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Proof step 56 from proof multg-at-ref (previous step)

i→n((i1 + 1) * (countrefs2(ints2) + 1)) ≤ 1 + # refs0, 0 ≤ i1, 0 < i1, ∀ sk, i, ints0, st, ty, r0, r, refs, ints1. ints#(ints1) < ints#((i1 + 1) + ints2) → (i→n(countrefs2(ints1)) ≤ # refs → r = r0 ∨ (addarraymultlist(r, ty, st, refs, ints1, ints0, i)[r0 -- sk]) = st[r0 -- sk] ∨ r0 ∈ refs) ⊦ r0 = r1, r = r0, ints2 = [], (addarraymultlist (r1, ty, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, i1 ' + ints2, ints0, i), restn(i→n(i1 * (countrefs2(ints2) + 1)), refs0), ints2, ints0 .rest, i) [r0 -- sk ] ) = st[r0 -- sk], r0 ∈ refs0

User applied rule apply induction

And obtained the following new goals

Proof step 26:
⊦ ints#(i ' + ints) = 1 + i→n(i) + ints#(ints)
Proof step 46:
i→n((i1 + 1) * (countrefs2(ints2) + 1)) ≤ 1 + # refs0, 0 ≤ i1, 0 < i1, ∀ sk, i, ints0, st, ty, r0, r, refs, ints1. ints#(ints1) < ints#((i1 + 1) + ints2) → (i→n(countrefs2(ints1)) ≤ # refs → r = r0 ∨ (addarraymultlist(r, ty, st, refs, ints1, ints0, i)[r0 -- sk]) = st[r0 -- sk] ∨ r0 ∈ refs) ⊦ i→n(countrefs2(ints2)) ≤ # restn(i→n(countrefs2(i1 ' + ints2)), refs0), r0 = r1, r = r0, ints2 = [], (addarraymultlist (r1, ty, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, i1 ' + ints2, ints0, i), restn(i→n(i1 * (countrefs2(ints2) + 1)), refs0), ints2, ints0 .rest, i) [r0 -- sk ] ) = st[r0 -- sk], r0 ∈ refs0
Proof step 55:
r1 = r0 ∨ (addarraymultlist (r1, ty, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, i1 ' + ints2, ints0, i), restn(i→n(countrefs2(i1 ' + ints2)), refs0), ints2, ints0 .rest, i) [r0 -- sk ] ) = (addarraymultlist (r, ty, st[r -- i1 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, i1 ' + ints2, ints0, i) [r0 -- sk ] ) ∨ r0 ∈ restn(i→n(countrefs2(i1 ' + ints2)), refs0), i→n((i1 + 1) * (countrefs2(ints2) + 1)) ≤ 1 + # refs0, 0 ≤ i1, 0 < i1, ∀ sk, i, ints0, st, ty, r0, r, refs, ints1. ints#(ints1) < ints#((i1 + 1) + ints2) → (i→n(countrefs2(ints1)) ≤ # refs → r = r0 ∨ (addarraymultlist(r, ty, st, refs, ints1, ints0, i)[r0 -- sk]) = st[r0 -- sk] ∨ r0 ∈ refs) ⊦ r0 = r1, r = r0, ints2 = [], (addarraymultlist (r1, ty, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)][r -- _type, typeval(mktype_from_dims(ty, (1 + n→i(# ints2)) + i))][r -- _length, intval(ints0 .first)], refs0, i1 ' + ints2, ints0, i), restn(i→n(i1 * (countrefs2(ints2) + 1)), refs0), ints2, ints0 .rest, i) [r0 -- sk ] ) = st[r0 -- sk], r0 ∈ refs0

Rule argument

[substlist-arg: [sk, i, ints0, st, ty, r0, r, refs, ints1] → [sk, i, ints0 .rest, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)] [r -- _type, typeval (mktype_from_dims(ty, (1 + n→i(# ints2)) + i)) ] [r -- _length, intval(ints0 .first) ], refs0, i1 ' + ints2, ints0, i), ty, r0, r1, restn(i→n(countrefs2(i1 ' + ints2)), refs0), ints2] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + ints = i ' + ints], [CSIMPSEQ : ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m + n ≤ m ↔ n = 0], [CSIMPSEQ : ⊦ m + n = 0 ↔ m = 0 ∧ n = 0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (sk, i, ints0 .rest, addarraymultlist (r, ty, st[r -- i1 ', refval(r1)] [r -- _type, typeval (mktype_from_dims(ty, (1 + n→i(# ints2)) + i)) ] [r -- _length, intval(ints0 .first) ], refs0, i1 ' + ints2, ints0, i), ty, r0, r1, restn(i→n(countrefs2(i1 ' + ints2)), refs0), ints2)]]