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Proof step 45 from proof multi-at-other (previous step)

# refs = 1 + i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), refs) = r0 ' + refs1, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, ∀ sk, st, ty, acc, 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 → (addmultiarrays(refs0, i, acc, ty, ints, st)[r -- sk]) = st[r -- sk])) ⊦ (addmultiarrays (firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs), i2, r0 ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st)) [r -- sk ] ) = st[r -- sk], 0 = i0, (~1) = i2, r ∈ refs, 0 ∈ (i1 ' + ints1) .butlast

User applied rule apply induction

And obtained the following new goals

Proof step 8:
⊦ countrefs(ints) ≠ 0
Proof step 44:
¬ r ∈ firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs) ∧ # firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs) = i→n(countrefs(i2 ' + (i0 ' + i1 ' + ints1) .rest)) ∧ i0 ' + i1 ' + ints1 ≠ [] ∧ 0 ≤ i2 ∧ posints(i0 ' + i1 ' + ints1) ∧ ¬ 0 ∈ (i0 ' + i1 ' + ints1) .butlast ∧ i2 ≤ (i0 ' + i1 ' + ints1) .first → (addmultiarrays (firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs), i2, r0 ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st)) [r -- sk ] ) = (addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st)[r -- sk]), # refs = 1 + i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), refs) = r0 ' + refs1, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, ∀ sk, st, ty, acc, 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 → (addmultiarrays(refs0, i, acc, ty, ints, st)[r -- sk]) = st[r -- sk])) ⊦ (addmultiarrays (firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs), i2, r0 ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st)) [r -- sk ] ) = st[r -- sk], 0 = i0, (~1) = i2, r ∈ refs, 0 ∈ (i1 ' + ints1) .butlast

Rule argument

[substlist-arg: [sk, st, ty, acc, ints, i, r, refs0] → [sk, addmultiarrays (r0 ' + refs1, i1, [], ty, i1 ' + ints1, st), ty, r0 ' + acc, i0 ' + i1 ' + ints1, i2, r, firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs)] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ # firstn(n, refs) = n], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ ¬ m + n < m], [CSIMPSEQ : ⊦ i→n(i) = 0 ↔ i = 0], [CSIMPFORWARD : ⊦ m < n → n ≠ 0], [CSIMPFORWARD : ⊦ m < n ∧ m = n0 → n0 < n], [CSIMPSEQ : ⊦ ¬ m ≤ n ↔ n < m], [CSIMPSEQ : n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m + n ≤ m ↔ n = 0])], [EXTRATERMS : (sk, addmultiarrays (r0 ' + refs1, i1, [], ty, i1 ' + ints1, st), ty, r0 ' + acc, i0 ' + i1 ' + ints1, i2, r, firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs))]]