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Proof step 7 from proof inv-gc-update-index (previous step)

# ax3 = # ndx2, # ax1 = # ndx, abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, elems nx ∩ jro0 = ∅, range fi2 ∩ ax1.toSet = ∅, range fi2 ∩ ax3.toSet = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, ax1 ≠ [], jmaxino1 ≠ 0, pmaxino ≠ 0, fi0.inos < pmaxino, fi2.inos < jmaxino1, p_os < pmaxino, jro1 < jmaxino1, fi2[ndx2.keys, ax3].inos < jmaxino, inv(gstore \ jro0), inv(gstore0), inv(gstore2 \ jro0), key-cons(ndx), key-cons(ndx2), nodes-cons(ax3 + ax1, ndx2 + ndx, gstore2, jro0), nodes-cons(ax1, ndx, gstore, jro0), refsize-cons(gstore0, range fi0), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), ri-inj(fi0), ri-inj(fi2), pmaxino ≤ jmaxino, pmaxino ≤ jmaxino1, jmaxino1 ≤ jmaxino, ndx.keys.toSet ⊆ dom fi2, fi0.lebs ⊆ dom gstore0, fi2.lebs ⊆ dom gstore2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, ndx2.keys.toSet ⊆ dom fi2, ¬ dups(ax1), ¬ dups(ax3), ¬ dups(ax3 + ax1), ¬ dups(nx), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ⟪gjournal_index_store#(ndx.head.key, ax1.head, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore); ax0 := ax1.tail; ndx0 := ndx.tail⟫ ( (∃ ax2, ndx3. # ax2 = # ndx3 ∧ ax3 + ax1 = ax2 + ax0 ∧ ndx2 + ndx = ndx3 + ndx0 ∧ fi = fi2[ndx3.keys, ax2]) ∧ dom fi = dom fi2 ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ jmaxino1 ≤ jmaxino ∧ fi.inos < jmaxino ∧ refsize-cons(gstore, range fi) ∧ ri-inj(fi) ∧ fi.lebs ⊆ dom gstore ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ # ax0 < # ax1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 8:
# ax1 = # ndx, # ax3 = # ndx2, abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], ndx = nd + ndx1, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax1.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, fi2[ndx2.keys, ax3].inos < jmaxino, jro1 < jmaxino1, p_os < pmaxino, fi2.inos < jmaxino1, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), key-cons(ndx2), key-cons(ndx), nodes-cons(ax1, ndx, gstore, jro0), nodes-cons(ax3 + ax1, ndx2 + ndx, gstore2, jro0), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore2, range fi2), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi0), jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore2, fi0.lebs ⊆ dom gstore0, ndx.keys.toSet ⊆ dom fi2, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0) ⊦ ⟪gjournal_index_store#(nd.key, ax1.head, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore); ax0 := ax1.tail; ndx0 := ndx1⟫ ( (∃ ax2, ndx3. # ax2 = # ndx3 ∧ ax3 + ax1 = ax2 + ax0 ∧ ndx2 + ndx = ndx3 + ndx0 ∧ fi = fi2[ndx3.keys, ax2]) ∧ dom fi = dom fi2 ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ jmaxino1 ≤ jmaxino ∧ fi.inos < jmaxino ∧ refsize-cons(gstore, range fi) ∧ ri-inj(fi) ∧ fi.lebs ⊆ dom gstore ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ # ax0 < # ax1), ax1 = [], jmaxino1 = 0, ndx = [], pmaxino = 0, dups(ax1), dups(ax3), dups(ax3 + ax1), dups(nx)

Rule argument

spec: list-data, inst: node-list name: elim-list-c seq: ndx ≠ [] ⊦ nd = ndx.head ∧ ndx0 = ndx.tail ↔ ndx = nd + ndx0 subst: [ndx, nd, ndx0] → [ndx, nd, ndx1]

Additional information

Used simplifier rules: ⊦ # [] = 0; (s)% ⊦ # ax = 0 ↔ ax = []; (s)%ndx ≠ [] ⊦ nd = ndx.head ∧ ndx0 = ndx.tail ↔ ndx = nd + ndx0; (elim) Used terms: ndx, nd, ndx1