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Proof step 9 from proof multg-at-ref (previous step)

i→n(countrefs2(ints2)) ≤ # refs0, ∀ sk, i, ints0, st, ty, r0, r, refs, ints1. ints#(ints1) < ints#(1 + ints2) → (i→n(countrefs2(ints1)) ≤ # refs → r = r0 ∨ (addarraymultlist(r, ty, st, refs, ints1, ints0, i)[r0 -- sk]) = st[r0 -- sk] ∨ r0 ∈ refs) ⊦ r = r0, ints2 = [], (addarraymultlist (r1, ty, st[r -- 0 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, ints2, ints0 .rest, i) [r0 -- sk ] ) = st[r0 -- sk], r0 = r1, r0 ∈ refs0

User applied rule apply induction

And obtained the following new goals

Rule argument

[substlist-arg: [sk, i, ints0, st, ty, r0, r, refs, ints1] → [sk, i, ints0 .rest, st[r -- 0 ', refval(r1)] [r -- _type, typeval (mktype_from_dims(ty, (1 + n→i(# ints2)) + i)) ] [r -- _length, intval(ints0 .first) ], ty, r0, r1, refs0, ints2] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ≠ r0 → (st[r0 -- sk0, val][r -- sk]) = st[r -- sk]], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + ints = i ' + ints], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m + n ≤ m ↔ n = 0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (sk, i, ints0 .rest, st[r -- 0 ', refval(r1)] [r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i)) ] [r -- _length, intval(ints0 .first) ], ty, r0, r1, refs0, ints2)]]