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Proof step 186 from proof inv-gc-copy (previous step)

# abs-log(nx0, gstore ↓) = # ndx + 1, abs-log(nx0, gstore ↓) = ax, abs-log(nx1, gstore2 ↓) = [], dom gstore1 = dom gstore3 ∪ jro, fi = fi2, ndx0 = nd ' + ndx, nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx], nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro3 = ∅, nx0.toSet ∩ dom gstore3 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, jmaxino3 ≠ 0, pmaxino ≠ 0, jro2 < jmaxino3, fi2.inos < jmaxino0, p_os < pmaxino, fi2.inos < jmaxino3, fi1.inos < pmaxino, inv(gstore \ jro3), inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), key-cons(ndx), key-cons(nd.key, nd), key-cons(ndx1), new(abs-log(nx0, gstore ↓), nodes(gstore2 ↓, jro0)), nodes-cons(abs-log(nx0, gstore ↓), nd ' + ndx, gstore, jro3), refsize-cons(gstore, range fi2), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2), refsize-cons(gstore1, range fi1), ri-inj(fi2), ri-inj(fi1), nd.key ∈ fi2, nd.flashsize + ndx.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, dom gstore3 ⊆ dom gstore, dom gstore2 ⊆ dom gstore3, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, fi1.lebs ⊆ dom gstore1, dom gstore2 ⊆ dom gstore, fi2.lebs ⊆ dom gstore, ¬ dups(abs-log(nx0, gstore ↓)), ¬ dups(nx0), ¬ dups(nx1), ¬ dups(ndx.keys + ndx1.keys), ¬ dups(ndx.keys), ¬ dups(ndx1.keys), ¬ nd.key ∈ ndx.keys, ¬ nd.key ∈ ndx1.keys, jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), ndx1 = [] ∨ LEB_SIZE < (nd.flashsize + ndx.flashsize) + ndx1.head.flashsize ⊦ ⟪gjournal_gc_update_index#(jvalid0, fi1, jro3, gstore1, nx0, pmaxino, p_os, jvalid; ax, ndx0, jmaxino0, fi, gstore)⟫ ( jmaxino3 ≤ jmaxino0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].inos < jmaxino0 ∧ fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]) ∧ range fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)] ⊆ dom nodes(gstore ↓, jro3) ∧ dom gstore2 ⊆ dom gstore ∧ new(abs-log(nx0, gstore ↓), nodes(gstore2 ↓, jro0)) ∧ ¬ dups(abs-log(nx0, gstore ↓)) ∧ # abs-log(nx0, gstore ↓) = # ndx + 1 ∧ fi = fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx])

