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Proof step 389 from proof refine-gc (previous step)

gstore2[n1] = groupblock(gndx, lax, 0), err0 = ESUCCESS, fs = nodes(gstore2 ↓, jro0), gstore = gstore2, log = abs-log(nx, gstore2 ↓), remove-refsize0(gstore1) = gstore1, ri0 = ri, elems nx ∩ (jro0 ++ n1) = ∅, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx = dom(gstore -- n1) ∪ (jro0 ++ n1), dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, jro1.keys.inodes?, p_os.keys.inodes?, fi1.inos < pmaxino, p_os < pmaxino, jro1 < jmaxino, ri.inos < jmaxino, inv((gstore -- n1) \ (jro0 ++ n1)), inv(gstore2 \ jro0), inv(gstore1), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), refsize-cons(gstore -- n1, range ri), 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, ri.lebs ⊆ dom(gstore -- n1), fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore -- n1, jro0 ++ n1), jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid5 → jvalid0, ⌊err0⌋ ∨ err0 = ESUCCESS ⊦ ⟨aubifs_core_journal_gc#(fi1, jro1.keys, p_os.keys; ri, fs, log; jvalid5)⟩ (ri0 = ri ∧ log = abs-log(nx, (gstore -- n1)↓) ∧ fs = nodes((gstore -- n1)↓, jro0 ++ n1)), jmaxino = 0, pmaxino = 0, dups(nx), n1 ∈ nx

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 390:
gstore2[n1] = groupblock(gndx, lax, 0), fs = nodes(gstore2 ↓, jro0), log = abs-log(nx, gstore2 ↓), remove-refsize0(gstore1) = gstore1, ri0 = ri, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx = jro0 ∪ dom gstore2, #(jro0 ∪ dom gstore2) ≠ 0, jmaxino ≠ 0, pmaxino ≠ 0, jro0 ∪ dom gstore2 ≠ ∅, p_os.keys.inodes?, jro1.keys.inodes?, ri.inos < jmaxino, jro1 < jmaxino, p_os < pmaxino, fi1.inos < pmaxino, inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range ri), refsize-cons(gstore1, range fi1), refsize-cons(gstore2 -- n1, range ri), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(ri), ri-inj(fi1), n1 ∈ jro0 ∪ dom gstore2, n1 ∈ gstore1, n1 ∈ gstore2, pmaxino ≤ jmaxino, ri.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, ¬ dups(nx), ¬ n1 ∈ nx, jvalid5 → jvalid0, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2 -- n1, jro0 ++ n1) ⊦ ⟨aubifs_core_journal_gc#(fi1, jro1.keys, p_os.keys; ri, fs, log; jvalid5)⟩ (ri0 = ri ∧ log = abs-log(nx, gstore2 ↓) ∧ fs = nodes(gstore2 ↓, jro0) \ (dom nodes(gstore2 ↓, jro0) | n1)), ∃ adr. adr.lnum = n1 ∧ adr ∈ ri

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ ns ≠ ∅ → # ns ≠ 0; (forward) from specification set-basic[nat-set]%ne-01: ⊦ n ∈ ns → ns ≠ ∅; (forward) from specification set-basic[nat-set]% ⊦ n ∈ ns → ns ++ n = ns; (s)% ⊦ ns ++ n = ns ↔ n ∈ ns; (s)% ⊦ ns ++ n ++ n = ns ++ n; (s)%⊦ nodes(gstore -- n, ginval ++ n) = nodes(gstore, ginval) \ (dom nodes(gstore, ginval) | n); (s)% ⊦ ¬ n ∈ nx → abs-log(nx, gstore -- n) = abs-log(nx, gstore); (s)% ⊦ (gstore -- n)↓ = gstore ↓ -- n; (s)% ⊦ ¬ ⌊ESUCCESS⌋ ; (s)% ⊦ inv(gstore \ ns) → inv(gstore \ (ns ++ n)); (s)% ⊦ n ∈ gstore → (gstore \ (ns ++ n)) -- n = gstore \ (ns ++ n); (s)% ⊦ (gstore -- n) \ ns = (gstore \ ns) -- n; (s)% ⊦ n ∈ ri.lebs ↔ (∃ adr. adr.lnum = n ∧ adr ∈ ri); (s)% ⊦ ns ⊆ ns0 -- n ↔ ns ⊆ ns0 ∧ ¬ n ∈ ns; (ws)%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]% ⊦ n ∈ elems nx ↔ n ∈ nx; (s)% ⊦ ns ∩ (ns0 ++ n) = ∅ ↔ ns ∩ ns0 = ∅ ∧ ¬ n ∈ ns; (ws)%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]% ⊦ (ns ∪ (ns0 -- n)) ++ n = (ns ∪ ns0) ++ n; (s)% ⊦ (ns ++ n) ∪ ns0 = (ns ∪ ns0) ++ n; (s)% ⊦ dom(gstore -- n) = dom gstore -- n; (s)