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Proof step 13 from proof absexpt-vols-write (previous step)

# vols[v] = # avols[v], v0 ∈ vols[v, vols[v][m, ebae]] ↔ v0 ∈ avols, apebs[vols[v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m2, boolvar), m < # avols[v], v ∈ vols, v ∈ avols, v0 ∈ avols → # vols[v, vols[v][m, ebae]][v0] = # avols[v0] ∧ (∀ m1. (v0 ≠ v ∨ m1 ≠ m) ∧ m1 < # vols[v, vols[v][m, ebae]][v0] → (vols[v, vols[v][m, ebae]][v0][m1] = unmapped → ¬ avols[v0][m1].mapped?) ∧ ( vols[v, vols[v][m, ebae]][v0][m1] ≠ unmapped → avols[v0][m1].mapped? ∧ avols[v0][m1].data = apebs[vols[v, vols[v][m, ebae]][v0][m1].pnum].data ∧ apebs[vols[v, vols[v][m, ebae]][v0][m1].pnum].fill ≤ avols[v0][m1].fill) ∧ (avols[v0][m1] = erased → ¬ v0 × m1 ∈ q)), ∀ v1. (v1 ∈ vols[v, vols[v][m, ebae]] ↔ v1 ∈ avols) ∧ ( v1 ∈ vols[v, vols[v][m, ebae]] → # vols[v, vols[v][m, ebae]][v1] = # avols[v1] ∧ (∀ m1. (v1 ≠ v ∨ m1 ≠ m) ∧ m1 < # vols[v, vols[v][m, ebae]][v1] → (vols[v, vols[v][m, ebae]][v1][m1] = unmapped → ¬ avols[v1][m1].mapped?) ∧ ( vols[v, vols[v][m, ebae]][v1][m1] ≠ unmapped → avols[v1][m1].mapped? ∧ avols[v1][m1].data = apebs[vols[v, vols[v][m, ebae]][v1][m1].pnum].data ∧ apebs[vols[v, vols[v][m, ebae]][v1][m1].pnum].fill ≤ avols[v1][m1].fill) ∧ (avols[v1][m1] = erased → ¬ v1 × m1 ∈ q))), ∀ m1. m1 ≠ m ∧ m1 < # avols[v] → (vols[v][m1] = unmapped → ¬ avols[v][m1].mapped?) ∧ ( vols[v][m1] ≠ unmapped → avols[v][m1].mapped? ∧ avols[v][m1].data = apebs[vols[v][m1].pnum].data ∧ apebs[vols[v][m1].pnum].fill ≤ avols[v][m1].fill) ∧ (avols[v][m1] = erased → ¬ v × m1 ∈ q) ⊦ v0 = v, (v0 ∈ vols ↔ v0 ∈ avols) ∧ ( v0 ∈ vols → # vols[v0] = # avols[v0] ∧ ( (v0 ≠ v ∨ m0 ≠ m) ∧ m0 < # vols[v0] → (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ (vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = buf ∧ m2 ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
# vols[v0] = # avols[v0], # vols[v] = # avols[v], apebs[vols[v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m2, boolvar), m0 < # avols[v0], m < # avols[v], v0 ∈ avols, v0 ∈ vols, v ∈ vols, v ∈ avols, ∀ m1. m1 < # avols[v0] → (vols[v0][m1] = unmapped → ¬ avols[v0][m1].mapped?) ∧ ( vols[v0][m1] ≠ unmapped → avols[v0][m1].mapped? ∧ avols[v0][m1].data = apebs[vols[v0][m1].pnum].data ∧ apebs[vols[v0][m1].pnum].fill ≤ avols[v0][m1].fill) ∧ (avols[v0][m1] = erased → ¬ v0 × m1 ∈ q), ∀ v1. (v1 ∈ vols[v, vols[v][m, ebae]] ↔ v1 ∈ avols) ∧ ( v1 ∈ vols[v, vols[v][m, ebae]] → # vols[v, vols[v][m, ebae]][v1] = # avols[v1] ∧ (∀ m1. (v1 ≠ v ∨ m1 ≠ m) ∧ m1 < # vols[v, vols[v][m, ebae]][v1] → (vols[v, vols[v][m, ebae]][v1][m1] = unmapped → ¬ avols[v1][m1].mapped?) ∧ ( vols[v, vols[v][m, ebae]][v1][m1] ≠ unmapped → avols[v1][m1].mapped? ∧ avols[v1][m1].data = apebs[vols[v, vols[v][m, ebae]][v1][m1].pnum].data ∧ apebs[vols[v, vols[v][m, ebae]][v1][m1].pnum].fill ≤ avols[v1][m1].fill) ∧ (avols[v1][m1] = erased → ¬ v1 × m1 ∈ q))), ∀ m1. m1 ≠ m ∧ m1 < # avols[v] → (vols[v][m1] = unmapped → ¬ avols[v][m1].mapped?) ∧ ( vols[v][m1] ≠ unmapped → avols[v][m1].mapped? ∧ avols[v][m1].data = apebs[vols[v][m1].pnum].data ∧ apebs[vols[v][m1].pnum].fill ≤ avols[v][m1].fill) ∧ (avols[v][m1] = erased → ¬ v × m1 ∈ q) ⊦ v0 = v, (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ (vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = buf ∧ m2 ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ v ≠ v0 → vols[v0, ebatbl][v] = vols[v]; (s)% ⊦ v ≠ v0 → (v ∈ vols[v0, ebatbl] ↔ v ∈ vols); (s)