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Proof step 50 from proof putarray-addarray-one (previous step)

r0 -- sk ∈ addarray(r0, ty, n→i(# vals) + 1, st) ↔ r0 -- sk ∈ addarray(r0, ty, refval(jvmref) ' + vals, st), is_fskey(sk), is_reftype(ty) ⊦ (addarray(r0, ty, n→i(# vals) + 1, st)[r0 -- sk]) = (addarray(r0, ty, refval(jvmref) ' + vals, st)[r0 -- sk])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ is_fskey(sk) ∧ sk ≠ _type ∧ sk ≠ _length → (addarray(r, ty, i, st)[r0 -- sk]) = st[r0 -- sk]
Proof step 39:
⊦ r -- _type ∈ addarray(r, ty, i, st)
Proof step 40:
⊦ r -- _type ∈ addarray(r, ty, vals, st)
Proof step 41:
⊦ (addarray(r, ty, i, st)[r -- _type]) = typeval(mkarraytype(ty))
Proof step 42:
⊦ (addarray(r, ty, vals, st)[r -- _type]) = typeval(mkarraytype(ty))
Proof step 43:
⊦ r -- _length ∈ addarray(r, ty, i, st)
Proof step 44:
⊦ r -- _length ∈ addarray(r, ty, vals, st)
Proof step 45:
⊦ (addarray(r, ty, i, st)[r -- _length]) = intval(i)
Proof step 46:
⊦ (addarray(r, ty, vals, st)[r -- _length]) = intval(n→i(# vals))
Proof step 49:
is_fskey(sk) ∧ sk ≠ _type ∧ sk ≠ _length, r0 -- sk ∈ addarray(r0, ty, n→i(# vals) + 1, st) ↔ r0 -- sk ∈ addarray(r0, ty, refval(jvmref) ' + vals, st), is_fskey(sk), is_reftype(ty) ⊦ st[r0 -- sk] = (addarray(r0, ty, refval(jvmref) ' + vals, st)[r0 -- sk])

Rule argument

[crewrite-arg: at-field-1dim , F , [sk, r0, st, i, ty, r] → [sk, r0, st, n→i(# vals) + 1, ty, r0] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ ¬ is_fskey(sk) ↔ is_indexkey(sk)], [CSIMPSEQ : ⊦ is_indexkey(sk) → ¬ is_fskey(sk)], [CSIMPSEQ : ⊦ typeval(ty) = typeval(ty0) ↔ ty = ty0], [CSIMPSEQ : ⊦ # (vals + vals0) = # vals + # vals0], [CSIMPSEQ : ⊦ # val ' = 0 +1], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1], [CSIMPSEQ : ⊦ n→i(m + n) = n→i(m) + n→i(n)], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ intval(i) = intval(i0) ↔ i = i0])], [EXTRATERMS : (sk, r0, st, n→i(# vals) + 1, ty, r0)]]