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Proof step 111 from proof inv-gc-copy (previous step)

# ax = # ndx1, nd ' + ndx2 = ndx0 + ndx3, LEB_SIZE = n1 + n0, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, ax0 = [], nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore1) = gstore1, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, jmaxino3 ≠ 0, ndx0 ≠ [], nx ≠ [], pmaxino ≠ 0, jro2 < jmaxino3, p_os < pmaxino, fi2.inos < jmaxino3, fi1.inos < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), key-cons(ndx1), key-cons(nd.key, nd), key-cons(ndx2), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range fi2), refsize-cons(gstore1, range fi1), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi1), ri-inj(fi2[ndx1.keys, ax]), nd.key ∈ fi2, nx.last ∈ gstore3, nd ∈ ndx0, ndx0.flashsize ≤ n1, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, pmaxino ≤ jmaxino0, n0 ≤ LEB_SIZE, nd.flashsize ≤ n0, nd.flashsize ≤ LEB_SIZE, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, ndx1.keys.toSet ⊆ dom fi2, ndx2.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), dom gstore2 ⊆ dom gstore3, ¬ dups(ndx2.keys), ¬ dups(ndx1.keys), ¬ dups(ndx1.keys + ndx2.keys), ¬ dups(ax), ¬ dups(nx), ¬ dups(nx1), ¬ nd.key ∈ ndx1.keys, ¬ nd.key ∈ ndx2.keys, jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), ndx3 = [] ∨ n1 < ndx0.flashsize + ndx3.head.flashsize ⊦ ⟪gjournal_journal_addn#(ndx0, jvalid2, fi2[ndx1.keys, ax], fi1, gstore1, pmaxino, p_os; ax0, jvalid1, jmaxino1, jro, gstore3, nx, jvalid0; err); fi := fi2[ndx1.keys, ax]; if err = ESUCCESS then gjournal_gc_update_index#(jvalid1, fi1, jro, gstore1, nx, pmaxino, p_os, jvalid0; ax0, ndx0, jmaxino0, fi, gstore3)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (∃ ndx. ndx1 + nd ' + ndx2 = ndx + ndx3) ∧ jmaxino3 ≤ jmaxino0 ∧ (jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore3 ∧ refsize-cons(gstore3, range fi) ∧ inv(gstore3 \ jro) ∧ elems nx ∩ jro = ∅ ∧ ¬ dups(nx) ∧ dom gstore1 ∪ elems nx = dom gstore3 ∪ jro ∧ ri-inj(fi) ∧ ndx3.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore3 ↓, jro) ∧ dom gstore2 ⊆ dom gstore3 ∧ (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax, ndx0, ndx. ndx1 + nd ' + ndx2 = ndx0 + ndx ∧ (err = ESUCCESS → ndx = ndx3) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax) ∧ # ax = # ndx0 ∧ fi = fi2[ndx0.keys, ax] ∧ nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx0] ∧ abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax) ∧ ¬ # ndx2 < (err = ESUCCESS ⊃ # ndx3;0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 112:
⊦ ndx = [] ↔ ¬(∃ nd, ndx0. ndx = nd ' + ndx0)
Proof step 113:
# ax = # ndx1, nd ' + ndx2 = ndx0 + ndx3, LEB_SIZE = n1 + n0, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, ax0 = [], nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore1) = gstore1, elems nx ∩ jro = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, fi2.inos < jmaxino3, p_os < pmaxino, jro2 < jmaxino3, inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(gstore3[nx.last]), key-cons(ndx2), key-cons(nd.key, nd), key-cons(ndx1), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), ri-inj(fi2), nd ∈ ndx0, nx.last ∈ gstore3, nd.key ∈ fi2, nd.flashsize ≤ LEB_SIZE, nd.flashsize ≤ n0, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino0, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, ndx0.flashsize ≤ n1, dom gstore2 ⊆ dom gstore3, range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx2.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi1.lebs ⊆ dom gstore1, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid1 → valid-journal-head(nx, gstore3, jro), ∃ nd, ndx1. ndx0 = nd ' + ndx1, ndx3 = [] ∨ n1 < ndx0.flashsize + ndx3.head.flashsize ⊦ ⟪gjournal_journal_addn#(ndx0, jvalid2, fi2[ndx1.keys, ax], fi1, gstore1, pmaxino, p_os; ax0, jvalid1, jmaxino1, jro, gstore3, nx, jvalid0; err); fi := fi2[ndx1.keys, ax]; if err = ESUCCESS then gjournal_gc_update_index#(jvalid1, fi1, jro, gstore1, nx, pmaxino, p_os, jvalid0; ax0, ndx0, jmaxino0, fi, gstore3)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (∃ ndx. ndx1 + nd ' + ndx2 = ndx + ndx3) ∧ jmaxino3 ≤ jmaxino0 ∧ (jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore3 ∧ refsize-cons(gstore3, range fi) ∧ inv(gstore3 \ jro) ∧ elems nx ∩ jro = ∅ ∧ ¬ dups(nx) ∧ dom gstore1 ∪ elems nx = dom gstore3 ∪ jro ∧ ri-inj(fi) ∧ ndx3.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore3 ↓, jro) ∧ dom gstore2 ⊆ dom gstore3 ∧ (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax, ndx0, ndx. ndx1 + nd ' + ndx2 = ndx0 + ndx ∧ (err = ESUCCESS → ndx = ndx3) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax) ∧ # ax = # ndx0 ∧ fi = fi2[ndx0.keys, ax] ∧ nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx0] ∧ abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax) ∧ ¬ # ndx2 < (err = ESUCCESS ⊃ # ndx3;0)), jmaxino3 = 0, nx = [], pmaxino = 0, dups(ndx2.keys), dups(ndx1.keys), dups(ndx1.keys + ndx2.keys), dups(ax), dups(nx), dups(nx1), nd.key ∈ ndx1.keys, nd.key ∈ ndx2.keys

Rule argument

spec: list, inst: node-list name: elim-first seq: ⊦ ndx = [] ↔ ¬(∃ nd, ndx0. ndx = nd ' + ndx0) subst: [ndx] → [ndx0] flags: <all paths>

Additional information

Used terms: ndx0