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Proof step 25 from proof valid-eraseq-stronger-next-append-lref (previous step)

valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) ↔ lref = v0 × m1, apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), apebs[m2] = mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0), subarray(apebs, 0, m + 1)[m2] = mkpeb(aehdr1, avhdr1, buf1, m4, boolvar1), m2 + 1 < # apebs, m2 < m, m < # apebs, m2 < # apebs, avhdr.sqn < valid-sqns(v, n, apebs, m).max, 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-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) → avhdr0.sqn ≠ valid-sqns(v0, m1, apebs, m).max ⊦ m2 = m, valid-sqns(v, n, apebs, m) = ∅, m2 ∈ q0, ( valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) → avhdr1.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (v = v0 ∧ n = m1 ⊃ {avhdr.sqn} ;∅)).max) ∧ (valid-peb(v0, m1, mkpeb(aehdr1, avhdr1, buf1, m4, boolvar1)) ↔ valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]
Proof step 27:
valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) ↔ lref = v0 × m1, apebs[0 + m2] = mkpeb(aehdr1, avhdr1, buf1, m4, boolvar1), apebs[m2] = mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0), apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), avhdr.sqn < valid-sqns(v, n, apebs, m).max, m2 < # apebs, m < # apebs, m2 < m, m2 + 1 < # apebs, m2 < m + 1, valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) '), valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) → avhdr0.sqn ≠ valid-sqns(v0, m1, apebs, m).max ⊦ m2 = m, valid-sqns(v, n, apebs, m) = ∅, m2 ∈ q0, ( valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) → avhdr1.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (v = v0 ∧ n = m1 ⊃ {avhdr.sqn} ;∅)).max) ∧ (valid-peb(v0, m1, mkpeb(aehdr1, avhdr1, buf1, m4, boolvar1)) ↔ valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)))

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] → [m2, m + 1, 0, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m ≠ n → (m < n + 1 ↔ m < n); (s)% ⊦ n ≠ n0 → (n < n0 ↔ ¬ n0 < n); (ss)% ⊦ m < n → ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%# apebs = n ⊦ subarray(apebs, 0, n) = apebs; (s) Used terms: m2, m + 1, 0, apebs