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Proof step 17 from proof multi-fieldin (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, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, r, ints, i, fs, refs0. # refs0 < # (r1 + refs2) → ( true → ( fs ' ≠ _type ∧ fs ' ≠ _length ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- fs ' ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- fs ' ∈ st))) ⊦ r -- fs ' ∈ 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 -- fs ' ∈ st, 0 = i0, (~1) = i2, fs ' = _length, fs ' = _type, 0 ∈ (i1 ' + ints1) .butlast

User applied rule apply induction

And obtained the following new goals

Proof step 11:
# refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r1 ' + refs2) = r0 ' + refs1, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, r, ints, i, fs, refs0. # refs0 < # (r1 + refs2) → ( true → ( fs ' ≠ _type ∧ fs ' ≠ _length ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- fs ' ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- fs ' ∈ st))) ⊦ # (r1 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)) < # (r1 + refs2), r -- fs ' ∈ 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 -- fs ' ∈ st, 0 = i0, (~1) = i2, fs ' = _length, fs ' = _type, 0 ∈ (i1 ' + ints1) .butlast
Proof step 16:
fs ' ≠ _type ∧ fs ' ≠ _length ∧ # (r1 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)) = 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 → ( r -- fs ' ∈ 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 -- fs ' ∈ addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, 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, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, r, ints, i, fs, refs0. # refs0 < # (r1 + refs2) → ( true → ( fs ' ≠ _type ∧ fs ' ≠ _length ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- fs ' ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- fs ' ∈ st))) ⊦ r -- fs ' ∈ 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 -- fs ' ∈ st, 0 = i0, (~1) = i2, fs ' = _length, fs ' = _type, 0 ∈ (i1 ' + ints1) .butlast

Rule argument

[substlist-arg: [st, ty, acc, r, ints, i, fs, refs0] → [addmultiarrays (r0 ' + refs1, i1, [], ty, i1 ' + ints1, st), ty, r0 ' + acc, r, i0 ' + i1 ' + ints1, i2, fs, r1 ' + firstn (i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (addmultiarrays (r0 ' + refs1, i1, [], ty, i1 ' + ints1, st), ty, r0 ' + acc, r, i0 ' + i1 ' + ints1, i2, fs, r1 ' + firstn (i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2))]]