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Proof step 9 from proof getarray-union-03 (previous step)

0 < j, 0 ≤ i, r ∈ st, j + i ≤ st[r -- _length] .intval, r -- _length ∈ st, r -- i ' ∈ st .keys, ∀ i0. ¬ i0 < i ∧ ¬ i + j ≤ i0 → r -- i0 ' ∈ st, ∀ st0, i, st, r, j0. i→n(j0) < i→n(j) → ( true → ( r -- _length ∈ st ∧ 0 ≤ i ∧ j0 ≤ st[r -- _length] .intval - i ∧ (∀ i1. i ≤ i1 ∧ i1 < i + j0 → r -- i1 ' ∈ st) → getarray(r, i, j0, st0 ∪ st) = getarray(r, i, j0, st))) ⊦ getarray(r, i + 1, j - 1, st0 ∪ st) = getarray(r, i + 1, j - 1, st)

User applied rule apply induction

And obtained the following new goals

Proof step 8:
r -- _length ∈ st ∧ 0 ≤ i + 1 ∧ j - 1 ≤ st[r -- _length] .intval - i + 1 ∧ (∀ i1. i + 1 ≤ i1 ∧ i1 < (i + 1) + (j - 1) → r -- i1 ' ∈ st) → getarray(r, i + 1, j - 1, st0 ∪ st) = getarray(r, i + 1, j - 1, st), 0 < j, 0 ≤ i, r ∈ st, j + i ≤ st[r -- _length] .intval, r -- _length ∈ st, r -- i ' ∈ st .keys, ∀ i0. ¬ i0 < i ∧ ¬ i + j ≤ i0 → r -- i0 ' ∈ st, ∀ st0, i, st, r, j0. i→n(j0) < i→n(j) → ( true → ( r -- _length ∈ st ∧ 0 ≤ i ∧ j0 ≤ st[r -- _length] .intval - i ∧ (∀ i1. i ≤ i1 ∧ i1 < i + j0 → r -- i1 ' ∈ st) → getarray(r, i, j0, st0 ∪ st) = getarray(r, i, j0, st))) ⊦ getarray(r, i + 1, j - 1, st0 ∪ st) = getarray(r, i + 1, j - 1, st)

Rule argument

[substlist-arg: [st0, i, st, r, j0] → [st0, i + 1, st, r, j - 1]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i < j → (¬ ¬ j ≤ i ↔ false)])], [EXTRATERMS : (st0, i + 1, st, r, j - 1)]]