[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 10 from proof prepost-write-once (previous step)

# apebs0 = # wlar, LEB_SIZE = (n1 + m0) + m1, vols0[v][m] = unmapped, apebs = apebs0, q = q0, m < # vols0[v], bflipset-cons(bflipset, wlar), is-aligned(m0, EB_PAGE_SIZE), is-aligned(m1, EB_PAGE_SIZE), ubi-io-inv(apebs0), unique-sqns(apebs0), valid-ebatbl(apebs0, wlar, vols0, vtblpnum2), valid-eraseq(q0), valid-eraseq(wlar, q0), valid-eraseq(apebs0, q0), valid-global-sqn(apebs0, sqnum), valid-vtbl(apebs0, vols0, wlar, vtblpnum2), valid-wl(apebs0, wlar), v ∈ vols0, m0 + m1 ≤ LEB_SIZE, m1 + m2 ≤ # buf, m1 ≤ LEB_SIZE, ¬ VTBL_VOLID ∈ vols0, ¬ vtblpnum2 ∈ q0 ⊦ ⟪ubi_wl_get_peb#(; vtblpnum, wlar, q, apebs; err); vols := vols0; if err = ESUCCESS then let m3 = ? in {ubi_next_sqnum#(; sqnum; m3); aubi_io_write_vidhdr#(vtblpnum, avidhdr(v, m, m3, 0, 0); apebs; err); if err = ESUCCESS then {if m1 ≠ 0 then aubi_io_write_data#(vtblpnum, m0, m2, m1, buf; apebs; err); if err = ESUCCESS then {vols[v][m] := ⌜vtblpnum⌝ ; wlar[vtblpnum] := wl-entry(wlar[vtblpnum].ec, used)} else ubi_wl_put_peb#(v × m, vtblpnum; wlar, q); skip} else ubi_wl_put_peb#(none, vtblpnum; wlar, q)}⟫ ( # apebs = # wlar ∧ unique-sqns(apebs) ∧ valid-global-sqn(apebs, sqnum) ∧ valid-wl(apebs, wlar) ∧ valid-eraseq(q) ∧ valid-eraseq(wlar, q) ∧ valid-eraseq(apebs, q) ∧ ¬ VTBL_VOLID ∈ vols ∧ bflipset-cons(bflipset, wlar) ∧ valid-ebatbl(apebs, wlar, vols, vtblpnum2) ∧ valid-vtbl(apebs, vols, wlar, vtblpnum2) ∧ ubi-io-inv(apebs) ∧ id(apebs, apebs0, mapped-pebs(vols0)) ∧ q.lebrefs ⊆ q0.lebrefs ++ (v × m) ∧ (m1 = 0 → q.lebrefs ⊆ q0.lebrefs) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err ≠ ESUCCESS → vols = vols0) ∧ ( err = ESUCCESS → (∃ n. vols = vols0[v, vols0[v][m, ⌜n⌝ ]] ∧ (m1 ≠ 0 → apebs[n].fill = m0 + m1) ∧ (m1 = 0 → apebs[n].fill = 0) ∧ apebs[n].data = mkempbuf(m0) + subarray(buf, m2, m1) + mkempbuf(n1))))

