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Proof step 14 from proof addarray-addaray (previous step)

0 ≤ i ⊦ (r -- fs ' ∈ addarray(r, ty, i, st) ↔ r -- fs ' ∈ addarray(r, ty, mklist(initial_value(ty), i→n(i)), st)) ∧ (addarray(r, ty, i, st)[r -- fs ']) = (addarray(r, ty, mklist(initial_value(ty), i→n(i)), st)[r -- fs ']), fs ' ≠ _type ∧ fs ' ≠ _length

User applied rule case distinction

And obtained the following new goals

Proof step 5:
⊦ r -- _type ∈ addarray(r, ty, i, st)
Proof step 6:
⊦ r -- _type ∈ addarray(r, ty, vals, st)
Proof step 7:
⊦ (addarray(r, ty, i, st)[r -- _type]) = typeval(mkarraytype(ty))
Proof step 8:
⊦ (addarray(r, ty, vals, st)[r -- _type]) = typeval(mkarraytype(ty))
Proof step 9:
⊦ r -- _length ∈ addarray(r, ty, i, st)
Proof step 10:
⊦ r -- _length ∈ addarray(r0, ty, vals, st) ↔ r = r0 ∨ r -- _length ∈ st
Proof step 11:
⊦ r -- _length ∈ addarray(r, ty, vals, st)
Proof step 12:
⊦ (addarray(r, ty, i, st)[r -- _length]) = intval(i)
Proof step 13:
⊦ (addarray(r, ty, vals, st)[r -- _length]) = intval(n→i(# vals))

Rule argument

[CASEDARG : pos: right 2, ()]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ typeval(ty) = typeval(ty0) ↔ ty = ty0], [CSIMPSEQ : ⊦ # mklist(val, n) = n], [CSIMPSEQ : ⊦ 0 ≤ i → n→i(i→n(i)) = i], [CSIMPSEQ : ⊦ intval(i) = intval(i0) ↔ i = i0])]]