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Proof step 51 from proof inv-gc-copy (previous step)

# ax = # ndx1, 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, jmaxino3 ≠ 0, nx ≠ [], pmaxino ≠ 0, n0 < nd0.flashsize, 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(ndx3), key-cons(ndx), key-cons(ndx1), key-cons(nd0.key, nd0), 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), nd0.key ∈ fi2, nx.last ∈ gstore3, nd0.flashsize + ndx.flashsize ≤ LEB_SIZE, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino0, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, 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), ndx3.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi1.lebs ⊆ dom gstore1, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, ¬ dups(ndx3.keys), ¬ dups(ndx1.keys + ndx.keys), ¬ dups(ndx.keys + ndx3.keys), ¬ dups(ax), ¬ dups(nx), ¬ dups(nx1), ¬ dups(ndx1.keys + ndx.keys + ndx3.keys), ¬ dups(ndx1.keys), ¬ dups(ndx.keys), ¬ nd0.key ∈ ndx3.keys, ¬ nd0.key ∈ ndx.keys, ¬ nd0.key ∈ ndx1.keys, jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid1 → valid-journal-head(nx, gstore3, jro), ndx3 = [] ∨ LEB_SIZE < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ ⟪gjournal_journal_addn# (nd0 ' + ndx, 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 {ndx0 := nd0 ' + ndx; gjournal_gc_update_index#(jvalid1, fi1, jro, gstore1, nx, pmaxino, p_os, jvalid0; ax0, ndx0, jmaxino0, fi, gstore3)}⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ 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, ndx4. ndx1 + nd0 ' + ndx + ndx3 = ndx0 + ndx4 ∧ (err = ESUCCESS → ndx4 = 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) ∧ ¬ # ndx + # ndx3 < (err = ESUCCESS ⊃ # ndx3;0))