User applied rule insert program lemma

And obtained the following new goals

Proof step 187:
jro < jmaxino, p_os < pmaxino, pmaxino ≤ jmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, jmaxino ≠ 0, pmaxino ≠ 0, jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), inv-journal(nx, gstore, gstore0, jro0), ri-inj(fi), ri-inj(fi0), ndx.keys.toSet ⊆ dom fi, # ax = # ndx, key-cons(ndx), ax.toSet ∩ range fi = ∅, ¬ dups(ax), nodes-cons(ax, ndx, gstore, jro0), fi1 = fi, ndx0 = ndx, ax0 = ax, gstore1 = gstore, jro1 = jro0, nx0 = nx, jmaxino0 = jmaxino ⊦ ⟪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)
Proof step 188:
# abs-log(nx0, gstore ↓) = # ndx + 1, # ax = # ndx0, jvalid1 ↔ jvalid0, jvalid2 ↔ jvalid, abs-log(nx1, gstore2 ↓) = [], abs-log(nx0, gstore ↓) = ax, ax = ax, ax1 = ax, dom gstore1 = dom gstore3 ∪ jro, fi = fi2, fi = fi, fi3 = fi1, fi4 = fi, gstore = gstore, gstore4 = gstore1, gstore5 = gstore, jmaxino0 = jmaxino0, jmaxino1 = jmaxino0, jro3 = jro3, jro4 = fi2.inos, jro5 = jro3, jro6 = jro3, ndx0 = nd ' + ndx, ndx0 = ndx0, ndx2 = ndx0, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0), nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx], nx0 = nx0, nx2 = nx0, nx3 = nx0, p_os0 = p_os, pmaxino0 = pmaxino, remove-refsize0(gstore1) = gstore1, elems nx1 ∩ jro0 = ∅, nx0.toSet ∩ dom gstore3 = ∅, elems nx0 ∩ jro3 = ∅, ax.toSet ∩ range fi = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, fi2.inos < jmaxino3, jro2 < jmaxino3, fi2.inos < jmaxino0, p_os < pmaxino, fi.inos < jmaxino0, fi1.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), inv(gstore \ jro3), inv-journal(nx0, gstore, gstore1, jro3), key-cons(ndx1), key-cons(nd.key, nd), key-cons(ndx), key-cons(ndx0), new(abs-log(nx0, gstore ↓), nodes(gstore2 ↓, jro0)), nodes-cons(abs-log(nx0, gstore ↓), nd ' + ndx, gstore, jro3), nodes-cons(ax, ndx0, gstore, jro3), refsize-cons(gstore3, range fi2), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2), refsize-cons(gstore, range fi), refsize-cons(gstore1, range fi1), ri-inj(fi2), ri-inj(fi), ri-inj(fi1), nd.key ∈ fi2, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, nd.flashsize + ndx.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, fi2.lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), dom gstore2 ⊆ dom gstore3, dom gstore3 ⊆ dom gstore, fi.lebs ⊆ dom gstore, fi1.lebs ⊆ dom gstore1, ndx0.keys.toSet ⊆ dom fi, jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), ndx1 = [] ∨ LEB_SIZE < (nd.flashsize + ndx.flashsize) + ndx1.head.flashsize ⊦ [gjournal_gc_update_index#(jvalid0, fi1, jro3, gstore1, nx0, pmaxino, p_os, jvalid; ax, ndx0, jmaxino0, fi, gstore)] ( jro4 < jmaxino0 ∧ pmaxino0 ≤ jmaxino0 ∧ fi.inos < jmaxino0 ∧ jmaxino0 ≠ 0 ∧ (jvalid1 ∧ nx2 ≠ [] → valid-journal-head(nx2, gstore, jro5)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx2, gstore, gstore4, jro5) ∧ ri-inj(fi) ∧ fi = fi4[ndx2.keys, ax1] ∧ dom gstore = dom gstore5 ∧ nodes(gstore ↓, jro5) = nodes(gstore5 ↓, jro6) ∧ abs-log(nx2, gstore ↓) = abs-log(nx3, gstore5 ↓) ∧ jmaxino1 ≤ jmaxino0 → jmaxino3 ≤ jmaxino0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].inos < jmaxino0 ∧ fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]) ∧ range fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)] ⊆ dom nodes(gstore ↓, jro3) ∧ dom gstore2 ⊆ dom gstore ∧ new(abs-log(nx0, gstore ↓), nodes(gstore2 ↓, jro0)) ∧ ¬ dups(abs-log(nx0, gstore ↓)) ∧ # abs-log(nx0, gstore ↓) = # ndx + 1 ∧ fi = fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx]), jmaxino0 = 0, jmaxino3 = 0, pmaxino = 0, dups(ax), dups(abs-log(nx0, gstore ↓)), dups(nx0), dups(nx1), dups(ndx.keys + ndx1.keys), dups(ndx.keys), dups(ndx1.keys), nd.key ∈ ndx.keys, nd.key ∈ ndx1.keys

Rule argument

spec: <toplevel> name: inv-gc-update-index seq: jro < jmaxino, p_os < pmaxino, pmaxino ≤ jmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, jmaxino ≠ 0, pmaxino ≠ 0, jvalid ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), inv-journal(nx, gstore, gstore0, jro0), ri-inj(fi), ri-inj(fi0), ndx.keys.toSet ⊆ dom fi, # ax = # ndx, key-cons(ndx), ax.toSet ∩ range fi = ∅, ¬ dups(ax), nodes-cons(ax, ndx, gstore, jro0), fi1 = fi, ndx0 = ndx, ax0 = ax, gstore1 = gstore, jro1 = jro0, nx0 = nx, jmaxino0 = jmaxino ⊦ ⟪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) subst: [jro, jvalid0, p_os, pmaxino, nx, gstore0, jro0, fi0, jvalid, gstore, fi, jmaxino, ndx, ax, fi1, ndx0, ax0, gstore1, jro1, nx0, jmaxino0] → [fi2.inos, jvalid, p_os, pmaxino, nx0, gstore1, jro3, fi1, jvalid0, gstore, fi, jmaxino0, ndx0, ax, fi, ndx0, ax, gstore, jro3, nx0, jmaxino0] flags: List([[1,0,0]])

Additional information

Used simplifier rules: ⊦ ns < n ∧ n ≤ m → (ns < m ↔ true); (ss)%⊦ 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)% ⊦ (nd ' + ndx).keys = nd.key + ndx.keys; (s)% ⊦ key + kx = key ' + kx; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ ks ∪ ∅ = ks; (s)%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ ks ++ key ⊆ ks0 ↔ ks ⊆ ks0 ∧ key ∈ ks0; (ws)% ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ #(nd ' + ndx) = (# ndx)+1; (s)% ⊦ m +1 = m + 1; (s)% ⊦ key-cons(nd ' + ndx) ↔ ¬ ¬(key-cons(nd.key, nd) ∧ key-cons(ndx)); (s)% ⊦ as ⊆ dom fs ∧ new(ax, fs) → ax.toSet ∩ as = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ m ≤ 0 ↔ m = 0; (s)