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Proof step 17 from proof move-initvals (previous step)

0 ≤ i, i→n(i) < # vals1, ∀ val, st, ty, r, i, vals0. # vals0 < # vals1 + 1 → (true → (0 ≤ i ∧ i→n(i) < # vals0 → (addarray(r, ty, vals0, st)[r -- i ', val]) = addarray(r, ty, vals0[i→n(i), val], st))) ⊦ i = n→i(# vals1), (addarray(r, ty, vals1, st)[r -- i ', val][r -- n→i(# vals1) ', val0][r -- _length, intval(n→i(# vals1) + 1)]) = addarray(r, ty, vals1[i→n(i), val] + val0 ', st), vals1 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ vals ≠ [] → addarray(r, ty, vals, st) = (addarray(r, ty, vals .butlast, st)[r -- n→i((# vals) -1) ', vals .last][r -- _length, intval(n→i(# vals))])
Proof step 16:
vals1[i→n(i), val] + val0 ' ≠ [], 0 ≤ i, i→n(i) < # vals1, ∀ val, st, ty, r, i, vals0. # vals0 < # vals1 + 1 → (true → (0 ≤ i ∧ i→n(i) < # vals0 → (addarray(r, ty, vals0, st)[r -- i ', val]) = addarray(r, ty, vals0[i→n(i), val], st))) ⊦ i = n→i(# vals1), (addarray(r, ty, vals1, st)[r -- i ', val][r -- n→i(# vals1) ', val0][r -- _length, intval(n→i(# vals1) + 1)]) = (addarray(r, ty, (vals1[i→n(i), val] + val0 ') .butlast, st)[r -- n→i((# (vals1[i→n(i), val] + val0 ')) -1) ', (vals1[i→n(i), val] + val0 ') .last] [r -- _length, intval(n→i(# (vals1[i→n(i), val] + val0 '))) ] ), vals1 = []

Rule argument

[crewrite-arg: add-lastrec , F , [st, vals, ty, r] → [st, vals1[i→n(i), val] + val0 ', ty, r] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ vals + vals0 ≠ [] ↔ vals = [] ∧ vals0 = []], [CSIMPSEQ : ⊦ val ' ≠ []], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1])], [EXTRATERMS : (st, vals1[i→n(i), val] + val0 ', ty, r)]]