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Proof step 18 from proof inv-gc-update-index (previous step)

# ax3 = # ndx2, # ax2 = # ndx1, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), 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 ∩ ax2.toSet = ∅, range fi2 ∩ ax3.toSet = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, 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(ndx2), key-cons(nd.key, nd), key-cons(ndx1), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), nodes-cons(ax2, ndx1, 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), @(m, m0, m1) ∈ gstore, nd.key ∈ fi2, pmaxino ≤ jmaxino, pmaxino ≤ jmaxino1, jmaxino1 ≤ jmaxino, ndx1.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(ax2), ¬ dups(ax3), ¬ dups(ax3 + ax2), ¬ dups(nx), ¬ @(m, m0, m1) ∈ ax3, ¬ @(m, m0, m1) ∈ ax2, ¬ @(m, m0, m1) ∈ fi2, ¬ m ∈ jro0, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ⟪gjournal_index_store#(nd.key, @(m, m0, m1), fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore); ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi = fi2[ndx.keys, ax]) ∧ 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)) ∧ ¬ # ax2 < # ax0)

User applied rule insert program lemma

And obtained the following new goals

Proof step 19:
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), ¬ adr ∈ fi, adr ∈ gstore, key-cons(key, gstore[adr].nd), gstore1 = gstore, fi1 = fi ⊦ ⟪gjournal_index_store#(key, adr, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid0; 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) ∧ gstore =○refsize gstore1 ∧ fi = fi1[key, adr])
Proof step 20:
# ax2 = # ndx1, # ax3 = # ndx2, jvalid1 ↔ jvalid2, jvalid3 ↔ jvalid, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), adr0 = @(m, m0, m1), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], fi = fi, fi3 = fi0, fi4 = fi, gstore = gstore, gstore3 = gstore0, gstore4 = gstore, jro2 = fi2[ndx2.keys, ax3].inos, jro3 = jro0, key0 = nd.key, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), nx0 = nx, p_os0 = p_os, pmaxino0 = pmaxino, remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax2.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, jro1 < jmaxino1, fi2.inos < jmaxino1, fi2[ndx2.keys, ax3].inos < jmaxino, p_os < pmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), inv-journal(nx, gstore, gstore0, jro0), key-cons(ndx1), key-cons(nd.key, nd), key-cons(ndx2), key-cons(nd.key, gstore[@(m, m0, m1)].nd), nodes-cons(ax2, ndx1, gstore, jro0), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi), ri-inj(fi0), nd.key ∈ fi2, @(m, m0, m1) ∈ gstore, jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ [gjournal_index_store#(nd.key, @(m, m0, m1), fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore)] ( jro2 < jmaxino ∧ pmaxino0 ≤ jmaxino ∧ fi.inos < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid1 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx0, gstore, gstore3, jro3) ∧ ri-inj(fi) ∧ gstore =○refsize gstore4 ∧ fi = fi4[key0, adr0] → ⟪ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi = fi2[ndx.keys, ax]) ∧ 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)) ∧ ¬ # ax2 < # ax0)), jmaxino = 0, jmaxino1 = 0, pmaxino = 0, dups(ax2), dups(ax3), dups(ax3 + ax2), dups(nx), @(m, m0, m1) ∈ fi, @(m, m0, m1) ∈ ax3, @(m, m0, m1) ∈ ax2, @(m, m0, m1) ∈ fi2, m ∈ jro0

Rule argument

spec: <toplevel> name: inv-store 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), ¬ adr ∈ fi, adr ∈ gstore, key-cons(key, gstore[adr].nd), gstore1 = gstore, fi1 = fi ⊦ ⟪gjournal_index_store# (key, adr, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid0 ; 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) ∧ gstore =○refsize gstore1 ∧ fi = fi1[key, adr]) subst: [jvalid0, p_os, pmaxino, nx, gstore0, jro0, fi0, adr, key, gstore, fi, jmaxino, jro, jvalid, gstore1, fi1] → [jvalid, p_os, pmaxino, nx, gstore0, jro0, fi0, @(m, m0, m1), nd.key, gstore, fi, jmaxino, fi2[ndx2.keys, ax3].inos, jvalid2, gstore, fi] flags: <all paths>

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)%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]% ⊦ # ndx.keys = # ndx; (s)%# ax = # kx ⊦ ri-inj(ri) ∧ ¬ dups(ax) ∧ ax.toSet ∩ range ri = ∅ → ri-inj(ri[kx, ax]); (s)%# kx = # ax ⊦ ¬ adr ∈ ax ∧ ¬ adr ∈ ri → ¬ adr ∈ ri[kx, ax]; (s)% ⊦ mkgnode(nd, m, boolvar, boolvar0).nd = nd; (s)% ⊦ m ≤ 0 ↔ m = 0; (s)