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

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]
Proof step 13:
apebs[m1] = mkpeb(aehdr, avhdr, buf, m2, boolvar), apebs[0 + m1] = mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0), m1 + 1 < # apebs, m1 < m, m < # apebs, m1 < # apebs, m1 < m + 1, valid-eraseq-stronger(subarray(apebs, 0, m), q0), ∀ v, m0. (valid-peb(v, m0, mkpeb(aehdr, avhdr, buf, m2, boolvar)) ↔ lref = v × m0) ∧ (lref = v × m0 → avhdr.sqn ≠ valid-sqns(v, m0, apebs, m).max), ∀ m, apebs. m < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) → valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0) ⊦ m1 ∈ q0, (valid-peb(v, m0, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) ↔ lref = v × m0) ∧ (lref = v × m0 → avhdr0.sqn ≠ (valid-sqns(v, m0, apebs, m) ∪ (valid-peb(v, m0, apebs[m]) ⊃ {apebs[m].vidhdr.sqn} ;∅)).max) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0)

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] → [m1, 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: m1, m + 1, 0, apebs