[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 13 from proof abs-lebrefs-subset-noterased-some (previous step)

# vols[v0] = # avols[v0], # vols[v] = # avols[v], avols[v0][m0] = erased, apebs[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), vols[v0][m0] = unmapped, apebs[vols[v][m].pnum] = mkpeb(aehdr0, avhdr0, buf0, m2, boolvar0), m0 < # avols[v0], m < # avols[v], v0 × m0 ∈ q1, v0 ∈ avols, v0 ∈ vols, v ∈ avols, v ∈ vols, q1.lebrefs ⊆ q0.lebrefs ++ (v × m), vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = buf0 ∧ m2 ≤ avols[v][m].fill, vols[v][m] = unmapped → ¬ avols[v][m].mapped?, ∀ m. m < # avols[v0] → (vols[v0][m] = unmapped → ¬ avols[v0][m].mapped?) ∧ ( vols[v0][m] ≠ unmapped → avols[v0][m].mapped? ∧ avols[v0][m].data = apebs[vols[v0][m].pnum].data ∧ apebs[vols[v0][m].pnum].fill ≤ avols[v0][m].fill) ∧ (avols[v0][m] = erased → ¬ v0 × 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))), ∀ 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) ⊦ avols[v][m] = erased, v0 × m0 ∈ q0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ s ⊆ s0 ↔ (∀ lref. lref ∈ s → lref ∈ s0)
Proof step 15:
# vols[v] = # avols[v], # vols[v0] = # avols[v0], apebs[vols[v][m].pnum] = mkpeb(aehdr0, avhdr0, buf0, m2, boolvar0), vols[v0][m0] = unmapped, apebs[unmapped.pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), avols[v0][m0] = erased, m < # avols[v], m0 < # avols[v0], v ∈ vols, v ∈ avols, v0 ∈ vols, v0 ∈ avols, v0 × m0 ∈ q1, vols[v][m] = unmapped → ¬ avols[v][m].mapped?, vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = buf0 ∧ m2 ≤ avols[v][m].fill, ∀ 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))), ∀ m. m < # avols[v0] → (vols[v0][m] = unmapped → ¬ avols[v0][m].mapped?) ∧ ( vols[v0][m] ≠ unmapped → avols[v0][m].mapped? ∧ avols[v0][m].data = apebs[vols[v0][m].pnum].data ∧ apebs[vols[v0][m].pnum].fill ≤ avols[v0][m].fill) ∧ (avols[v0][m] = erased → ¬ v0 × m ∈ q0), ∀ lref. lref ∈ q1.lebrefs → lref ∈ q0.lebrefs ++ (v × m) ⊦ avols[v][m] = erased, v0 × m0 ∈ q0

Rule argument

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

Additional information

Used terms: q0.lebrefs ++ (v × m), q1.lebrefs