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Proof step 10 from proof init-at-01 (previous step)

∀ j, i, r, st. j ≤ i ∧ i < j + n→i(# vals0) → (addarray(r, vals0, j, st)[r -- i ']) = vals0[i→n(i - j)] ⊦ j ≤ i ∧ i < j + n→i(# (val + vals0)) → (addarray(r, val + vals0, j, st)[r -- i ']) = (val + vals0)[i→n(i - j)]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 2:
⊦ addarray(r, val ' + vals, i, st) = (addarray(r, vals, i + 1, st)[r -- i ', val])
Proof step 9:
i ≤ n→i(# vals0) + j, j ≤ i, ∀ j0, i0, r0, st0. ¬ i0 < j0 ∧ ¬ j0 + n→i(# vals0) ≤ i0 → (addarray(r0, vals0, j0, st0)[r0 -- i0 ']) = vals0[i→n(i0 - j0)] ⊦ ((i = j ⊃ val; addarray(r, vals0, j + 1, st)[r -- i '])) = (val ' + vals0)[i→n(i - j)]

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ 0 ≤ n→i(n)], [CSIMPSEQ : ⊦ i ≤ i + j ↔ 0 ≤ j], [CSIMPFORWARD : ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i < j + 1 ↔ i ≤ j], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ n→i(m + n) = n→i(m) + n→i(n)], [CSIMPSEQ : ⊦ # val ' = 0 +1], [CSIMPSEQ : ⊦ # (vals + vals0) = # vals + # vals0], [CSIMPSEQ : ⊦ i ' = i0 ' ↔ i = i0], [CSIMPSEQ : ⊦ r -- sk = r0 -- sk0 ↔ ¬ ¬ (r = r0 ∧ sk = sk0)], [CSIMPSEQ : ⊦ (st[rk0, val][rk]) = ((rk = rk0 ⊃ val; st[rk]))], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1], [CSIMPSEQ : ⊦ val + vals = val ' + vals])]]