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Proof step 17 from proof valid-wl-next-valid-use (previous step)

# apebs = # wlar, apebs[m] = mkpeb(aehdr, avhdr, buf, m2, false), valid-sqns(v, n, apebs, m) = ∅, m < # wlar, valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m2, false)), ∀ n0. n0 < m → (wlar[n0].status = erroneous ↔ apebs[n0].bad) ∧ (wlar[n0].status = free → apebs[n0].vidhdr = empty) ∧ (wlar[n0].status ≠ erroneous ∧ wlar[n0].status ≠ erasing → apebs[n0].echdr = aechdr(wlar[n0].ec)) ∧ (wlar[n0].status = used → (∃ v0, m1. valid-peb(v0, m1, apebs[n0]) ∧ apebs[n0].vidhdr.sqn = valid-sqns(v0, m1, apebs, m).max)) ⊦ aehdr = aechdr(aehdr.ec) ∧ (∃ v0, m1. valid-peb(v0, m1, mkpeb(aechdr(aehdr.ec), avhdr, buf, m2, false)) ∧ avhdr.sqn = (valid-sqns(v0, m1, apebs, m) ∪ (v = v0 ∧ n = m1 ⊃ {avhdr.sqn} ;∅)).max)

Heuristic cut applied rule cut formula

And obtained the following new goals

Proof step 18:
# apebs = # wlar, apebs[m] = mkpeb(aehdr, avhdr, buf, m2, false), valid-sqns(v, n, apebs, m) = ∅, m < # wlar, valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m2, false)), ∀ n0. n0 < m → (wlar[n0].status = erroneous ↔ apebs[n0].bad) ∧ (wlar[n0].status = free → apebs[n0].vidhdr = empty) ∧ (wlar[n0].status ≠ erroneous ∧ wlar[n0].status ≠ erasing → apebs[n0].echdr = aechdr(wlar[n0].ec)) ∧ (wlar[n0].status = used → (∃ v0, m1. valid-peb(v0, m1, apebs[n0]) ∧ apebs[n0].vidhdr.sqn = valid-sqns(v0, m1, apebs, m).max)) ⊦ aehdr = aechdr(aehdr.ec), aehdr = aechdr(aehdr.ec) ∧ (∃ v0, m1. valid-peb(v0, m1, mkpeb(aechdr(aehdr.ec), avhdr, buf, m2, false)) ∧ avhdr.sqn = (valid-sqns(v0, m1, apebs, m) ∪ (v = v0 ∧ n = m1 ⊃ {avhdr.sqn} ;∅)).max)

Rule argument

fma: aehdr = aechdr(aehdr.ec)

Additional information

Used simplifier rules: ⊦ 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) Used formulas: aehdr = aechdr(aehdr.ec)