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Proof step 24 from proof multi-in-other (previous step)

# refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r1 ' + refs2) = r0 ' + refs1, r -- sk ∈ addmultiarrays (r1 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), 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), posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, sk, ints, i, r, refs0. # refs0 < # (r1 + refs2) → ( 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(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st) ↔ r -- sk ∈ st, 0 = i0, (~1) = i2, r = r1, 0 ∈ (i1 ' + ints1) .butlast, r ∈ refs2

User applied rule apply induction

And obtained the following new goals

Proof step 14:
⊦ # refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)) ∧ lastn(i→n(countrefs(i1 ' + ints1)), r1 ' + refs2) = r0 ' + refs1 ∧ 0 ≤ i2 → ¬ # refs2 ≤ # refs1
Proof step 23:
¬ r ∈ r0 ' + refs1 ∧ # (r0 ' + refs1) = i→n(countrefs(i1 ' + (i1 ' + ints1) .rest)) ∧ i1 ' + ints1 ≠ [] ∧ 0 ≤ i1 ∧ posints(i1 ' + ints1) ∧ ¬ 0 ∈ (i1 ' + ints1) .butlast ∧ i1 ≤ (i1 ' + ints1) .first → (r -- sk ∈ addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st) ↔ r -- sk ∈ st), # refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r1 ' + refs2) = r0 ' + refs1, r -- sk ∈ addmultiarrays (r1 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), 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), posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, sk, ints, i, r, refs0. # refs0 < # (r1 + refs2) → ( 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(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st) ↔ r -- sk ∈ st, 0 = i0, (~1) = i2, r = r1, 0 ∈ (i1 ' + ints1) .butlast, r ∈ refs2

Rule argument

[substlist-arg: [st, ty, acc, sk, ints, i, r, refs0] → [st, ty, [], sk, i1 ' + ints1, i1, r, r0 ' + refs1] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m0 * m * n = (m0 * m) * n], [CSIMPSEQ : ⊦ m * n = n * m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r + refs = r ' + refs], [CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1])], [EXTRATERMS : (st, ty, [], sk, i1 ' + ints1, i1, r, r0 ' + refs1)]]