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Proof step 20 from proof multi-length-first (previous step)

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

User applied rule apply induction

And obtained the following new goals

Proof step 13:
lastn(i→n(countrefs(i0 ' + ints1)), r ' + refs) = r0 ' + refs1, # refs = i→n(countrefs(i0 ' + ints1)) * i→n(i1) + i→n(countrefs(i0 ' + ints1)), 1 + # refs - i→n(countrefs(i0 ' + ints1)) = m + 1, 0 ≤ i1, 0 ≤ n→i(# acc) + i1, 0 ≤ i0, posints(ints1), ∀ st, ty, r, acc, ints, i, refs0. # refs0 < # refs → ( true → ( # refs0 + 1 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first ∧ n→i(# acc) + i = ints .first → (addmultiarrays(r ' + refs0, i, acc, ty, ints, st)[r -- _length]) = intval(ints .first))) ⊦ # firstn(m, refs) < # refs, (addmultiarrays (r ' + firstn(m, refs), i1, r0 ' + acc, ty, (n→i(# acc) + i1 + 1) ' + i0 ' + ints1, addmultiarrays(r0 ' + refs1, i0, [], ty, i0 ' + ints1, st)) [r -- _length ] ) = intval(n→i(# acc) + i1 + 1), (~1) = i1, (~1) = n→i(# acc) + i1, 0 ∈ (i0 ' + ints1) .butlast
Proof step 19:
# firstn(m, refs) + 1 = i→n(countrefs(i1 ' + ((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) .rest)) ∧ (n→i(# acc) + i1 + 1) ' + i0 ' + ints1 ≠ [] ∧ 0 ≤ i1 ∧ posints((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) ∧ ¬ 0 ∈ ((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) .butlast ∧ i1 ≤ ((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) .first ∧ n→i(# (r0 ' + acc)) + i1 = ((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) .first → (addmultiarrays (r ' + firstn(m, refs), i1, r0 ' + acc, ty, (n→i(# acc) + i1 + 1) ' + i0 ' + ints1, addmultiarrays(r0 ' + refs1, i0, [], ty, i0 ' + ints1, st)) [r -- _length ] ) = intval(((n→i(# acc) + i1 + 1) ' + i0 ' + ints1) .first), lastn(i→n(countrefs(i0 ' + ints1)), r ' + refs) = r0 ' + refs1, # refs = i→n(countrefs(i0 ' + ints1)) * i→n(i1) + i→n(countrefs(i0 ' + ints1)), 1 + # refs - i→n(countrefs(i0 ' + ints1)) = m + 1, 0 ≤ i1, 0 ≤ n→i(# acc) + i1, 0 ≤ i0, posints(ints1), ∀ st, ty, r, acc, ints, i, refs0. # refs0 < # refs → ( true → ( # refs0 + 1 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first ∧ n→i(# acc) + i = ints .first → (addmultiarrays(r ' + refs0, i, acc, ty, ints, st)[r -- _length]) = intval(ints .first))) ⊦ (addmultiarrays (r ' + firstn(m, refs), i1, r0 ' + acc, ty, (n→i(# acc) + i1 + 1) ' + i0 ' + ints1, addmultiarrays(r0 ' + refs1, i0, [], ty, i0 ' + ints1, st)) [r -- _length ] ) = intval(n→i(# acc) + i1 + 1), (~1) = i1, (~1) = n→i(# acc) + i1, 0 ∈ (i0 ' + ints1) .butlast

Rule argument

[substlist-arg: [st, ty, r, acc, ints, i, refs0] → [addmultiarrays (r0 ' + refs1, i0, [], ty, i0 ' + ints1, st), ty, r, r0 ' + acc, (n→i(# acc) + i1 + 1) ' + i0 ' + ints1, i1, firstn(m, refs)] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (addmultiarrays (r0 ' + refs1, i0, [], ty, i0 ' + ints1, st), ty, r, r0 ' + acc, (n→i(# acc) + i1 + 1) ' + i0 ' + ints1, i1, firstn(m, refs))]]