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Proof step 38 from proof inv-addn (previous step)

jvalid0 ↔ true, err0 = ESUCCESS ↔ # lax = # gndx, abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓), ax0 = lax.to-adr(nx.last), err = err0, fi = fi1, fi0 = fi2, gstore = gstore3, gstore1 = gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))] [nx.last, gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last] .ladrlist:= (gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last].ladrlist + lax) ], jro = jro0, nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro1), nx = nx0, remove-refsize0(gstore0) = gstore0, elems nx1 ∩ jro1 = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx0 ∩ jro0 = ∅, dom gstore0 ∪ elems nx0 = dom gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))] [nx.last, gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last] .ladrlist:= (gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last].ladrlist + lax) ] ∪ jro0, dom gstore0 ∪ elems nx1 = dom gstore2 ∪ jro1, dom gstore0 ∪ elems nx0 = dom gstore3 ∪ jro0, p_os < pmaxino, fi2.inos < jmaxino1, fi1.inos < pmaxino, inv ( gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))] [nx.last, gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last] .ladrlist:= (gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))][nx.last].ladrlist + lax) ] \ jro0), inv(gstore0), inv(gstore2 \ jro1), inv(gstore3 \ jro0), inv(gstore3[nx0.last]), inv-flashsize(prefixn(# lax, gndx), lax), refsize-cons(gstore2, range fi2), refsize-cons(gstore0, range fi1), refsize-cons(gstore3, range fi2), ri-inj(fi2), ri-inj(fi1), transaction(ndx, gndx), valid-journal-head(nx0, gstore3, jro0), nx0.last ∈ gstore3, nx.last ∈ gstore[nx.last, gstore[nx.last].gndlist:= (gstore[nx.last].gndlist + prefixn(# lax, gndx))], nx.last ∈ gstore, # lax ≤ # gndx, pmaxino ≤ jmaxino1, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore0, dom gstore2 ⊆ dom gstore3, ⌊err0⌋ ∨ err0 = ESUCCESS → nx.last ∈ gstore ∧ nx ≠ [], jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1), ⌊err0⌋ ∨ err0 = ESUCCESS ⊦ err = ESUCCESS, nx = [], nx0 = [], pmaxino = 0, dups(nx1), dups(nx0), dups(gstore[nx.last].ladrlist + lax), (false → valid-journal-head(nx0, gstore1, jro0)) ∧ fi2.lebs ⊆ dom gstore1 ∧ refsize-cons(gstore1, range fi2) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ dom gstore2 ⊆ dom gstore1 ∧ ( err = ESUCCESS → # ax = # ndx ∧ ¬ dups(ax) ∧ nodes-cons(ax, ndx, gstore1, jro0) ∧ abs-log(nx0, gstore1 ↓) = abs-log(nx1, gstore2 ↓) + ax ∧ nodes(gstore1 ↓, jro0) = nodes(gstore2 ↓, jro1)[ax, ndx] ∧ new(ax, nodes(gstore2 ↓, jro1))) ∧ (err ≠ ESUCCESS → abs-log(nx0, gstore1 ↓) = abs-log(nx1, gstore2 ↓) ∧ nodes(gstore1 ↓, jro0) = nodes(gstore2 ↓, jro1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ls-inv-reduce-jnlhead: ⊦ inv(gstore \ ns) ∧ valid-journal-head(nx, gstore, ns) ∧ nx ≠ [] → (inv(gstore[nx.last, gblock] \ ns) ↔ inv(gblock));% ⊦ (gblock.gndlist:= gndx).ref-size = gblock.ref-size; (s)% ⊦ (gblock.ladrlist:= lax).ref-size = gblock.ref-size; (s)%⊦ refsize-cons(gstore, as) → ( refsize-cons(gstore[n, gblock], as) ↔ ¬ n ∈ gstore ∨ gstore[n].ref-size = (as | n).size → gblock.ref-size = (as | n).size); (ss)% ⊦ # gndx = 0 ↔ gndx = []; (s)%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]%a: ⊦ ¬ dups(lax + lax0) → ¬ dups(lax) ∧ ¬ dups(lax0); (forward) from specification list-dup[laddress-list]%fwd-inv-loghead: ⊦ nx ≠ [] ∧ inv(gstore \ ginval) ∧ ginval ∩ elems nx = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval → inv(gstore[nx.last]); (forward) from specification apersistence-base%fwd-in-gstore-loghead: ⊦ nx ≠ [] ∧ ginval ∩ elems nx = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval → nx.last ∈ gstore; (forward) from specification apersistence-base%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]%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]%sub-trans: ⊦ ns ⊆ ns0 ∧ ns0 ⊆ ns1 → ns ⊆ ns1; (forward) from specification set-basic[nat-set]%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%⊦ nx ≠ [] ∧ ¬ dups(nx) → abs-log(nx, gstore[nx.last, gblock]) = abs-log(nx.butlast, gstore) + gblock.ladrlist.to-adr(nx.last); (s)% ⊦ gstore[m, gblock] ↓ = gstore ↓[m, gblock ↓]; (s)% ⊦ ⌊err⌋ → err ≠ ESUCCESS; (s)% ⊦ n ∈ gstore[n, gblock]; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ n ∈ gstore → dom gstore ++ n = dom gstore; (s)% ⊦ dom gstore[n, gblock] = dom gstore ++ n; (s)% ⊦ gstore[n, gblock][n, gblock0] = gstore[n, gblock0]; (s)% ⊦ (gblock.gndlist:= gndx).ladrlist = gblock.ladrlist; (s)% ⊦ gstore[n, gblock][n] = gblock; (s)