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Proof step 25 from proof getarray-offset (previous step)

j0 ≤ j, 1 < j, 0 < j, 0 ≤ j0, 0 ≤ j, 1 + j0 < j, ∀ j1, j0, r, i, st. i→n(j1) < i→n(j) → (j0 ≤ j1 ∧ 0 ≤ j0 → getarray(r, i, j1, st) = getarray(r, i, j0, st) + getarray(r, i + j0, j1 - j0, st)) ⊦ getarray(r, i, j - 1, st) = getarray(r, i, j0, st) + getarray(r, i + j0, j - j0 + 1, st)

User applied rule apply induction

And obtained the following new goals

Proof step 22:
⊦ getarray(r, i, j, st) = [] ↔ j ≤ 0
Proof step 23:
⊦ j ≤ 0 → getarray(r, i, j, st) = []
Proof step 24:
⊦ j ≤ 0 → getarray(r, i, j, st) = []

Rule argument

[substlist-arg: [j1, j0, r, i, st] → [j - 1, j0, r, i, st]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i - j - i0 = i - j + i0], [CSIMPSEQ : ⊦ i + j < j ↔ i < 0], [CSIMPSEQ : ⊦ vals + [] = vals], [CSIMPSEQ : ⊦ i - j + i = ~ j], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ i + 1 ≤ j ↔ i < j], [CSIMPSEQ : ⊦ i ≤ j - i0 ↔ i + i0 ≤ j], [CSIMPSEQ : ⊦ 1 < i → i→n(i - 1) = i→n(i) -1], [CSIMPSEQ : ⊦ n -1 = n - 1], [CSIMPSEQ : ⊦ i < 0 ∧ 0 ≤ j → ¬ j < i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + 0 = i], [CSIMPSEQ : ⊦ i - j ≤ i0 ↔ i ≤ j + i0], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1], [CSIMPSEQ : ⊦ [] + vals = vals], [CSIMPSEQ : ⊦ ~ (i - j) = j - i], [CSIMPSEQ : ⊦ 0 - i = ~ i], [CSIMPSEQ : i < 0 ⊦ ((* i)) + j = j - ~ i], [CSIMPSEQ : ⊦ i ≠ j → (i ≤ j ↔ i < j)], [CSIMPSEQ : ⊦ i ≤ j → (j ≤ i ↔ i = j)], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ 1 ≤ i ↔ 0 < i], [CSIMPSEQ : 0 ≤ i ⊦ ((* m)) ≤ i→n(i) ↔ n→i(m) ≤ i], [CSIMPSEQ : ⊦ i < j → ¬ j < i], [CSIMPSEQ : n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j)], [CSIMPSEQ : ⊦ 1 < i → abs(i) = i], [CSIMPSEQ : ⊦ ¬ i < i], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (j - 1, j0, r, i, st)]]