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Proof step 9 from proof putarray-add-same-02 (previous step)

∀ st, ty, r, vals0. # vals0 < # (vals1 + val ') → (true → putArray(vals0, r, 0, addarray(r, initial_value(ty), n→i(# vals0), st)) = addarray(r, vals0, 0, st)) ⊦ putArray(vals1, r, 0, addarray(r, initial_value(ty), n→i(# vals1), st)[r -- n→i(# vals1) ', val]) = (addarray(r, vals1, 0, st)[r -- n→i(# vals1) ', val])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ rk .ref ≠ r ∨ is_fskey(rk .key) ∨ rk .key .index < j ∨ n→i(# vals) + j ≤ rk .key .index → (putArray(vals, r, j, st)[rk, val]) = putArray(vals, r, j, st[rk, val])
Proof step 8:
(r -- n→i(# vals1) ') .ref ≠ r ∨ is_fskey((r -- n→i(# vals1) ') .key) ∨ (r -- n→i(# vals1) ') .key .index < 0 ∨ n→i(# vals1) + 0 ≤ (r -- n→i(# vals1) ') .key .index, ∀ st, ty, r, vals0. # vals0 < # (vals1 + val ') → (true → putArray(vals0, r, 0, addarray(r, initial_value(ty), n→i(# vals0), st)) = addarray(r, vals0, 0, st)) ⊦ (putArray(vals1, r, 0, addarray(r, initial_value(ty), n→i(# vals1), st))[r -- n→i(# vals1) ', val]) = (addarray(r, vals1, 0, st)[r -- n→i(# vals1) ', val])

Rule argument

[speclemmaarg: , rk .ref ≠ r ∨ is_fskey(rk .key) ∨ rk .key .index < j ∨ n→i(# vals) + j ≤ rk .key .index → (putArray(vals, r, j, st)[rk, val]) = putArray(vals, r, j, st[rk, val]) ,[val, rk, st, j, r, vals] → [val, r -- n→i(# vals1) ', addarray(r, initial_value(ty), n→i(# vals1), st), 0, r, vals1] ,T ,storearrays , ,putar-rew ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (r -- sk) .ref = r], [CSIMPSEQ : ⊦ (r -- sk) .key = sk], [CSIMPSEQ : ⊦ ¬ is_fskey(i ')], [CSIMPSEQ : ⊦ i ' .index = i], [CSIMPSEQ : ⊦ ¬ n→i(n) < 0], [CSIMPSEQ : ⊦ i + 0 = i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ n→i(m) ≤ n→i(n) ↔ m ≤ n], [CSIMPSEQ : ⊦ ¬ m ≤ n ↔ n < m], [CSIMPSEQ : ⊦ ¬ n < n])], [EXTRATERMS : (val, r -- n→i(# vals1) ', addarray(r, initial_value(ty), n→i(# vals1), st), 0, r, vals1)]]