[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 64 from proof prepost-atomic-leb-change (previous step)

# apebs0 = # wlar, LEB_SIZE = n4 + align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), LEB_SIZE = n2 + n0, LEB_SIZE = n3 + align↑(datasize(buf, n0), EB_PAGE_SIZE), apebs0[n1] = mkpeb(aehdr, empty, mkempbuf(LEB_SIZE), 0, false), wlar[n1] = wl-entry(m1, free), m0 = VTBL_LNUM, q0 = q, v = VTBL_VOLID, vols0 = vols, vtblpnum0 = vtblpnum1, vtblpnum2 = n1, aehdr.valid?, n1 < # wlar, bflipset-cons(bflipset, wlar), is-aligned(n0, EB_PAGE_SIZE), ubi-io-inv(apebs0[n1, mkpeb(aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), mkempbuf(LEB_SIZE), 0, false)]), ubi-io-inv(apebs0), ubi-io-inv (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ]), unique-sqns(apebs0), valid-ebatbl(apebs0, wlar, vols, vtblpnum1), valid-eraseq(apebs0, q), valid-eraseq(wlar, q), valid-eraseq(q), valid-global-sqn(apebs0, sqnum), valid-vtbl(apebs0, vols, wlar, vtblpnum1), valid-wl(apebs0, wlar), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE) ≤ LEB_SIZE, align↑(datasize(buf, n0), EB_PAGE_SIZE) ≤ # buf, align↑(datasize(buf, n0), EB_PAGE_SIZE) ≤ LEB_SIZE, n0 ≤ LEB_SIZE, n0 ≤ # buf, v ∈ vols ∧ m0 < # vols[v] ∨ v = VTBL_VOLID ∧ m0 = VTBL_LNUM ⊦ ⟪ubi_wl_put_peb#(VTBL_VOLID × VTBL_LNUM, vtblpnum0; wlar, q); bflipset := bflipset -- vtblpnum0; vtblpnum0 := vtblpnum2; wlar[vtblpnum2] := wl-entry(wlar[vtblpnum2].ec, used)⟫ ( # apebs0 = # wlar ∧ unique-sqns (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ]) ∧ valid-global-sqn (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], sqnum + 1) ∧ valid-wl (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], wlar) ∧ valid-eraseq(q) ∧ valid-eraseq(wlar, q) ∧ valid-eraseq (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], q) ∧ ¬ VTBL_VOLID ∈ vols ∧ bflipset-cons(bflipset, wlar) ∧ ( v ∈ vols0 ∨ is-encoded(to-vtbl(vols0), subarray(buf, 0, LEB_SIZE)) ∧ n0 = LEB_SIZE → valid-ebatbl (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], wlar, vols, vtblpnum0) ∧ valid-vtbl (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], vols, wlar, vtblpnum0)) ∧ ( v = VTBL_VOLID ∧ m0 = VTBL_LNUM → (∀ v0, m. ¬ v0 ∈ vols0 ∧ is-encoded(to-vtbl(vols0)[v0, m], subarray(buf, 0, LEB_SIZE)) ∧ n0 = LEB_SIZE → valid-ebatbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], wlar, vols[v0, fill(mkebatbl(m), unmapped)], vtblpnum0) ∧ valid-vtbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], vols[v0, fill(mkebatbl(m), unmapped)], wlar, vtblpnum0))) ∧ (v ∈ vols0 ∧ vols0[v][m0] ≠ unmapped → q = q0 + eq-entry(vols0[v][m0].pnum, v × m0) ') ∧ (v = VTBL_VOLID ∧ m0 = VTBL_LNUM → q = q0 + eq-entry(vtblpnum1, VTBL_VOLID × VTBL_LNUM) ') ∧ id (apebs0 [n1, mkpeb (aehdr, avidhdr(v, m0, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], apebs0, mapped-pebs(vols0)) ∧ (v = VTBL_VOLID ∧ m0 = VTBL_LNUM → vols = vols0) ∧ (v ∈ vols0 → vols = vols0[v, vols0[v][m0, ⌜n1⌝ ]]) ∧ q.lebrefs ⊆ q0.lebrefs ++ (v × m0)), vtblpnum1 = n1, vtblpnum1 ∈ q, VTBL_VOLID ∈ vols, ∃ buf0. buf = mkempbuf(n0) + buf0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 65:
# apebs0 = # wlar, LEB_SIZE = n3 + align↑(datasize(buf, n0), EB_PAGE_SIZE), LEB_SIZE = n2 + n0, LEB_SIZE = n4 + align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), wlar[n1] = wl-entry(m1, free), apebs0[n1] = mkpeb(aehdr, empty, mkempbuf(LEB_SIZE), 0, false), q0 = q, vtblpnum1 ≠ n1, aehdr.valid?, n1 < # wlar, bflipset-cons(bflipset, wlar), is-aligned(n0, EB_PAGE_SIZE), ubi-io-inv (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ]), ubi-io-inv(apebs0), ubi-io-inv (apebs0[n1, mkpeb(aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), mkempbuf(LEB_SIZE), 0, false)]), unique-sqns(apebs0), valid-ebatbl(apebs0, wlar, vols, vtblpnum1), valid-eraseq(q), valid-eraseq(wlar, q), valid-eraseq(apebs0, q), valid-global-sqn(apebs0, sqnum), valid-vtbl(apebs0, vols, wlar, vtblpnum1), valid-wl(apebs0, wlar), n0 ≤ # buf, n0 ≤ LEB_SIZE, align↑(datasize(buf, n0), EB_PAGE_SIZE) ≤ LEB_SIZE, align↑(datasize(buf, n0), EB_PAGE_SIZE) ≤ # buf, align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE) ≤ LEB_SIZE, ¬ VTBL_VOLID ∈ vols, ¬ vtblpnum1 ∈ q ⊦ ⟪ubi_wl_put_peb#(VTBL_VOLID × VTBL_LNUM, vtblpnum1; wlar, q); bflipset := bflipset -- vtblpnum1; vtblpnum0 := n1; wlar[n1] := wl-entry(wlar[n1].ec, used)⟫ ( # apebs0 = # wlar ∧ unique-sqns (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ]) ∧ valid-global-sqn (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], sqnum + 1) ∧ valid-wl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], wlar) ∧ valid-eraseq(q0 + eq-entry(vtblpnum1, VTBL_VOLID × VTBL_LNUM) ') ∧ valid-eraseq(wlar, q0 + eq-entry(vtblpnum1, VTBL_VOLID × VTBL_LNUM) ') ∧ valid-eraseq (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], q0 + eq-entry(vtblpnum1, VTBL_VOLID × VTBL_LNUM) ') ∧ bflipset-cons(bflipset, wlar) ∧ ( is-encoded(to-vtbl(vols), subarray(buf, 0, LEB_SIZE)) ∧ n0 = LEB_SIZE → valid-ebatbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], wlar, vols, vtblpnum0) ∧ valid-vtbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], vols, wlar, vtblpnum0)) ∧ (∀ v0, m. ¬ v0 ∈ vols ∧ is-encoded(to-vtbl(vols)[v0, m], subarray(buf, 0, LEB_SIZE)) ∧ n0 = LEB_SIZE → valid-ebatbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], wlar, vols[v0, fill(mkebatbl(m), unmapped)], vtblpnum0) ∧ valid-vtbl (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, LEB_SIZE), checksum(buf, datasize(buf, LEB_SIZE))), subarray(buf, 0, LEB_SIZE), align↑(datasize(buf, LEB_SIZE), EB_PAGE_SIZE), false) ], vols[v0, fill(mkebatbl(m), unmapped)], wlar, vtblpnum0)) ∧ q = q0 + eq-entry(vtblpnum1, VTBL_VOLID × VTBL_LNUM) ' ∧ id (apebs0 [n1, mkpeb (aehdr, avidhdr(VTBL_VOLID, VTBL_LNUM, sqnum, datasize(buf, n0), checksum(buf, datasize(buf, n0))), subarray(buf, 0, n0) + mkempbuf(n2), align↑(datasize(buf, n0), EB_PAGE_SIZE), false) ], apebs0, mapped-pebs(vols))), ∃ buf0. buf = mkempbuf(n0) + buf0

Rule argument

Additional information

Used simplifier rules: ls-array-append-length0: # buf0 = 0 ⊦ buf + buf0 = buf;% ⊦ s0 ⊆ s0 ++ lref; (s)% ⊦ s1 ++ lref ⊆ s2 ++ lref ↔ s1 ⊆ s2 ++ lref; (s)%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base]% ⊦ (q + eq-entry(n, v × m) ').lebrefs = q.lebrefs ++ (v × m); (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m = m + n ↔ n = 0; (s)% ⊦ # mkempbuf(n) = n; (s)