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Proof step 10 from proof abs-unmap (previous step)

# vols[v0] = # avols[v0], v0 ∈ vols ↔ v0 ∈ avols, vols[v][m] = ⌜n⌝ , apebs[vols[v, vols[v][m, unmapped]][v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), apebs[vols[v0][m0].pnum] = mkpeb(aehdr0, avhdr0, buf0, m2, boolvar0), m < # vols[v], v ∈ vols, m0 < # avols[v0] → (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ (vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = buf0 ∧ m2 ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q) ⊦ (v0 ∈ vols[v, vols[v][m, unmapped]] ↔ v0 ∈ avols[v, avols[v][m, unmapped]]) ∧ ( v0 ∈ vols[v, vols[v][m, unmapped]] → # vols[v, vols[v][m, unmapped]][v0] = # avols[v, avols[v][m, unmapped]][v0] ∧ ( m0 < # vols[v, vols[v][m, unmapped]][v0] → (vols[v, vols[v][m, unmapped]][v0][m0] = unmapped → ¬ avols[v, avols[v][m, unmapped]][v0][m0].mapped?) ∧ ( vols[v, vols[v][m, unmapped]][v0][m0] ≠ unmapped → avols[v, avols[v][m, unmapped]][v0][m0].mapped? ∧ avols[v, avols[v][m, unmapped]][v0][m0].data = buf ∧ m1 ≤ avols[v, avols[v][m, unmapped]][v0][m0].fill) ∧ (avols[v, avols[v][m, unmapped]][v0][m0] = erased → ¬ v0 × m0 ∈ q + eq-entry(n, v × m) ')))

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 11:
# vols[v0] = # avols[v0], v0 ∈ vols ↔ v0 ∈ avols, apebs[vols[v0][m0].pnum] = mkpeb(aehdr0, avhdr0, buf0, m2, boolvar0), apebs[vols[v, vols[v][m, unmapped]][v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), vols[v][m] = ⌜n⌝ , v0 = v, m < # vols[v], v ∈ vols, m0 < # avols[v0] → (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ (vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = buf0 ∧ m2 ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q) ⊦ (v0 ∈ vols[v, vols[v][m, unmapped]] ↔ v0 ∈ avols[v, avols[v][m, unmapped]]) ∧ ( v0 ∈ vols[v, vols[v][m, unmapped]] → # vols[v, vols[v][m, unmapped]][v0] = # avols[v, avols[v][m, unmapped]][v0] ∧ ( m0 < # vols[v, vols[v][m, unmapped]][v0] → (vols[v, vols[v][m, unmapped]][v0][m0] = unmapped → ¬ avols[v, avols[v][m, unmapped]][v0][m0].mapped?) ∧ ( vols[v, vols[v][m, unmapped]][v0][m0] ≠ unmapped → avols[v, avols[v][m, unmapped]][v0][m0].mapped? ∧ avols[v, avols[v][m, unmapped]][v0][m0].data = buf ∧ m1 ≤ avols[v, avols[v][m, unmapped]][v0][m0].fill) ∧ (avols[v, avols[v][m, unmapped]][v0][m0] = erased → ¬ v0 × m0 ∈ q + eq-entry(n, v × m) ')))

Rule argument

fma: v0 = v

Additional information

Used simplifier rules: ⊦ v ≠ v0 → vols[v0, ebatbl][v] = vols[v]; (s)% ⊦ v ≠ v0 → (v ∈ vols[v0, ebatbl] ↔ v ∈ vols); (s)% ⊦ v ≠ v0 → (v ∈ avols[v0, evol] ↔ v ∈ avols); (s)% ⊦ v ≠ v0 → avols[v0, evol][v] = avols[v]; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%⊦ mkpeb(aehdr, avhdr, buf, m, boolvar) = mkpeb(aehdr0, avhdr0, buf0, m0, boolvar0) ↔ aehdr = aehdr0 ∧ avhdr = avhdr0 ∧ buf = buf0 ∧ m = m0 ∧ (boolvar ↔ boolvar0); (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ ¬ lref ∈ q → (lref ∈ q + eq-entry(n, lref0) ' ↔ lref = lref0); (s)% ⊦ v × m = v0 × m0 ↔ v = v0 ∧ m = m0; (ws)%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base] Used formulas: v0 = v