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Proof step 8 from proof inv-gc-copy (previous step)

jvalid2 ↔ jvalid4, jvalid5 ↔ jvalid7, jvalid6 ↔ jvalid3, err = ESUCCESS, err = err0, err1 = ESUCCESS, fi1 = fi2, gstore1 = gstore2, jmaxino = 0, jmaxino2 = jmaxino3, jmaxino4 = jmaxino3, jro1 = jro0, jro3 = jro0, ndx2 = ndx4, ndx3 = ndx, ndx5 = ndx, nx0 = nx1, remove-refsize0(gstore0) = gstore0, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore0 ∪ elems nx1 = dom gstore2 ∪ jro0, fi.inos < jmaxino0, fi0.inos < pmaxino, fi2.inos < jmaxino3, p_os < pmaxino, jro2 < jmaxino3, inv(gstore0), inv(gstore2 \ jro0), inv-journal(nx, gstore, gstore0, jro), key-cons(ndx), refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), refsize-cons(gstore2, range fi2), ri-inj(fi), ri-inj(fi0), ri-inj(fi2), pmaxino ≤ jmaxino3, jmaxino0 ≥ jmaxino4, dom gstore2 ⊆ dom gstore, range fi ⊆ dom nodes(gstore ↓, jro), ndx2.keys.toSet ⊆ dom fi, fi.lebs ⊆ dom gstore, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx.keys.toSet ⊆ dom fi2, fi0.lebs ⊆ dom gstore0, fi2.lebs ⊆ dom gstore2, jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), ∃ ax, ndx0, ndx1. ndx5 = ndx0 + ndx1 ∧ (err = ESUCCESS → ndx1 = ndx2) ∧ new(ax, nodes(gstore2 ↓, jro3)) ∧ ¬ dups(ax) ∧ # ax = # ndx0 ∧ fi = fi2[ndx0.keys, ax] ∧ nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro3)[ax, ndx0] ∧ abs-log(nx, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax, ∃ ndx0. ndx5 = ndx0 + ndx2, err = ESUCCESS ∨ ⌊err⌋ ⊦ ⟪index_is_log_empty#(fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; jvalid); if ¬ jvalid then {index_get_block_free_size#(fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; jmaxino); if jmaxino < ndx2.head.flashsize then jmaxino := 0}; let ndx = [], ax0 = [], n = ? in {index_get_leb_size(fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; n); gjournal_split_nodes#(n - jmaxino; ndx2, ndx); if ndx = [] then err := EINVAL else {gjournal_journal_addn#(ndx, jvalid2, fi, fi0, gstore0, pmaxino, p_os; ax0, jvalid1, jmaxino1, jro, gstore, nx, jvalid0; err); if err = ESUCCESS then gjournal_gc_update_index#(jvalid1, fi0, jro, gstore0, nx, pmaxino, p_os, jvalid0; ax0, ndx, jmaxino0, fi, gstore)}}⟫ ( ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (∃ ndx0. ndx5 = ndx0 + ndx2) ∧ jmaxino0 ≥ jmaxino4 ∧ (jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ ri-inj(fi) ∧ ndx2.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore ↓, jro) ∧ dom gstore2 ⊆ dom gstore ∧ (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax, ndx0, ndx1. ndx5 = ndx0 + ndx1 ∧ (err = ESUCCESS → ndx1 = ndx2) ∧ new(ax, nodes(gstore2 ↓, jro3)) ∧ ¬ dups(ax) ∧ # ax = # ndx0 ∧ fi = fi2[ndx0.keys, ax] ∧ nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro3)[ax, ndx0] ∧ abs-log(nx, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax)) ∧ (err = ESUCCESS ⊃ # ndx2;0) < (err0 = ESUCCESS ⊃ # ndx4;0)), jmaxino3 = 0, ndx2 = [], pmaxino = 0, dups(ndx.keys), dups(nx1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
