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Proof step 185 from proof refine-add-gnds (previous step)

# gndx0 = # lax0, # buf0 = gndx0.flashsize, # lpt0 = # avol, # lpt1 = # avol, # lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ] = # avol, (elems nx ∪ dom gstore0) ++ n1 = dom gstore ∪ ginval, avol[n1] = mapped(buf1), lpt1[n1] = mklp(m1, # buf1, LP_GROUP_NODES, m2), gstore[n1] = groupblock(gndx, lax, m1), keyset(gch) = gc-heap(lpt1, nx + n1 '), keyset(gch) = gc-heap (lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], nx + n1 '), pstate = journal((nx + n1 ').last), remove-refsize0(gstore0) = gstore0, remove-refsize0(istore0) = istore0, ginval ∩ elems nx = ∅, buffered((nx + n1 ').last).isbuf?, ginval < # avol, n1 < # avol, n < gndx0.flashsize, buffered((nx + n1 ').last).leb < # lpt1, abs-gstore(avol, lpt1, gstore, ginval), abs-istore(avol, lpt1, istore), abs-nodes(buf0, # buf1, gndx0, lax0, false), awbuf-inv(avol, buffered(n1)), awbuf-inv(avol[n1, mapped(avol[n1].data + subarray(buf0, 0, n))], buffered(n1)), bin-heap-inv(gch, lpt1), bin-heap-inv (gch, lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ]), buf-ro-rw-mode(false, buffered(n1), lpt1, avol), buf-ro-rw-mode (true, buffered(n1), lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], avol[n1, mapped(avol[n1].data + subarray(buf0, 0, n))]), bufs-log-only(buffered(n1), nx + n1 ', lpt1), bufs-log-only (buffered(n1), nx + n1 ', lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ]), flash-lpt-cons(lpt0, istore0, gstore0), freelist-cons(lpt1, freelist), freelist-cons (lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], freelist), inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), inv(groupblock(gndx, lax, m1)), is-aligned(# buf1 + n, EB_PAGE_SIZE), log-group-block(nx + n1 ', lpt1), log-group-block (nx + n1 ', lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ]), lpt-free-unref(lpt0), lpt-free-unref(lpt1), lpt-free-unref (lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ]), lpt-leb-size(lpt0), lpt-leb-size(lpt1), lpt-leb-size (lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ]), lpt-ram-flash(lpt1, lpt0, nx + n1 '), lpt-ram-flash (lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], lpt0, nx + n1 '), n1 ∈ gstore, # buf1 + gndx0.flashsize ≤ LEB_SIZE, istore0 ⊑ istore, ⌊err0⌋ , isbuf1 → synced(avol) ⊦ ⟨apersistence_add_gnds#(gndx0, adr0, istore, istore0, ginval, gstore0, nx + n1 ', m0, ns0; ax0, gstore, pstate, isbuf1; err1)⟩ ( abs-istore (avol[n1, mapped(avol[n1].data + subarray(buf0, 0, n))], lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], istore) ∧ abs-gstore (avol[n1, mapped(avol[n1].data + subarray(buf0, 0, n))], lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ], gstore, ginval) ∧ ginval < # lpt1 [buffered((nx + n1 ').last).leb, mklp (lpt1[buffered((nx + n1 ').last).leb].ref-size, LEB_SIZE, lpt1[buffered((nx + n1 ').last).leb].flags, lpt1[buffered((nx + n1 ').last).leb].gcheapidx) ] ∧ pstate = readonly ∧ lax0.to-adr(n1) = ax0 ∧ err0 = err1), nx + n1 ' = [], gndx0 = [], n = 0, dups(nx), n1 ∈ ginval, n1 ∈ nx

