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Proof step 19 from proof valid-eraseq-stronger-prev-append-lref (previous step)

apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), eqent = eq-entry(m3, lref), m3 + 1 < # apebs, m0 + 1 < # apebs, m < # apebs, m0 < m, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < # apebs, m3 < m, m3 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, apebs[m0]), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), ∀ apebs, m, aehdr, avhdr, buf, m1, boolvar, m0, v, n. apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar) ∧ m0 + 1 < # apebs ∧ m < # apebs ∧ m0 < m ∧ valid-sqns(v, n, apebs, m).max < avhdr.sqn ∧ m0 < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) ∧ valid-peb(v, n, apebs[m0]) ∧ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)) → valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) ') ∨ m0 ∈ q0, ∀ v0, m2. (valid-peb(v0, m2, apebs[m3]) ↔ lref = v0 × m2) ∧ (lref = v0 × m2 → apebs[m3].vidhdr.sqn ≠ valid-sqns(v0, m2, apebs, m).max) ⊦ m3 ∈ q0, m0 ∈ eqent ', m0 ∈ q0, ¬ m3 ∈ q0 + eq-entry(m0, v × n) ' ∧ (valid-peb(v0, m2, subarray(apebs, 0, m + 1)[m3]) ↔ lref = v0 × m2) ∧ (lref = v0 × m2 → subarray(apebs, 0, m + 1)[m3].vidhdr.sqn ≠ (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {avhdr.sqn} ;∅)).max) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) ')

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), m3 < # apebs, m0 + 1 < # apebs, m < # apebs, m0 < m, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < # apebs, m3 < m, m3 + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, apebs[m0]), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), ∀ apebs, m, aehdr, avhdr, buf, m1, boolvar, m0, v, n. apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar) ∧ m0 + 1 < # apebs ∧ m < # apebs ∧ m0 < m ∧ valid-sqns(v, n, apebs, m).max < avhdr.sqn ∧ m0 < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) ∧ valid-peb(v, n, apebs[m0]) ∧ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)) → valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) ') ∨ m0 ∈ q0, ∀ v0, m2. (valid-peb(v0, m2, apebs[m3]) ↔ lref = v0 × m2) ∧ (lref = v0 × m2 → apebs[m3].vidhdr.sqn ≠ valid-sqns(v0, m2, apebs, m).max) ⊦ m0 = m3, m0 ∈ q0, m3 ∈ q0, (valid-peb(v0, m2, subarray(apebs, 0, m + 1)[m3]) ↔ lref = v0 × m2) ∧ (lref = v0 × m2 → subarray(apebs, 0, m + 1)[m3].vidhdr.sqn ≠ (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {avhdr.sqn} ;∅)).max) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) ')

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base]% ⊦ n ≠ n0 → (n ∈ q + eq-entry(n0, lref) ' ↔ n ∈ q); (s)% ⊦ n ∈ eq-entry(m, lref) ' ↔ n = m; (s)