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Proof step 9 from proof valid-eraseq-stronger-next-append-lref (previous step)

subarray(apebs, 0, m + 1)[m] = mkpeb(aehdr0, avhdr0, buf0, m1, boolvar0), apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), m < # apebs ⊦ valid-peb(v, n, mkpeb(aehdr0, avhdr0, buf0, m1, boolvar0)) ∧ avhdr0.sqn ≠ (valid-sqns(v, n, apebs, m) ∪ (valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊃ {avhdr.sqn} ;∅)).max ↔ 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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-copy, inst: apebs name: subarray-at seq: ⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0] subst: [n0, n, m, apebs] → [m, m + 1, 0, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%⊦ mkpeb(aehdr, avhdr, buf, m, boolvar) = mkpeb(aehdr0, avhdr0, buf0, m0, boolvar0) ↔ aehdr = aehdr0 ∧ avhdr = avhdr0 ∧ buf = buf0 ∧ m = m0 ∧ (boolvar ↔ boolvar0); (s)% ⊦ ¬ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, true)); (s)%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)% ⊦ bag ∪ {n} = bag ++ n; (s)%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]% ⊦ m = (bag ++ m).max ↔ ¬(bag ≠ ∅ ∧ m < bag.max); (s)% ⊦ bag ≠ ∅ → (bag ++ m).max = (bag.max > m ⊃ bag.max;m); (s)% ⊦ m > n ↔ n < m; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → ¬ n < m; (s)%# apebs = n ⊦ subarray(apebs, 0, n) = apebs; (s) Used terms: m, m + 1, 0, apebs