# ax = # ndx1, abs-log(nx, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore0) = gstore0, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, dom gstore0 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore0 ∪ elems nx = dom gstore ∪ jro, jmaxino3 ≠ 0, ndx4 ≠ [], pmaxino ≠ 0, jro2 < jmaxino3, p_os < pmaxino, fi2.inos < jmaxino3, fi0.inos < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro), key-cons(ndx1), key-cons(ndx4), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range fi2), refsize-cons(gstore0, range fi0), refsize-cons(gstore, range fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi0), ri-inj(fi2[ndx1.keys, ax]), jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, pmaxino ≤ jmaxino0, fi2.lebs ⊆ dom gstore2, fi0.lebs ⊆ dom gstore0, ndx1.keys.toSet ⊆ dom fi2, ndx4.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), fi2[ndx1.keys, ax].lebs ⊆ dom gstore, range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore ↓, jro), dom gstore2 ⊆ dom gstore, fi2.lebs ⊆ dom gstore, ¬ dups(ax), ¬ dups(nx), ¬ dups(nx1), ¬ dups(ndx1.keys + ndx4.keys), ¬ dups(ndx1.keys), ¬ dups(ndx4.keys), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro) ⊦ ⟪index_is_log_empty#(fi2[ndx1.keys, ax], fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; jvalid); fi := fi2[ndx1.keys, ax]; ndx2 := ndx4; jmaxino := 0; if ¬ jvalid then {index_get_block_free_size#(fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; jmaxino); if jmaxino < ndx2.head.flashsize then jmaxino := 0}; let ndx = [], ax0 = [], n = ? in {index_get_leb_size(fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; n); ndx0 := ndx2; gjournal_split_nodes#(n - jmaxino; ndx2, ndx); if ndx = [] then err := EINVAL else {gjournal_journal_addn#(ndx, jvalid2, fi, fi0, gstore0, pmaxino, p_os; ax0, jvalid1, jmaxino1, jro, gstore, nx, jvalid0; err); if err = ESUCCESS then gjournal_gc_update_index#(jvalid1, fi0, jro, gstore0, nx, pmaxino, p_os, jvalid0; ax0, ndx, jmaxino0, fi, gstore)}}⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (∃ ndx3. ndx1 + ndx0 = ndx3 + ndx2) ∧ jmaxino3 ≤ jmaxino0 ∧ (jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv(gstore \ jro) ∧ elems nx ∩ jro = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro ∧ ri-inj(fi) ∧ ndx2.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore ↓, jro) ∧ dom gstore2 ⊆ dom gstore ∧ (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax, ndx3, ndx4. ndx1 + ndx0 = ndx3 + ndx4 ∧ (err = ESUCCESS → ndx4 = ndx2) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax) ∧ # ax = # ndx3 ∧ fi = fi2[ndx3.keys, ax] ∧ nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx3] ∧ abs-log(nx, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax) ∧ (err = ESUCCESS ⊃ # ndx2;0) < # ndx4)

Rule argument

Additional information

Used simplifier rules: ⊦ ndx + ndx0 = ndx1 + ndx0 ↔ ndx = ndx1; (s)%a: ⊦ ¬ dups(kx + kx0) → ¬ dups(kx) ∧ ¬ dups(kx0); (forward) from specification list-dup[key-list-set]%sub-trans: ⊦ ns ⊆ ns0 ∧ ns0 ⊆ ns1 → ns ⊆ ns1; (forward) from specification set-basic[nat-set]%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%⊦ 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)% ⊦ key-cons(ndx0 + ndx1) ↔ key-cons(ndx0) ∧ key-cons(ndx1); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ ks ⊆ ks0 → ks ⊆ ks0 ∪ ks1; (s)%# kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet; (s)% ⊦ # ndx.keys = # ndx; (s)%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]% ⊦ ks ∪ ks0 ⊆ ks1 ↔ ks ⊆ ks1 ∧ ks0 ⊆ ks1; (ws)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ (ndx0 + ndx1).keys = ndx0.keys + ndx1.keys; (s)