User applied rule insert program lemma

And obtained the following new goals

Proof step 52:
p_os < pmaxino, pmaxino ≤ jmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, pmaxino ≠ 0, jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro), fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), inv-journal(nx, gstore, gstore0, jro), ri-inj(fi), ri-inj(fi0), gstore1 = gstore, nx0 = nx, jro0 = jro ⊦ ⟪gjournal_journal_addn#(ndx, jvalid0, fi, fi0, gstore0, pmaxino, p_os; ax, jvalid, jmaxino0, jro, gstore, nx, jvalid1; err)⟫ ( (jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ dom gstore1 ⊆ dom gstore ∧ (nx.toSet \ nx0.toSet) ∩ dom gstore1 = ∅ ∧ ( err = ESUCCESS → # ax = # ndx ∧ ¬ dups(ax) ∧ nodes-cons(ax, ndx, gstore, jro) ∧ abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) + ax ∧ nodes(gstore ↓, jro) = nodes(gstore1 ↓, jro0)[ax, ndx] ∧ new(ax, nodes(gstore1 ↓, jro0))) ∧ (err ≠ ESUCCESS → abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) ∧ nodes(gstore ↓, jro) = nodes(gstore1 ↓, jro0)))
Proof step 53:
# ax = # ndx1, jvalid3 ↔ jvalid2, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, ax0 = [], fi3 = fi2[ndx1.keys, ax], fi4 = fi1, gstore3 = gstore3, gstore4 = gstore1, gstore5 = gstore3, jmaxino2 = jmaxino0, jro = jro, jro1 = jro, ndx2 = nd0 ' + ndx, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], nx = nx, nx2 = nx, p_os0 = p_os, pmaxino0 = pmaxino, 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, jro2 < jmaxino3, fi2.inos < jmaxino3, n0 < nd0.flashsize, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), inv-journal(nx, gstore3, gstore1, jro), key-cons(nd0.key, nd0), key-cons(ndx1), key-cons(ndx), key-cons(ndx3), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), nx.last ∈ gstore3, nd0.key ∈ fi2, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, ndx3.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), dom gstore2 ⊆ dom gstore3, fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, fi1.lebs ⊆ dom gstore1, jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro), ndx3 = [] ∨ LEB_SIZE < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ [gjournal_journal_addn# (nd0 ' + ndx, jvalid2, fi2[ndx1.keys, ax], fi1, gstore1, pmaxino, p_os; ax0, jvalid1, jmaxino1, jro, gstore3, nx, jvalid0; err)] ( (jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro)) ∧ fi3.lebs ⊆ dom gstore3 ∧ refsize-cons(gstore3, range fi3) ∧ inv-journal(nx, gstore3, gstore4, jro) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ dom gstore5 ⊆ dom gstore3 ∧ (nx.toSet \ nx2.toSet) ∩ dom gstore5 = ∅ ∧ ( err = ESUCCESS → # ax0 = # ndx2 ∧ ¬ dups(ax0) ∧ nodes-cons(ax0, ndx2, gstore3, jro) ∧ abs-log(nx, gstore3 ↓) = abs-log(nx2, gstore5 ↓) + ax0 ∧ nodes(gstore3 ↓, jro) = nodes(gstore5 ↓, jro1)[ax0, ndx2] ∧ new(ax0, nodes(gstore5 ↓, jro1))) ∧ (err ≠ ESUCCESS → abs-log(nx, gstore3 ↓) = abs-log(nx2, gstore5 ↓) ∧ nodes(gstore3 ↓, jro) = nodes(gstore5 ↓, jro1)) → ⟪fi := fi2[ndx1.keys, ax]; if err = ESUCCESS then {ndx0 := nd0 ' + ndx; gjournal_gc_update_index#(jvalid1, fi1, jro, gstore1, nx, pmaxino, p_os, jvalid0; ax0, ndx0, jmaxino0, fi, gstore3)}⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ 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, ndx4. ndx1 + nd0 ' + ndx + ndx3 = ndx0 + ndx4 ∧ (err = ESUCCESS → ndx4 = 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) ∧ ¬ # ndx + # ndx3 < (err = ESUCCESS ⊃ # ndx3;0))), jmaxino3 = 0, nx = [], pmaxino = 0, dups(ndx3.keys), dups(ndx1.keys + ndx.keys), dups(ndx.keys + ndx3.keys), dups(ax), dups(nx), dups(nx1), dups(ndx1.keys + ndx.keys + ndx3.keys), dups(ndx1.keys), dups(ndx.keys), nd0.key ∈ ndx3.keys, nd0.key ∈ ndx.keys, nd0.key ∈ ndx1.keys

Rule argument

spec: <toplevel> name: inv-addn seq: p_os < pmaxino, pmaxino ≤ jmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, pmaxino ≠ 0, jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro), fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), inv-journal(nx, gstore, gstore0, jro), ri-inj(fi), ri-inj(fi0), gstore1 = gstore, nx0 = nx, jro0 = jro ⊦ ⟪gjournal_journal_addn# (ndx, jvalid0, fi, fi0, gstore0, pmaxino, p_os ; ax, jvalid, jmaxino0, jro, gstore, nx, jvalid1 ; err)⟫ ( (jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ dom gstore1 ⊆ dom gstore ∧ (nx.toSet \ nx0.toSet) ∩ dom gstore1 = ∅ ∧ ( err = ESUCCESS → # ax = # ndx ∧ ¬ dups(ax) ∧ nodes-cons(ax, ndx, gstore, jro) ∧ abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) + ax ∧ nodes(gstore ↓, jro) = nodes(gstore1 ↓, jro0)[ax, ndx] ∧ new(ax, nodes(gstore1 ↓, jro0))) ∧ ( err ≠ ESUCCESS → abs-log(nx, gstore ↓) = abs-log(nx0, gstore1 ↓) ∧ nodes(gstore ↓, jro) = nodes(gstore1 ↓, jro0))) subst: [jmaxino, p_os, pmaxino, gstore0, fi0, fi, jvalid0, ndx, jvalid1, nx, gstore, jro, jmaxino0, jvalid, ax, err, gstore1, nx0, jro0] → [jmaxino0, p_os, pmaxino, gstore1, fi1, fi2[ndx1.keys, ax], jvalid2, nd0 ' + ndx, jvalid0, nx, gstore3, jro, jmaxino1, jvalid1, ax0, err, gstore3, nx, jro] flags: List([[1,0,0]])

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)