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Proof step 11 from proof valid-eraseq-stronger-next-append-lref (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), eqent.pnum < # apebs, eqent.pnum < m, m < # apebs, eqent.pnum + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), ∀ v0, m1. (valid-peb(v0, m1, apebs[eqent.pnum]) ↔ eqent.lebref = v0 × m1) ∧ (eqent.lebref = v0 × m1 → apebs[eqent.pnum].vidhdr.sqn ≠ valid-sqns(v0, m1, apebs, m).max), ∀ apebs, m, aehdr, avhdr, buf, m0, boolvar, v, n. apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar) ∧ m < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) → ( valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max) ⊦ ¬ eqent.pnum ∈ q0 + eq-entry(m, v × n) ' ∧ (∀ v0, m1. (valid-peb(v0, m1, subarray(apebs, 0, m + 1)[eqent.pnum]) ↔ eqent.lebref = v0 × m1) ∧ ( eqent.lebref = v0 × m1 → subarray(apebs, 0, m + 1)[eqent.pnum].vidhdr.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (valid-peb(v0, m1, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊃ {avhdr.sqn} ;∅)).max)) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max, eqent.pnum ∈ q0

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);% ⊦ m +1 = m + 1; (s)% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)% ⊦ m < n → n ≠ 0; (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%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n → ¬ n < m; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ n + 0 = n; (s)% ⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]; (s)%⊦ 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)%n ≤ # apebs ⊦ valid-sqns(v, m, subarray(apebs, 0, n)) = valid-sqns(v, m, apebs, n); (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ # subarray(apebs, m, n) = n; (s)%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base]% ⊦ eqent + q = eqent ' + q; (s)