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(ax, nodes(gstore ↓, jro0)), node-cons(nd2.key, nd2), node-cons(nd1.key, nd1), nodes-cons(ax, 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) ⊦ ∃ err0. ⌊err0⌋ ∧ ⟨err := err0; adr0 := ?, adr := ?; jvalid := ?⟩ ( abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) ∧ nodes(gstore ↓, jro0) = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ adr3 = adr0 ∧ ax[1] = adr ∧ ESUCCESS = err), ⟨choose adr1, adr2 with new(adr1 ' + adr2 ', fs) ∧ ¬ dups(adr1 ' + adr2 ') in {adr0 := adr1, adr := adr2; fs := fs[adr0, nd2][adr, nd1], log := (log + adr0) + adr, err := ESUCCESS} ifnone err := ENOSPC; jvalid := ?⟩ (log = abs-log(nx0, gstore1 ↓) ∧ fs = nodes(gstore1 ↓, jro) ∧ (jvalid → jvalid1) ∧ adr3 = adr0 ∧ ax[1] = adr ∧ ESUCCESS = err), ax = [], jmaxino1 = 0, pmaxino = 0, dups(nx0), dups(nx), dups(ax)

Rule argument

spec: list-data, inst: address-list-set name: elim-list-c seq: ax ≠ [] ⊦ adr = ax.head ∧ ax0 = ax.tail ↔ ax = adr + ax0 subst: [ax, adr, ax0] → [ax, adr3, ax0]

Additional information

Used simplifier rules: ax ≠ [] ⊦ adr = ax.head ∧ ax0 = ax.tail ↔ ax = adr + ax0; (elim) Used terms: ax, adr3, ax0