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Proof step 22 from proof get-alleq (previous step)

0 < i1, ∀ j0, r, i, st, r0, i0, st0. i→n(j0) < i→n(i1 + 1) → (true → (getarray(r, i, j0, st) = getarray(r0, i0, j0, st0) ↔ (∀ j1. 0 ≤ j1 ∧ j1 < j0 → st[r -- (i + j1) '] = st0[r0 -- (i0 + j1) ']))) ⊦ getarray(r, i, i1, st) = getarray(r0, i0, i1, st0) ∧ st[r -- (i + i1) '] = st0[r0 -- (i0 + i1) '] ↔ (∀ j. j ≤ i1 ∧ 0 ≤ j → st[r -- (i + j) '] = st0[r0 -- (i0 + j) '])

User applied rule apply induction

And obtained the following new goals

Proof step 21:
getarray(r, i, i1, st) = getarray(r0, i0, i1, st0) ↔ (∀ j1. 0 ≤ j1 ∧ j1 < i1 → st[r -- (i + j1) '] = st0[r0 -- (i0 + j1) ']), 0 < i1, ∀ j0, r, i, st, r0, i0, st0. i→n(j0) < i→n(i1 + 1) → (true → (getarray(r, i, j0, st) = getarray(r0, i0, j0, st0) ↔ (∀ j1. 0 ≤ j1 ∧ j1 < j0 → st[r -- (i + j1) '] = st0[r0 -- (i0 + j1) ']))) ⊦ getarray(r, i, i1, st) = getarray(r0, i0, i1, st0) ∧ st[r -- (i + i1) '] = st0[r0 -- (i0 + i1) '] ↔ (∀ j. j ≤ i1 ∧ 0 ≤ j → st[r -- (i + j) '] = st0[r0 -- (i0 + j) '])

Rule argument

[substlist-arg: [j0, r, i, st, r0, i0, st0] → [i1, r, i, st, r0, i0, st0] ]

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 : (i1, r, i, st, r0, i0, st0)]]