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Proof step 37 from proof get-add-02 (previous step)

0 ≤ i, i + j ≤ n→i(# vals), 1 < j, 0 < j, ∀ ty, st, r, vals, i, j0. i→n(j0) < i→n(j) → ( i + j0 ≤ n→i(# vals) ∧ 0 ≤ i ∧ 0 ≤ j0 → getarray(r, i, j0, addarray(r, vals, 0, st[r -- _type, typeval(mkarraytype(ty))][r -- _length, intval(n→i(# vals))])) = sublist(i→n(i), i→n(j0), vals)) ⊦ 0 = j, getarray(r, i, j - 1, addarray(r, vals, 0, st[r -- _type, typeval(mkarraytype(ty))][r -- _length, intval(n→i(# vals))])) + (addarray(r, vals, 0, st)[r -- (i + j - 1) ']) ' = sublist(i→n(i), i→n(j), vals)

User applied rule apply induction

And obtained the following new goals

Proof step 36:
getarray(r, i, j - 1, addarray(r, vals, 0, st[r -- _type, typeval(mkarraytype(ty))][r -- _length, intval(n→i(# vals))])) = sublist(i→n(i), i→n(j - 1), vals), 0 ≤ i, i + j ≤ n→i(# vals), 1 < j, 0 < j, ∀ ty, st, r, vals, i, j0. i→n(j0) < i→n(j) → ( i + j0 ≤ n→i(# vals) ∧ 0 ≤ i ∧ 0 ≤ j0 → getarray(r, i, j0, addarray(r, vals, 0, st[r -- _type, typeval(mkarraytype(ty))][r -- _length, intval(n→i(# vals))])) = sublist(i→n(i), i→n(j0), vals)) ⊦ 0 = j, getarray(r, i, j - 1, addarray(r, vals, 0, st[r -- _type, typeval(mkarraytype(ty))][r -- _length, intval(n→i(# vals))])) + (addarray(r, vals, 0, st)[r -- (i + j - 1) ']) ' = sublist(i→n(i), i→n(j), vals)

Rule argument

[substlist-arg: [ty, st, r, vals, i, j0] → [ty, st, r, vals, i, j - 1] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ i < i], [CSIMPSEQ : ⊦ 1 < i → abs(i) = i], [CSIMPSEQ : ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j)], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0], [CSIMPSEQ : 0 ≤ i ⊦ ((* m)) ≤ i→n(i) ↔ n→i(m) ≤ i], [CSIMPSEQ : ⊦ n -1 = n - 1], [CSIMPSEQ : ⊦ 1 < i → i→n(i - 1) = i→n(i) -1], [CSIMPSEQ : ⊦ i + (j - i0) = i + j - i0], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i - j ≤ i0 ↔ i ≤ j + i0], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ i ≤ j - i0 ↔ i + i0 ≤ j], [CSIMPSEQ : ⊦ 1 ≤ i ↔ 0 < i], [CSIMPSEQ : ⊦ i ≠ i0 → (¬ i < i0 ↔ i0 < i)], [CSIMPSEQ : ⊦ i < j → ¬ j < i], [CSIMPFORWARD : ⊦ i < j ∧ j ≤ j0 → i < j0], [CSIMPSEQ : ⊦ n→i(m) + 1 < n→i(n) ↔ m + 1 < n], [CSIMPSEQ : ⊦ ¬ m + n < m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (ty, st, r, vals, i, j - 1)]]