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Proof step 44 from proof multi-at-other (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
⊦ 0 ≤ i2 → i→n(countrefs(i2 ' + i1 ' + ints1)) = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + 1
Proof step 43:
# refs = 1 + i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), refs) = r0 ' + refs1, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, ¬ r ∈ firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs) ∧ # refs - i→n(countrefs(i1 ' + ints1)) = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + 1 → (addmultiarrays (firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs), 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)[r -- sk]), ∀ sk, st, ty, acc, ints, i, r, refs0. # refs0 < # refs → ( true → ( ¬ r ∈ refs0 ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (addmultiarrays(refs0, i, acc, ty, ints, st)[r -- sk]) = st[r -- sk])) ⊦ (addmultiarrays (firstn(# refs - i→n(countrefs(i1 ' + ints1)), refs), i2, r0 ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st)) [r -- sk ] ) = st[r -- sk], 0 = i0, (~1) = i2, r ∈ refs, 0 ∈ (i1 ' + ints1) .butlast

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i < j → ¬ j < i], [CSIMPSEQ : ⊦ (i ' + ints) .first = i], [CSIMPSEQ : ⊦ i ≠ i0 → (i ∈ i0 ' + ints ↔ i ∈ ints)], [CSIMPSEQ : ⊦ (ints + i ' + ints0) .butlast = ints + (i ' + ints0) .butlast], [CSIMPSEQ : ⊦ 0 ≤ i → ( posints(i ' + ints) ↔ posints(ints))], [CSIMPSEQ : ⊦ 0 < i → ( posints(i ' + ints) ↔ posints(ints))], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i ' ≠ []], [CSIMPSEQ : ⊦ ¬ ints + ints0 ≠ [] ↔ ints = [] ∧ ints0 = []], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ (i ' + ints) .rest = ints], [CSIMPSEQ : ⊦ # firstn(n, refs) = n])]]