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Proof step 33 from proof refine-create-volume (previous step)

# apebs0 = # wlar0, # apebs = # wlar, apebs0[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), m < n, abs(avols, apebs, vols, q), bflipset-cons(bflipset0, wlar0), bflipset-cons(bflipset, wlar), ebm-cons(avols), id(apebs0, apebs, mapped-pebs(vols)), ubi-io-inv(apebs0), ubi-io-inv(apebs), unique-sqns(apebs0), unique-sqns(apebs), valid-ebatbl(apebs0, wlar0, vols[v, fill(mkebatbl(n), unmapped)], vtblpnum0), valid-ebatbl(apebs, wlar, vols, vtblpnum), valid-eraseq(q0), valid-eraseq(wlar0, q0), valid-eraseq(apebs0, q0), valid-eraseq(q), valid-eraseq(wlar, q), valid-eraseq(apebs, q), valid-global-sqn(apebs0, sqnum0), valid-global-sqn(apebs, sqnum), valid-vtbl(apebs0, vols[v, fill(mkebatbl(n), unmapped)], wlar0, vtblpnum0), valid-vtbl(apebs, vols, wlar, vtblpnum), valid-wl(apebs0, wlar0), valid-wl(apebs, wlar), v × m ∈ q0, q0.lebrefs ⊆ ∅ ++ (VTBL_VOLID × VTBL_LNUM), ∀ v. (v ∈ vols ↔ v ∈ avols) ∧ ( v ∈ vols → # vols[v] = # avols[v] ∧ (∀ m. m < # vols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ ( vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs0[vols[v][m].pnum].data ∧ apebs0[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q))) ⊦ VTBL_VOLID = v, VTBL_VOLID ∈ vols, v ∈ avols, v ∈ vols

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ s ⊆ s0 ↔ (∀ lref. lref ∈ s → lref ∈ s0)
Proof step 35:
# apebs = # wlar, # apebs0 = # wlar0, apebs0[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), m < n, abs(avols, apebs, vols, q), bflipset-cons(bflipset, wlar), bflipset-cons(bflipset0, wlar0), ebm-cons(avols), id(apebs0, apebs, mapped-pebs(vols)), ubi-io-inv(apebs), ubi-io-inv(apebs0), unique-sqns(apebs), unique-sqns(apebs0), valid-ebatbl(apebs, wlar, vols, vtblpnum), valid-ebatbl(apebs0, wlar0, vols[v, fill(mkebatbl(n), unmapped)], vtblpnum0), valid-eraseq(apebs, q), valid-eraseq(wlar, q), valid-eraseq(q), valid-eraseq(apebs0, q0), valid-eraseq(wlar0, q0), valid-eraseq(q0), valid-global-sqn(apebs, sqnum), valid-global-sqn(apebs0, sqnum0), valid-vtbl(apebs, vols, wlar, vtblpnum), valid-vtbl(apebs0, vols[v, fill(mkebatbl(n), unmapped)], wlar0, vtblpnum0), valid-wl(apebs, wlar), valid-wl(apebs0, wlar0), v × m ∈ q0, ∀ v. (v ∈ vols ↔ v ∈ avols) ∧ ( v ∈ vols → # vols[v] = # avols[v] ∧ (∀ m. m < # vols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ ( vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs0[vols[v][m].pnum].data ∧ apebs0[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q))), ∀ lref. lref ∈ q0.lebrefs → lref ∈ ∅ ++ (VTBL_VOLID × VTBL_LNUM) ⊦ VTBL_VOLID = v, v ∈ vols, v ∈ avols, VTBL_VOLID ∈ vols

Rule argument

spec: set-basic, inst: lebrefset name: Subset seq: ⊦ s ⊆ s0 ↔ (∀ lref. lref ∈ s → lref ∈ s0) subst: [s0, s] → [∅ ++ (VTBL_VOLID × VTBL_LNUM), q0.lebrefs] flags: <all paths>

Additional information

Used terms: ∅ ++ (VTBL_VOLID × VTBL_LNUM), q0.lebrefs