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Proof step 25 from proof multi-in-other (previous step)

¬ r ∈ r1 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2) ∧ # (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 -- sk ∈ 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 -- sk ∈ 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, sk, ints, i, r, refs0. # refs0 < # (r1 + refs2) → ( true → ( ¬ r ∈ refs0 ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- sk ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- sk ∈ st))) ⊦ r -- sk ∈ 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 -- sk ∈ st, 0 = i0, (~1) = i2, r = r1, 0 ∈ (i1 ' + ints1) .butlast, r ∈ refs2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
⊦ # refs = n + m ∧ ¬ r ∈ refs → ¬ r ∈ firstn(n, refs)
Proof step 13:
⊦ 0 ≤ i2 → i→n(countrefs(i2 ' + i1 ' + ints1)) = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + 1
Proof step 24:
# refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r1 ' + refs2) = r0 ' + refs1, r -- sk ∈ 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 -- sk ∈ addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st), posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ st, ty, acc, sk, ints, i, r, refs0. # refs0 < # (r1 + refs2) → ( true → ( ¬ r ∈ refs0 ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → (r -- sk ∈ addmultiarrays(refs0, i, acc, ty, ints, st) ↔ r -- sk ∈ st))) ⊦ r -- sk ∈ addmultiarrays(r0 ' + refs1, i1, [], ty, i1 ' + ints1, st) ↔ r -- sk ∈ st, 0 = i0, (~1) = i2, r = r1, 0 ∈ (i1 ' + ints1) .butlast, r ∈ refs2

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 : ⊦ posints(i ' + ints) ↔ 0 ≤ i ∧ 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 : ⊦ m + n = m + n0 ↔ n = n0], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ (i ' + ints) .rest = ints], [CSIMPSEQ : ⊦ # firstn(n, refs) = n], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r ≠ r0 → (r ∈ r0 ' + refs ↔ r ∈ refs)], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m0 * m * n = (m0 * m) * n], [CSIMPSEQ : ⊦ m * n = n * m])]]