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Proof step 8 from proof abs-lebrefs-subset (previous step)

# vols[v] = # avols[v], avols[v][m] = erased, apebs[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), vols[v][m] = unmapped, m < # avols[v], v × m ∈ q1, v ∈ avols, v ∈ vols, q1.lebrefs ⊆ q0.lebrefs, ∀ m. m < # avols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ (vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ 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 = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q0))) ⊦ v × m ∈ q0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ s ⊆ s0 ↔ (∀ lref. lref ∈ s → lref ∈ s0)
Proof step 10:
# vols[v] = # avols[v], vols[v][m] = unmapped, apebs[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), avols[v][m] = erased, m < # avols[v], v ∈ vols, v ∈ avols, v × m ∈ q1, ∀ 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 = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q0))), ∀ m. m < # avols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ (vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q0), ∀ lref. lref ∈ q1.lebrefs → lref ∈ q0.lebrefs ⊦ v × m ∈ q0

Rule argument

spec: set-basic, inst: lebrefset name: Subset seq: ⊦ s ⊆ s0 ↔ (∀ lref. lref ∈ s → lref ∈ s0) subst: [s0, s] → [q0.lebrefs, q1.lebrefs] flags: <all paths>

Additional information

Used terms: q0.lebrefs, q1.lebrefs