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Proof step 38 from proof multi-in-other (previous step)

# refs = i→n(countrefs(i ' + ints0)), posints(ints0), 0 < i0, i ≤ i0, 0 < i, ∀ st, ty, acc, sk, ints, i, r, refs0. # refs0 < # refs → ( true → ( ¬ r ∈ refs0 ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- sk ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- sk ∈ st))) ⊦ r -- sk ∈ addmultiarrays(refs, i, acc, ty, i0 ' + ints0, st) ↔ r -- sk ∈ st, ints0 = [], 0 = i0, i = 0, r ∈ refs, 0 ∈ ints0 .butlast

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ i > 0 ∧ # ints > 1 → addmultiarrays(refs, i, acc, ty, ints, st) = addmultiarrays (firstn(# refs - i→n(countrefs(ints .rest)), refs), i - 1, lastn(i→n(countrefs(ints .rest)), refs) .first + acc, ty, ints, addmultiarrays(lastn(i→n(countrefs(ints .rest)), refs), ints .rest .first, [], ty, ints .rest, st))
Proof step 37:
i > 0 ∧ # (i0 ' + ints0) > 1, # refs = i→n(countrefs(i ' + ints0)), posints(ints0), 0 < i0, i ≤ i0, 0 < i, ∀ st, ty, acc, sk, ints, i, r, refs0. # refs0 < # refs → ( true → ( ¬ r ∈ refs0 ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- sk ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- sk ∈ st))) ⊦ r -- sk ∈ addmultiarrays (firstn(# refs - i→n(countrefs((i0 ' + ints0) .rest)), refs), i - 1, lastn(i→n(countrefs((i0 ' + ints0) .rest)), refs) .first + acc, ty, i0 ' + ints0, addmultiarrays(lastn(i→n(countrefs((i0 ' + ints0) .rest)), refs), (i0 ' + ints0) .rest .first, [], ty, (i0 ' + ints0) .rest, st)) ↔ r -- sk ∈ st, ints0 = [], 0 = i0, i = 0, r ∈ refs, 0 ∈ ints0 .butlast

Rule argument

[crewrite-arg: addmultiarrays-rec , F , [st, ints, ty, acc, i, refs] → [st, i0 ' + ints0, ty, acc, i, refs] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i > j ↔ j < i], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ i ≠ j → (i ≤ j ↔ i < j)], [CSIMPSEQ : ⊦ i < j → ¬ j < i], [CSIMPSEQ : ⊦ # (ints + ints0) = # ints + # ints0], [CSIMPSEQ : ⊦ # i ' = 0 +1], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ m > n ↔ n < m], [CSIMPSEQ : ⊦ 1 < m ↔ ¬ (m = 0 ∨ m = 1)], [CSIMPSEQ : ⊦ m + 1 ≠ 0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ 1 = m + 1 ↔ m = 0], [CSIMPSEQ : ⊦ # ints = 0 ↔ ints = []])], [EXTRATERMS : (st, i0 ' + ints0, ty, acc, i, refs)]]