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Proof step 198 from proof addarray-put-01 (previous step)

0 ≤ i0 + j, 0 ≤ i0, 0 ≤ j ⊦ rk = r -- j ', ( rk ∈ addarray(r, ty, i0 + j + 1, st) ↔ rk ∈ addarray(r, ty, ((j = 0 ⊃ []; mklist(initial_value(ty), i→n(j)))) + val ' + mklist(initial_value(ty), i→n(i0)), st)) ∧ (addarray(r, ty, i0 + j + 1, st)[rk]) = (addarray(r, ty, ((j = 0 ⊃ []; mklist(initial_value(ty), i→n(j)))) + val ' + mklist(initial_value(ty), i→n(i0)), st)[rk])

User applied rule structural induction

And obtained the following new goals

Proof step 197:
0 ≤ i0 + j, 0 ≤ i0, 0 ≤ j ⊦ r0 -- sk = r -- j ', ( r0 -- sk ∈ addarray(r, ty, i0 + j + 1, st) ↔ r0 -- sk ∈ addarray(r, ty, ((j = 0 ⊃ []; mklist(initial_value(ty), i→n(j)))) + val ' + mklist(initial_value(ty), i→n(i0)), st)) ∧ (addarray(r, ty, i0 + j + 1, st)[r0 -- sk]) = (addarray(r, ty, ((j = 0 ⊃ []; mklist(initial_value(ty), i→n(j)))) + val ' + mklist(initial_value(ty), i→n(i0)), st)[r0 -- sk])

Rule argument

[VARWITHVARSEQSARG : rk, ()]

Additional information

[[CONCRETESIMPRULES : ([CSIMPGEN : refkey generated by --;])]]