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Proof step 12 from proof multi-mode (previous step)

∀ i, ints0, refs, st, ty, r, ints1. ints#(ints1) < ints#(1 + ints2) → (true → (addarraymultlist(r, ty, st, refs, ints1, ints0, i)[_mode]) = st[_mode]) ⊦ (addarraymultlist (refs .first, ty, st[r -- 0 ', refval(refs .first)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs .rest, ints2, ints0 .rest, i) [_mode ] ) = st[_mode], ints2 = []

User applied rule apply induction

And obtained the following new goals

Rule argument

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

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ _mode ≠ r -- i '], [CSIMPSEQ : ⊦ r -- _type ≠ _mode], [CSIMPSEQ : ⊦ _mode ≠ r -- _length], [CSIMPSEQ : ⊦ (st[rk0, val][rk]) = ((rk = rk0 ⊃ val; st[rk]))], [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 : (i, ints0 .rest, refs .rest, st[r -- 0 ', refval(refs .first)] [r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i)) ] [r -- _length, intval(ints0 .first) ], ty, refs .first, ints2)]]