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Proof step 19 from proof multi-length-first (previous step)

# 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
⊦ 0 ≤ i2 → i→n(countrefs(i2 ' + i1 ' + ints1)) = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + 1
Proof step 18:
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, m = i→n(countrefs(i0 ' + ints1)) * i→n(i1) ∧ posints((n→i(# acc) + i1 + 1) ' + i0 ' + ints1)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = i + i0 ↔ j = i0], [CSIMPSEQ : ⊦ n→i(m + n) = n→i(m) + n→i(n)], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ ¬ n→i(n) + 1 < 0], [CSIMPSEQ : ⊦ i + j < j ↔ i < 0], [CSIMPSEQ : ⊦ (i ' + ints) .first = i], [CSIMPSEQ : ⊦ ((* j)) = i0 + ((* j0)) ↔ j - j0 = i0], [CSIMPSEQ : ⊦ ¬ i ∈ ints → (i ∈ i0 ' + ints ↔ i = i0)], [CSIMPSEQ : ⊦ (ints + i ' + ints0) .butlast = ints + (i ' + ints0) .butlast], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i ' ≠ []], [CSIMPSEQ : ⊦ ¬ ints + ints0 ≠ [] ↔ ints = [] ∧ ints0 = []], [CSIMPSEQ : ⊦ m + n = m + n0 ↔ n = n0], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ (i ' + ints) .rest = ints], [CSIMPSEQ : ⊦ # firstn(n, refs) = n], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m * n = n * m], [CSIMPSEQ : ⊦ m0 * m * n = (m0 * m) * n], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0])]]