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 186:
# lpt1 = # avol, # lpt0 = # avol, # buf0 = gndx0.flashsize, # gndx0 = # lax0, (elems nx ∪ dom gstore0) ++ n1 = dom gstore ∪ ginval, gstore[n1] = groupblock(gndx, lax, m1), lpt1[n1] = mklp(m1, # buf1, LP_GROUP_NODES, m2), avol[n1] = mapped(buf1), keyset(gch) = gc-heap(lpt1, nx + n1 '), pstate = journal(n1), remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, ginval ∩ elems nx = ∅, gndx0 ≠ [], n ≠ 0, n < gndx0.flashsize, ginval < # avol, n1 < # avol, abs-gstore(avol, lpt1, gstore, ginval), abs-istore(avol, lpt1, istore), abs-nodes(buf0, # buf1, gndx0, lax0, false), awbuf-inv(avol[n1, mapped(buf1 + subarray(buf0, 0, n))], buffered(n1)), awbuf-inv(avol, buffered(n1)), bin-heap-inv(gch, lpt1), buf-ro-rw-mode(false, buffered(n1), lpt1, avol), bufs-log-only(buffered(n1), nx + n1 ', lpt1), flash-lpt-cons(lpt0, istore0, gstore0), freelist-cons(lpt1, freelist), inv(groupblock(gndx, lax, m1)), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), is-aligned(# buf1 + n, EB_PAGE_SIZE), log-group-block(nx + n1 ', lpt1), lpt-free-unref(lpt1), lpt-free-unref(lpt0), lpt-leb-size(lpt1), lpt-leb-size(lpt0), lpt-ram-flash(lpt1, lpt0, nx + n1 '), n1 ∈ gstore, # buf1 + gndx0.flashsize ≤ LEB_SIZE, istore0 ⊑ istore, ⌊err0⌋ , ¬ dups(nx), ¬ n1 ∈ nx, ¬ n1 ∈ ginval, isbuf1 → synced(avol) ⊦ ⟨apersistence_add_gnds#(gndx0, adr0, istore, istore0, ginval, gstore0, nx + n1 ', m0, ns0; ax0, gstore, pstate, isbuf1; err1)⟩ ( abs-gstore (avol[n1, mapped(avol[n1].data + subarray(buf0, 0, n))], lpt1[n1, mklp(lpt1[n1].ref-size, LEB_SIZE, lpt1[n1].flags, lpt1[n1].gcheapidx)], gstore, ginval) ∧ pstate = readonly ∧ lax0.to-adr(n1) = ax0 ∧ err0 = err1)

Rule argument

Additional information

Used simplifier rules: ⊦ LP_FREE ≠ LP_GROUP_NODES; (s)%n < # lpt ⊦ lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2) ∧ log-group-block(nx, lpt) → log-group-block(nx, lpt[n, mklp(m0, m1, LP_GROUP_NODES, m2)]); (s)%⊦ freelist-cons(lpt, freelist) ∧ lpt[n] = mklp(m0, m1, lpf, m2) → freelist-cons(lpt[n, mklp(n0, n1, lpf, n2)], freelist); (s)%n < # lpt ⊦ lpt[n] = mklp(m0, m1, lpf0, m2) → ( bin-heap-inv(gch, lpt[n, mklp(n0, n1, lpf1, m2)]) ↔ bin-heap-inv(gch, lpt)); (s)% ⊦ # avol[n, aleb] = # avol; (s)% ⊦ mapped(buf).data = buf; (s)%n < # lpt ⊦ lpt[n] = mklp(m0, m1, LP_GROUP_NODES, m2) → gc-heap(lpt[n, mklp(m0, n1, LP_GROUP_NODES, n2)], nx) = gc-heap(lpt, nx); (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%inv-buf-in-vol: ⊦ awbuf-inv(avol, buffered(n)) → n < # avol; (forward) from specification awbuf-inv%lpf ≠ LP_INDEX_NODES ⊦ lpt[n] = mklp(n0, n1, lpf, n2) → ( abs-istore(avol[n, aleb], lpt, istore) ↔ abs-istore(avol, lpt, istore)); (s)%lpf ≠ LP_INDEX_NODES, lp.flags ≠ LP_INDEX_NODES ⊦ lpt[n] = mklp(n0, n1, lpf, n2) → ( abs-istore(avol, lpt[n, lp], istore) ↔ abs-istore(avol, lpt, istore)); (s)% ⊦ LP_GROUP_NODES ≠ LP_INDEX_NODES; (s)% ⊦ buffered(m).isbuf?; (s)%⊦ buf-ro-rw-mode (true, buffered(n), lpt[n, mklp(m0, LEB_SIZE, lpf, m1)], avol) ↔ n < # lpt; (s)%lpt[n].flags = lp.flags ⊦ bufs-log-only(bleb, nx, lpt) → bufs-log-only(bleb, nx, lpt[n, lp]); (s)% ⊦ mklp(n2, n3, lpf, m).gcheapidx = m; (s)% ⊦ mklp(n2, n3, lpf, m).ref-size = n2; (s)% ⊦ mklp(n2, n3, lpf, m).flags = lpf; (s)%⊦ lpt-free-unref(lpt) → ( lpt-free-unref(lpt[n, lp]) ↔ n < # lpt ∧ lp.flags = LP_FREE → lp.ref-size = 0); (s)% ⊦ m ≤ m; (s)% ⊦ mklp(n2, n3, lpf, m).size = n3; (s)%⊦ lpt-leb-size(lpt) → (lpt-leb-size(lpt[n, lp]) ↔ # lpt ≤ n ∨ lp.size ≤ LEB_SIZE); (s)%⊦ lpt-ram-flash(lpt[n, lp], lpt0, nx + n ') ↔ lpt-ram-flash(lpt, lpt0, nx + n '); (s)% ⊦ nx + n ' ≠ []; (s)% ⊦ # lpt[n, lp] = # lpt; (s)% ⊦ buffered(m).leb = m; (s)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ (nx + n ').last = n; (s)