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Proof step 27 from proof putarray-addarray-01 (previous step)

0 ≤ i, 0 ≤ i0 + n→i(# vals), 0 ≤ i0 + i + n→i(# vals), 0 < i0, i1 < i ∨ n→i(# vals) + i ≤ i1 ⊦ (addarray(r, ty, i0 + i + n→i(# vals), st)[r -- i1 ']) = (addarray(r, ty, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), st)[r -- i1 ']), ¬ i1 < i ∧ ¬ n→i(# vals) + i ≤ i1, ¬ i1 < 0 ∧ ¬ i + n→i(# vals) + i0 ≤ i1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ i < 0 ∨ n→i(# vals) ≤ i → (addarray(r, ty, vals, st)[r -- i ']) = st[r -- i ']
Proof step 19:
⊦ 0 ≤ j ∧ j < i → (addarray(r, ty, i, st)[r -- j ']) = initial_value(ty)
Proof step 26:
i1 < 0 ∨ n→i(# (mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)))) ≤ i1, 0 ≤ i, 0 ≤ i0 + n→i(# vals), 0 ≤ i0 + i + n→i(# vals), 0 < i0, i1 < i ∨ n→i(# vals) + i ≤ i1 ⊦ (addarray(r, ty, i0 + i + n→i(# vals), st)[r -- i1 ']) = st[r -- i1 '], ¬ i1 < i ∧ ¬ n→i(# vals) + i ≤ i1, ¬ i1 < 0 ∧ ¬ i + n→i(# vals) + i0 ≤ i1

Rule argument

[crewrite-arg: add-at-other-index , F , [i, st, vals, ty, r] → [i1, st, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), ty, r] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ # (vals + vals0) = # vals + # vals0], [CSIMPSEQ : ⊦ # mklist(val, n) = n], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1], [CSIMPSEQ : ⊦ n→i(m + n) = n→i(m) + n→i(n)], [CSIMPSEQ : ⊦ 0 ≤ i → n→i(i→n(i)) = i], [CSIMPSEQ : ⊦ 0 < i → n→i(i→n(i)) = i], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ i < j → (¬ ¬ j ≤ i ↔ false)])], [EXTRATERMS : (i1, st, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), ty, r)]]