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Proof step 19 from proof valid-eraseq-stronger-next-append-lref (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
subarray(apebs, 0, m + 1)[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, 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)), ∀ v0, m1. (valid-peb(v0, m1, mkpeb(aehdr0, avhdr0, buf0, m3, boolvar0)) ↔ lref = v0 × m1) ∧ (lref = v0 × m1 → avhdr0.sqn ≠ valid-sqns(v0, m1, apebs, m).max) ⊦ valid-sqns(v, n, apebs, m) = ∅, m2 ∈ q0, ¬ m2 ∈ q0 + eq-entry(m, v × n) ' ∧ (valid-peb(v0, m1, mkpeb(aehdr1, avhdr1, buf1, m4, boolvar1)) ↔ lref = v0 × m1) ∧ (lref = v0 × m1 → avhdr1.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (v = v0 ∧ n = m1 ⊃ {avhdr.sqn} ;∅)).max)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)