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Proof step 15 from proof Comp-get-01 (previous step)

0 < i0, ∀ i. ¬ i0 < i ∧ ¬ i < 0 → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) '], ∀ j0, st, r1, i1, r2, i2. i→n(j0) < i→n(i0 + 1) → ((∀ i. ¬ i < 0 ∧ ¬ j0 ≤ i → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) ']) → getarray(r1, i1, j0, st) = getarray(r2, i2, j0, st)) ⊦ getarray(r1, i1, i0, st) + st[r1 -- (i1 + i0) '] ' = getarray(r2, i2, i0, st) + st[r2 -- (i2 + i0) '] '

User applied rule apply induction

And obtained the following new goals

Proof step 12:
0 < i0, ∀ i. ¬ i0 < i ∧ ¬ i < 0 → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) '], ∀ j0, st, r1, i1, r2, i2. i→n(j0) < i→n(i0 + 1) → ((∀ i. ¬ i < 0 ∧ ¬ j0 ≤ i → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) ']) → getarray(r1, i1, j0, st) = getarray(r2, i2, j0, st)) ⊦ ∀ i. ¬ i < 0 ∧ ¬ i0 ≤ i → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) '], getarray(r1, i1, i0, st) + st[r1 -- (i1 + i0) '] ' = getarray(r2, i2, i0, st) + st[r2 -- (i2 + i0) '] '
Proof step 14:
getarray(r1, i1, i0, st) = getarray(r2, i2, i0, st), 0 < i0, ∀ i. ¬ i0 < i ∧ ¬ i < 0 → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) '], ∀ j0, st, r1, i1, r2, i2. i→n(j0) < i→n(i0 + 1) → ((∀ i. ¬ i < 0 ∧ ¬ j0 ≤ i → st[r1 -- (i1 + i) '] = st[r2 -- (i2 + i) ']) → getarray(r1, i1, j0, st) = getarray(r2, i2, j0, st)) ⊦ getarray(r1, i1, i0, st) + st[r1 -- (i1 + i0) '] ' = getarray(r2, i2, i0, st) + st[r2 -- (i2 + i0) '] '

Rule argument

[substlist-arg: [j0, st, r1, i1, r2, i2] → [i0, st, r1, i1, r2, i2]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j)], [CSIMPSEQ : ⊦ 0 < j → abs(j) = j], [CSIMPSEQ : ⊦ ¬ i < i], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (i0, st, r1, i1, r2, i2)]]