User applied rule insert program lemma

And obtained the following new goals

Proof step 11:
ubi-cons(apebs, vols, wlar, q, bflipset, vtblpnum, sqnum), ubi-io-inv(apebs), wlar0 = wlar, q0 = q, apebs0 = apebs ⊦ ⟪ubi_wl_get_peb#(; n, wlar, q, apebs; err)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err ≠ ESUCCESS → ubi-cons(apebs, vols, wlar, q, bflipset, vtblpnum, sqnum)) ∧ ubi-io-inv(apebs) ∧ q.lebrefs ⊆ q0.lebrefs ∧ id(apebs, apebs0, mapped-pebs(vols)) ∧ ( err = ESUCCESS → n < # apebs ∧ wlar[n].status = free ∧ (apebs0 = apebs ∧ wlar0 = wlar ∧ q0 = q ∨ id○(apebs0, apebs, n) ∧ id○(wlar0, wlar, n) ∧ q0 ≠ [] ∧ q0 = eq-entry(n, q0.head.lebref) + q) ∧ ubi-cons(apebs, vols, wlar, q, bflipset, vtblpnum, sqnum)))
Proof step 12:
# apebs0 = # wlar, LEB_SIZE = (n1 + m0) + m1, vols0[v][m] = unmapped, apebs = apebs0, apebs = apebs, apebs1 = apebs, bflipset0 = bflipset, q = q0, q = q, q1 = q, sqnum0 = sqnum, vols1 = vols0, vtblpnum0 = vtblpnum2, wlar = wlar, wlar1 = wlar, m < # vols0[v], bflipset-cons(bflipset, wlar), is-aligned(m1, EB_PAGE_SIZE), is-aligned(m0, EB_PAGE_SIZE), ubi-cons(apebs, vols0, wlar, q, bflipset, vtblpnum2, sqnum), ubi-io-inv(apebs0), ubi-io-inv(apebs), unique-sqns(apebs0), valid-ebatbl(apebs0, wlar, vols0, vtblpnum2), valid-eraseq(apebs0, q0), valid-eraseq(wlar, q0), valid-eraseq(q0), valid-global-sqn(apebs0, sqnum), valid-vtbl(apebs0, vols0, wlar, vtblpnum2), valid-wl(apebs0, wlar), v ∈ vols0, m1 ≤ LEB_SIZE, m1 + m2 ≤ # buf, m0 + m1 ≤ LEB_SIZE ⊦ [ubi_wl_get_peb#(; vtblpnum, wlar, q, apebs; err)] ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err ≠ ESUCCESS → ubi-cons(apebs, vols1, wlar, q, bflipset0, vtblpnum0, sqnum0)) ∧ ubi-io-inv(apebs) ∧ q.lebrefs ⊆ q1.lebrefs ∧ id(apebs, apebs1, mapped-pebs(vols1)) ∧ ( err = ESUCCESS → vtblpnum < # apebs ∧ wlar[vtblpnum].status = free ∧ ( apebs1 = apebs ∧ wlar1 = wlar ∧ q1 = q ∨ id○(apebs1, apebs, vtblpnum) ∧ id○(wlar1, wlar, vtblpnum) ∧ q1 ≠ [] ∧ q1 = eq-entry(vtblpnum, q1.head.lebref) + q) ∧ ubi-cons(apebs, vols1, wlar, q, bflipset0, vtblpnum0, sqnum0)) → ⟪vols := vols0; if err = ESUCCESS then let m3 = ? in {ubi_next_sqnum#(; sqnum; m3); aubi_io_write_vidhdr#(vtblpnum, avidhdr(v, m, m3, 0, 0); apebs; err); if err = ESUCCESS then {if m1 ≠ 0 then aubi_io_write_data#(vtblpnum, m0, m2, m1, buf; apebs; err); if err = ESUCCESS then {vols[v][m] := ⌜vtblpnum⌝ ; wlar[vtblpnum] := wl-entry(wlar[vtblpnum].ec, used)} else ubi_wl_put_peb#(v × m, vtblpnum; wlar, q); skip} else ubi_wl_put_peb#(none, vtblpnum; wlar, q)}⟫ ( # apebs = # wlar ∧ unique-sqns(apebs) ∧ valid-global-sqn(apebs, sqnum) ∧ valid-wl(apebs, wlar) ∧ valid-eraseq(q) ∧ valid-eraseq(wlar, q) ∧ valid-eraseq(apebs, q) ∧ ¬ VTBL_VOLID ∈ vols ∧ bflipset-cons(bflipset, wlar) ∧ valid-ebatbl(apebs, wlar, vols, vtblpnum2) ∧ valid-vtbl(apebs, vols, wlar, vtblpnum2) ∧ ubi-io-inv(apebs) ∧ id(apebs, apebs0, mapped-pebs(vols0)) ∧ q.lebrefs ⊆ q0.lebrefs ++ (v × m) ∧ (m1 = 0 → q.lebrefs ⊆ q0.lebrefs) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err ≠ ESUCCESS → vols = vols0) ∧ ( err = ESUCCESS → (∃ n. vols = vols0[v, vols0[v][m, ⌜n⌝ ]] ∧ (m1 ≠ 0 → apebs[n].fill = m0 + m1) ∧ (m1 = 0 → apebs[n].fill = 0) ∧ apebs[n].data = mkempbuf(m0) + subarray(buf, m2, m1) + mkempbuf(n1))))), VTBL_VOLID ∈ vols0, vtblpnum2 ∈ q0

Rule argument

spec: <toplevel> name: prepost-get-peb seq: ubi-cons(apebs, vols, wlar, q, bflipset, vtblpnum, sqnum), ubi-io-inv(apebs), wlar0 = wlar, q0 = q, apebs0 = apebs ⊦ ⟪ubi_wl_get_peb#(; n, wlar, q, apebs; err)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err ≠ ESUCCESS → ubi-cons (apebs, vols, wlar, q, bflipset, vtblpnum, sqnum)) ∧ ubi-io-inv(apebs) ∧ q.lebrefs ⊆ q0.lebrefs ∧ id(apebs, apebs0, mapped-pebs(vols)) ∧ ( err = ESUCCESS → n < # apebs ∧ wlar[n].status = free ∧ ( apebs0 = apebs ∧ wlar0 = wlar ∧ q0 = q ∨ id○(apebs0, apebs, n) ∧ id○(wlar0, wlar, n) ∧ q0 ≠ [] ∧ q0 = eq-entry(n, q0.head.lebref) + q) ∧ ubi-cons (apebs, vols, wlar, q, bflipset, vtblpnum, sqnum))) subst: [apebs, q, wlar, n, err, vols, bflipset, vtblpnum, sqnum, wlar0, q0, apebs0] → [apebs, q, wlar, vtblpnum, err, vols0, bflipset, vtblpnum2, sqnum, wlar, q, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ubi-cons(apebs, vols, wlar, q, bflipset, n0, n1) ↔ # apebs = # wlar ∧ unique-sqns(apebs) ∧ valid-global-sqn(apebs, n1) ∧ valid-wl(apebs, wlar) ∧ valid-eraseq(q) ∧ valid-eraseq(wlar, q) ∧ valid-eraseq(apebs, q) ∧ ¬ VTBL_VOLID ∈ vols ∧ bflipset-cons(bflipset, wlar) ∧ valid-ebatbl(apebs, wlar, vols, n0) ∧ valid-vtbl(apebs, vols, wlar, n0); (s)