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Proof step 15 from proof putarray-addarray (previous step)

0 ≤ i, 0 ≤ j, i ≤ (n→i(# vals1) + j) + 1, j ≤ i, (~1) ≤ n→i(# vals1) + j, ∀ st, val, r, j, i, vals0. # vals0 < # (vals1 + val0 ') → ( 0 ≤ i ∧ 0 ≤ j ∧ i ≤ j + n→i(# vals0) ∧ j ≤ i ∧ 0 ≤ j + n→i(# vals0) → putArray(vals0, r, j, addarray(r, val, i, st)) = addarray(r, mklist(val, i→n(j)) + vals0, 0, st)) ⊦ putArray(vals1, r, j, addarray(r, val, i, st)[r -- (j + n→i(# vals1)) ', val0]) = (addarray(r, mklist(val, i→n(j)) + vals1, 0, st)[r -- (n→i(# vals1) + j) ', val0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 ≤ j ∧ 0 ≤ i ∧ i + n→i(# vals) ≤ j → putArray(vals, r, i, st[r -- j ', val]) = (putArray(vals, r, i, st)[r -- j ', val])
Proof step 14:
0 ≤ j + n→i(# vals1) ∧ 0 ≤ j ∧ j + n→i(# vals1) ≤ j + n→i(# vals1), 0 ≤ i, 0 ≤ j, i ≤ (n→i(# vals1) + j) + 1, j ≤ i, (~1) ≤ n→i(# vals1) + j, ∀ st, val, r, j, i, vals0. # vals0 < # (vals1 + val0 ') → ( 0 ≤ i ∧ 0 ≤ j ∧ i ≤ j + n→i(# vals0) ∧ j ≤ i ∧ 0 ≤ j + n→i(# vals0) → putArray(vals0, r, j, addarray(r, val, i, st)) = addarray(r, mklist(val, i→n(j)) + vals0, 0, st)) ⊦ (putArray(vals1, r, j, addarray(r, val, i, st))[r -- (j + n→i(# vals1)) ', val0]) = (addarray(r, mklist(val, i→n(j)) + vals1, 0, st)[r -- (n→i(# vals1) + j) ', val0])

Rule argument

[speclemmaarg: , 0 ≤ j ∧ 0 ≤ i ∧ i + n→i(# vals) ≤ j → putArray(vals, r, i, st[r -- j ', val]) = (putArray(vals, r, i, st)[r -- j ', val]) ,[val, j, st, i, r, vals] → [val0, j + n→i(# vals1), addarray(r, val, i, st), j, r, vals1] ,F ,storearrays , ,putArray-put-switch ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ i + j ≤ i0 + j ↔ i ≤ i0], [CSIMPSEQ : ⊦ ¬ i < i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i < 0 → ((~1) ≤ i ↔ i = (~1))], [CSIMPFORWARD : ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1], [CSIMPFORWARD : ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j], [CSIMPFORWARD : ⊦ i < i0 ∧ i = i1 → i1 < i0], [CSIMPFORWARD : ⊦ i < j ∧ j ≤ j0 → i < j0], [CSIMPFORWARD : ⊦ j < i ∧ i ≤ 0 → j < 0], [CSIMPSEQ : i + 1 = i0 ⊦ ((* i)) < j → (j ≤ ((* i0)) ↔ j = i0)], [CSIMPSEQ : ⊦ (~1) < i ↔ 0 ≤ i], [CSIMPSEQ : ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j], [CSIMPSEQ : ⊦ i ≤ j → (j ≤ i ↔ i = j)], [CSIMPSEQ : ⊦ i + 0 = i], [CSIMPSEQ : i < 0 ⊦ n→i(n) ≠ ((* i))])], [EXTRATERMS : (val0, j + n→i(# vals1), addarray(r, val, i, st), j, r, vals1)]]