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Proof step 28 from proof valid-eraseq-stronger-prev-append-lref (previous step)

valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) ↔ lref = v0 × m2, subarray(apebs, 0, m + 1)[m3] = mkpeb(aehdr1, avhdr1, buf1, m5, boolvar1), apebs[m3] = mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0), apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), m3 < # apebs, m0 + 1 < # apebs, m < # apebs, m0 < m, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < # apebs, m3 < m, m3 + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) '), valid-peb(v, n, apebs[m0]), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) → avhdr0.sqn ≠ valid-sqns(v0, m2, apebs, m).max ⊦ m0 = m3, m0 ∈ q0, m3 ∈ q0, (valid-peb(v0, m2, mkpeb(aehdr1, avhdr1, buf1, m5, boolvar1)) ↔ valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0))) ∧ ( valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) → avhdr1.sqn ≠ (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {avhdr.sqn} ;∅)).max)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]
Proof step 30:
valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) ↔ lref = v0 × m2, apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), apebs[m3] = mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0), apebs[0 + m3] = mkpeb(aehdr1, avhdr1, buf1, m5, boolvar1), m3 + 1 < # apebs, m3 < m, m0 < # apebs, valid-sqns(v, n, apebs, m).max < avhdr.sqn, m0 < m, m < # apebs, m0 + 1 < # apebs, m3 < # apebs, m3 < m + 1, valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v × n) '), valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v, n, apebs[m0]), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) → avhdr0.sqn ≠ valid-sqns(v0, m2, apebs, m).max ⊦ m0 = m3, m3 ∈ q0, m0 ∈ q0, (valid-peb(v0, m2, mkpeb(aehdr1, avhdr1, buf1, m5, boolvar1)) ↔ valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0))) ∧ ( valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, boolvar0)) → avhdr1.sqn ≠ (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {avhdr.sqn} ;∅)).max)

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] → [m3, m + 1, 0, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ 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: m3, m + 1, 0, apebs