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Proof step 113 from proof putarray-addarray-01 (previous step)

0 ≤ i, 0 ≤ i0 + n→i(# vals), 0 ≤ i0 + i + n→i(# vals), 0 < i0 ⊦ ( rk ∈ putArray(vals, r, i, addarray(r, ty, i0 + i + n→i(# vals), st)) ↔ rk = r -- _length ∨ rk ∈ addarray(r, ty, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), st)) ∧ (putArray(vals, r, i, addarray(r, ty, i0 + i + n→i(# vals), st))[rk]) = (addarray(r, ty, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), st)[r -- _length, intval(i0 + i + n→i(# vals))] [rk ] )

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 2:
⊦ r -- _length ∈ addarray(r, ty, i, st)
Proof step 3:
⊦ (addarray(r, ty, i, st)[r -- _length]) = intval(i)
Proof step 112:
0 ≤ i, 0 ≤ i0 + n→i(# vals), 0 ≤ i0 + i + n→i(# vals), 0 < i0 ⊦ rk = r -- _length, ( rk ∈ putArray(vals, r, i, addarray(r, ty, i0 + i + n→i(# vals), st)) ↔ rk = r -- _length ∨ rk ∈ addarray(r, ty, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), st)) ∧ (putArray(vals, r, i, addarray(r, ty, i0 + i + n→i(# vals), st))[rk]) = (addarray(r, ty, mklist(initial_value(ty), i→n(i)) + vals + mklist(initial_value(ty), i→n(i0)), st)[r -- _length, intval(i0 + i + n→i(# vals))] [rk ] )

Rule argument

[fma-arg: rk = r -- _length]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ rk ∈ putArray(vals, r, i, st) ↔ rk ∈ st ∨ rk .ref = r ∧ is_indexkey(rk .key) ∧ i ≤ rk .key .index ∧ rk .key .index < i + n→i(# vals)], [CSIMPSEQ : ⊦ (putArray(vals, r, i, st) [r0 -- _length ] ) = st[r0 -- _length]], [CSIMPSEQ : ⊦ (st[rk, val][rk]) = val], [CSIMPSEQ : ⊦ intval(i) = intval(i0) ↔ i = i0])], [EXTRAFMAS : (rk = r -- _length)]]