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Proof step 2 from proof inv-gc-update-index (previous step)

# ax = # ndx, ax0 = ax, fi1 = fi, gstore1 = gstore, jmaxino0 = jmaxino, jro1 = jro0, ndx0 = ndx, nx0 = nx, ax.toSet ∩ range fi = ∅, fi0.inos < pmaxino, fi.inos < jmaxino, p_os < pmaxino, jro < jmaxino, inv-journal(nx, gstore, gstore0, jro0), key-cons(ndx), nodes-cons(ax, ndx, gstore, jro0), refsize-cons(gstore0, range fi0), refsize-cons(gstore, range fi), ri-inj(fi0), ri-inj(fi), pmaxino ≤ jmaxino, ndx.keys.toSet ⊆ dom fi, fi0.lebs ⊆ dom gstore0, fi.lebs ⊆ dom gstore, jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ⟪gjournal_gc_update_index#(jvalid, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid0; ax, ndx, jmaxino, fi, gstore)⟫ ( jro < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi.inos < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx, gstore, gstore0, jro0) ∧ ri-inj(fi) ∧ fi = fi1[ndx0.keys, ax0] ∧ dom gstore = dom gstore1 ∧ nodes(gstore ↓, jro0) = nodes(gstore1 ↓, jro1) ∧ abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) ∧ jmaxino0 ≤ jmaxino), jmaxino = 0, pmaxino = 0, dups(ax)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
# ax = # ndx, ax0 = ax, fi1 = fi, gstore1 = gstore, jmaxino0 = jmaxino, ndx0 = ndx, remove-refsize0(gstore0) = gstore0, ax.toSet ∩ range fi = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ jro0, jmaxino ≠ 0, pmaxino ≠ 0, jro < jmaxino, p_os < pmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, inv(gstore \ jro0), inv(gstore0), key-cons(ndx), nodes-cons(ax, ndx, gstore, jro0), refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), ri-inj(fi), ri-inj(fi0), pmaxino ≤ jmaxino, fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, ndx.keys.toSet ⊆ dom fi, ¬ dups(ax), ¬ dups(nx), jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ⟪gjournal_gc_update_index#(jvalid, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid0; ax, ndx, jmaxino, fi, gstore)⟫ ( jro < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi1[ndx0.keys, ax0].inos < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ fi1[ndx0.keys, ax0].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi1[ndx0.keys, ax0]) ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ ri-inj(fi1[ndx0.keys, ax0]) ∧ fi = fi1[ndx0.keys, ax0] ∧ dom gstore = dom gstore1 ∧ nodes(gstore ↓, jro0) = nodes(gstore1 ↓, jro0) ∧ abs-log(nx, gstore ↓) = abs-log(nx, gstore1 ↓) ∧ jmaxino0 ≤ jmaxino)

Rule argument

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)