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Proof step 4 from proof valid-peb-in-sqnums-h (previous step)

apebs[n0] = mkpeb(aehdr, avhdr, buf, m0, boolvar), n0 < # apebs, n0 < n +1, valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)), n +1 ≤ # apebs, ∀ apebs, n0, aehdr, avhdr, buf, m0, boolvar, v, m. apebs[n0] = mkpeb(aehdr, avhdr, buf, m0, boolvar) ∧ n0 < n ∧ n0 < # apebs ∧ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ n ≤ # apebs → avhdr.sqn ∈ valid-sqns(v, m, apebs, n) ⊦ avhdr.sqn ∈ valid-sqns(v, m, apebs, n +1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
apebs[n0] = mkpeb(aehdr, avhdr, buf, m0, boolvar), n0 < # apebs, n < # apebs, valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)), n0 ≤ n, ∀ apebs, n0, aehdr, avhdr, buf, m0, boolvar, v, m. apebs[n0] = mkpeb(aehdr, avhdr, buf, m0, boolvar) ∧ n0 < n ∧ n0 < # apebs ∧ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ n ≤ # apebs → avhdr.sqn ∈ valid-sqns(v, m, apebs, n) ⊦ avhdr.sqn ∈ valid-sqns(v, m, apebs, n), avhdr.sqn ∈ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)

Rule argument

Additional information

Used simplifier rules: valid-sqns-rec: ⊦ 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} ;∅);% ⊦ m < n ↔ ¬ n ≤ m; (ss)%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]% ⊦ n ∈ bag ∪ bag2 ↔ n ∈ bag ∨ n ∈ bag2; (ws)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m +1 = m + 1; (s)