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Proof step 34 from proof multi-isarrayref (previous step)

# refs1 = i→n(countrefs(i0 ' + ints1)) * i→n(i1) + i→n(countrefs(i0 ' + ints1)), lastn(i→n(countrefs(i0 ' + ints1)), r0 ' + refs1) = r1 ' + refs, posints(ints1), 0 ≤ i0, 0 ≤ n→i(# acc) + i1, 0 ≤ i1, i→n(countrefs(i0 ' + ints1)) ≤ 1 + # refs1, r ∈ refs1, ∀ st, ty, ints, i, acc, r, refs0. # refs0 < # (r0 + refs1) → ( true → ( r ∈ refs0 ∧ n→i(# acc) + i = ints .first ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → is_arrayref(r, (addmultiarrays(refs0, i, acc, ty, ints, st)[r -- _length]) .intval, addmultiarrays(refs0, i, acc, ty, ints, st)))) ⊦ r = r0, (~1) = n→i(# acc) + i1, (~1) = i1, 0 ∈ (i0 ' + ints1) .butlast, is_arrayref (r, (addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)[r -- _length]) .intval, addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)), r ∈ firstn(i→n(countrefs(i0 ' + ints1)) * i→n(i1), refs1)

User applied rule apply induction

And obtained the following new goals

Proof step 25:
⊦ # 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 33:
r ∈ r1 ' + refs ∧ n→i(# []) + i0 = (i0 ' + ints1) .first ∧ # (r1 ' + refs) = i→n(countrefs(i0 ' + (i0 ' + ints1) .rest)) ∧ i0 ' + ints1 ≠ [] ∧ 0 ≤ i0 ∧ posints(i0 ' + ints1) ∧ ¬ 0 ∈ (i0 ' + ints1) .butlast ∧ i0 ≤ (i0 ' + ints1) .first → is_arrayref (r, (addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)[r -- _length]) .intval, addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)), # refs1 = i→n(countrefs(i0 ' + ints1)) * i→n(i1) + i→n(countrefs(i0 ' + ints1)), lastn(i→n(countrefs(i0 ' + ints1)), r0 ' + refs1) = r1 ' + refs, posints(ints1), 0 ≤ i0, 0 ≤ n→i(# acc) + i1, 0 ≤ i1, i→n(countrefs(i0 ' + ints1)) ≤ 1 + # refs1, r ∈ refs1, ∀ st, ty, ints, i, acc, r, refs0. # refs0 < # (r0 + refs1) → ( true → ( r ∈ refs0 ∧ n→i(# acc) + i = ints .first ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → is_arrayref(r, (addmultiarrays(refs0, i, acc, ty, ints, st)[r -- _length]) .intval, addmultiarrays(refs0, i, acc, ty, ints, st)))) ⊦ r = r0, (~1) = n→i(# acc) + i1, (~1) = i1, 0 ∈ (i0 ' + ints1) .butlast, is_arrayref (r, (addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)[r -- _length]) .intval, addmultiarrays(r1 ' + refs, i0, [], ty, i0 ' + ints1, st)), r ∈ firstn(i→n(countrefs(i0 ' + ints1)) * i→n(i1), refs1)

Rule argument

[substlist-arg: [st, ty, ints, i, acc, r, refs0] → [st, ty, i0 ' + ints1, i0, [], r, r1 ' + refs] ]

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, i0 ' + ints1, i0, [], r, r1 ' + refs)]]