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Proof step 51 from proof valid-wl-write-valid (previous step)

# apebs = # wlar, avhdr1.sqn = valid-sqns(v, m5, apebs).max, wlar[m] = wl-entry(m2, used), apebs[m] = mkpeb(aehdr, avhdr, buf0, m0, boolvar0), wlar[n] = wl-entry(m4, used), apebs[n] = mkpeb(aechdr(m4), avhdr1, buf2, m6, false), n < # wlar, m < # wlar, valid-peb(v, m5, mkpeb(aechdr(m4), avhdr1, buf2, m6, false)), valid-peb(mkpeb(aehdr, avhdr, buf1, m1, boolvar1)), valid-peb(mkpeb(aehdr, avhdr, buf0, m0, boolvar0)) ⊦ avhdr1.sqn = ( (valid-sqns(v, m5, apebs) \ (valid-peb(v, m5, mkpeb(aehdr, avhdr, buf0, m0, boolvar0)) ⊃ {avhdr.sqn} ;∅)) ∪ (valid-peb(v, m5, mkpeb(aehdr, avhdr, buf1, m1, boolvar1)) ⊃ {avhdr.sqn} ;∅) ).max, n = m

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 52:
# apebs = # wlar, avhdr1.sqn = valid-sqns(v, m5, apebs).max, apebs[n] = mkpeb(aechdr(m4), avhdr1, buf2, m6, false), wlar[n] = wl-entry(m4, used), apebs[m] = mkpeb(aehdr, avhdr, buf0, m0, boolvar0), wlar[m] = wl-entry(m2, used), m < # wlar, n < # wlar, valid-peb(mkpeb(aehdr, avhdr, buf0, m0, boolvar0)), valid-peb(mkpeb(aehdr, avhdr, buf1, m1, boolvar1)), valid-peb(v, m5, mkpeb(aechdr(m4), avhdr1, buf2, m6, false)) ⊦ avhdr1.sqn = ((valid-sqns(v, m5, apebs) \ ∅) ∪ (valid-peb(v, m5, mkpeb(aehdr, avhdr, buf1, m1, boolvar1)) ⊃ {avhdr.sqn} ;∅)).max, n = m, valid-peb(v, m5, mkpeb(aehdr, avhdr, buf0, m0, boolvar0))

Rule argument

pos: right 1

Additional information

Used simplifier rules: ⊦ bag \ {n} = bag -- n; (s)%⊦ valid-peb(apeb) → (valid-peb(v, m, apeb) ↔ v = apeb.vidhdr.vol ∧ m = apeb.vidhdr.leb); (ss)% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)%⊦ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) → avhdr.vol = v; (s)%⊦ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) → avhdr.leb = m; (s)% ⊦ bag ∪ {n} = bag ++ n; (s)% ⊦ (bag -- n) ++ n = (n ∈ bag ⊃ bag;bag ++ n); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%n < # apebs ⊦ valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ apebs[n] = mkpeb(aehdr, avhdr, buf, m0, boolvar) → avhdr.sqn ∈ valid-sqns(v, m, apebs); (ss)%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]