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Proof step 17 from proof valid-eraseq-stronger-prev-append-lref (previous step)

apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), m0 < # apebs, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < m, m < # apebs, m0 + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), eqent + q0), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v, n, apebs[m0]), ∀ 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 ⊦ valid-eraseq-stronger(subarray(apebs, 0, m + 1), (eqent + q0) + eq-entry(m0, v × n) '), m0 ∈ eqent + q0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), eqent.pnum < # apebs, eqent.pnum < m, m0 < # apebs, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < m, m < # apebs, m0 + 1 < # apebs, eqent.pnum + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v, n, apebs[m0]), ∀ v0, m2. (valid-peb(v0, m2, apebs[eqent.pnum]) ↔ eqent.lebref = v0 × m2) ∧ (eqent.lebref = v0 × m2 → apebs[eqent.pnum].vidhdr.sqn ≠ valid-sqns(v0, m2, apebs, m).max), ∀ 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 ⊦ m0 ∈ q0, m0 ∈ eqent ', eqent.pnum ∈ q0, ¬ eqent.pnum ∈ q0 + eq-entry(m0, v × n) ' ∧ (valid-peb(v0, m2, subarray(apebs, 0, m + 1)[eqent.pnum]) ↔ eqent.lebref = v0 × m2) ∧ ( eqent.lebref = v0 × m2 → subarray(apebs, 0, m + 1)[eqent.pnum].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: valid-eraseq-stronger-rec: ⊦ valid-eraseq-stronger(apebs, eqent ' + q) ↔ eqent.pnum < # apebs ∧ ¬ eqent.pnum ∈ q ∧ (∀ v, m. (valid-peb(v, m, apebs[eqent.pnum]) ↔ eqent.lebref = v × m) ∧ ( eqent.lebref = v × m → apebs[eqent.pnum].vidhdr.sqn ≠ valid-sqns(v, m, apebs).max)) ∧ valid-eraseq-stronger(apebs, q);% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)% ⊦ 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%⊦ n < # apebs → valid-sqns(v, m, apebs, n + 1) = valid-sqns(v, m, apebs, n) ∪ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅); (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ n ∈ q0 + q1 ↔ n ∈ q0 ∨ n ∈ q1; (ss)%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base]%n ≤ # apebs ⊦ valid-sqns(v, m, subarray(apebs, 0, n)) = valid-sqns(v, m, apebs, n); (s)% ⊦ m < n → m ≤ n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]; (s)% ⊦ # subarray(apebs, m, n) = n; (s)% ⊦ eqent + q = eqent ' + q; (s)