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Proof step 105 from proof refine-gc (previous step)

# lax = # gndx0, gstore2[n1] = groupblock(gndx0, lax, m0), fi = ri, fs = nodes(gstore2 ↓, jro0), gstore4 = gstore2, jmaxino2 = jmaxino, jro3 = jro0, log = abs-log(nx, gstore2 ↓), ndx = nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)], ndx0 = nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)], nx1 = nx, remove-refsize0(gstore1) = gstore1, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, jro1.keys.inodes?, p_os.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino, p_os < pmaxino, jro1 < jmaxino, inv(gstore2 \ jro0), inv(gstore1), inv(groupblock(gndx0, lax, m0)), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(fi1), ri-inj(ri), n1 ∈ gstore2, n1 ∈ gstore1, pmaxino ≤ jmaxino, range ri ⊆ dom nodes(gstore2 ↓, jro0), fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid5 → jvalid4 ⊦ ∀ ndx, jvalid3, jmaxino0, jmaxino, fi, jro0, gstore2, nx, jvalid4, err. jro1 < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi.inos < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0)) ∧ fi.lebs ⊆ dom gstore2 ∧ refsize-cons(gstore2, range fi) ∧ inv(gstore2 \ jro0) ∧ elems nx ∩ jro0 = ∅ ∧ ¬ dups(nx) ∧ dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0 ∧ ri-inj(fi) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ jmaxino2 ≤ jmaxino ∧ dom gstore4 ⊆ dom gstore2 ∧ (nx.toSet \ nx1.toSet) ∩ dom gstore4 = ∅ ∧ (∃ ax, ndx, ndx1. ndx0 = ndx + ndx1 ∧ (err = ESUCCESS → ndx1 = []) ∧ new(ax, nodes(gstore4 ↓, jro3)) ∧ # ax = # ndx ∧ ¬ dups(ax) ∧ fi = ri[ndx.keys, ax] ∧ nodes(gstore2 ↓, jro0) = nodes(gstore4 ↓, jro3)[ax, ndx] ∧ abs-log(nx, gstore2 ↓) = abs-log(nx1, gstore4 ↓) + ax) → [if err ≠ ESUCCESS then debug#("gjournal: failed to copy nodes of LEB " + n1.toString + " to journal head during GC"); fi0 := fi1; if err = ESUCCESS then index_sync#(fi, fi0, jro0, gstore2, gstore1, nx, pmaxino, p_os; ; jvalid4, err); if err = ESUCCESS then index_deallocate_gnd#(n1, fi, fi0, gstore1, nx, pmaxino, p_os, jvalid4; jro0, gstore2; err)] ( (jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0)) ∧ fi.lebs ⊆ dom gstore2 ∧ refsize-cons(gstore2, range fi) ∧ inv(gstore2 \ jro0) ∧ elems nx ∩ jro0 = ∅ ∧ dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0 → ⟨aubifs_core_journal_gc#(fi1, jro1.keys, p_os.keys; ri, fs, log; jvalid5)⟩ (fi = ri ∧ log = abs-log(nx, gstore2 ↓) ∧ fs = nodes(gstore2 ↓, jro0) ∧ (jvalid5 → jvalid4))), jmaxino = 0, m0 = 0, pmaxino = 0, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys), dups(nx), n1 ∈ nx, n1 ∈ jro0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 106:
# lax = # gndx0, # ax = # ndx3, ndx3 + ndx2 = nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)], gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore ↓) = abs-log(nx, gstore2 ↓) + ax, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx3], remove-refsize0(gstore1) = gstore1, elems nx ∩ jro0 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore2 = ∅, elems nx0 ∩ jro = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro, jmaxino ≠ 0, jmaxino0 ≠ 0, m0 ≠ 0, pmaxino ≠ 0, jro1.keys.inodes?, p_os.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino, p_os < pmaxino, jro1 < jmaxino, ri[ndx3.keys, ax].inos < jmaxino0, jro1 < jmaxino0, inv(gstore2 \ jro0), inv(gstore1), inv(groupblock(gndx0, lax, m0)), inv(gstore \ jro), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri), refsize-cons(gstore, range ri[ndx3.keys, ax]), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(fi1), ri-inj(ri), ri-inj(ri[ndx3.keys, ax]), n1 ∈ gstore2, n1 ∈ gstore1, pmaxino ≤ jmaxino, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino0, ri.lebs ⊆ dom gstore, range ri ⊆ dom nodes(gstore2 ↓, jro0), fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, dom gstore2 ⊆ dom gstore, ri[ndx3.keys, ax].lebs ⊆ dom gstore, ¬ dups(nx), ¬ dups(nx0), ¬ dups(ax), ¬ dups(ndx3.keys + ndx2.keys), ¬ dups(ndx3.keys), ¬ dups(ndx2.keys), ¬ n1 ∈ nx, ¬ n1 ∈ jro0, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid5 → jvalid4, err = ESUCCESS → ndx2 = [], jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro), err = ESUCCESS ∨ ⌊err⌋ ⊦ [if err ≠ ESUCCESS then debug#("gjournal: failed to copy nodes of LEB " + n1.toString + " to journal head during GC"); fi0 := fi1; fi2 := ri[ndx3.keys, ax]; if err = ESUCCESS then index_sync#(fi2, fi0, jro, gstore, gstore1, nx0, pmaxino, p_os; ; jvalid, err); if err = ESUCCESS then index_deallocate_gnd#(n1, fi2, fi0, gstore1, nx0, pmaxino, p_os, jvalid; jro, gstore; err)] ( (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro)) ∧ fi2.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2) ∧ inv(gstore \ jro) ∧ elems nx0 ∩ jro = ∅ ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro → ⟨log := abs-log(nx, gstore2 ↓); fs := nodes(gstore2 ↓, jro0); aubifs_core_journal_gc#(fi1, jro1.keys, p_os.keys; ri, fs, log; jvalid5)⟩ (fi2 = ri ∧ log = abs-log(nx0, gstore ↓) ∧ fs = nodes(gstore ↓, jro) ∧ (jvalid5 → jvalid)))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%a: ⊦ ¬ dups(kx + kx0) → ¬ dups(kx) ∧ ¬ dups(kx0); (forward) from specification list-dup[key-list-set]%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ (ndx0 + ndx1).keys = ndx0.keys + ndx1.keys; (s)%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