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Proof step 48 from proof push-corr (previous step)

i→n(i0) + i→n(i1) = (i→n(i0) + n) % # ar + # ar, 0 ≤ (i0 + i1) % n→i(# ar), 0 ≤ i0, i1 ≤ n→i(# ar), i0 < n→i(# ar), inv(mk(ar, n→i(# ar), i0, i1)), i1 < n→i(# ar), inv(mk(ar[(i→n(i0) + n) % # ar, a], n→i(# ar), i0, i1 + 1)), 0 ≤ i0 + i1, n < i→n(i1), 0 < i1, (i→n(i0) + n) % # ar < # ar ⊦ ar[(i→n(i0) + n) % # ar] = a, # ar = 1, i1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
n ≠ 0 ⊦ n ≤ m → m % n = (m - n) % n
Proof step 30:
¬ (# ar ≠ 0 ∧ # ar ≤ i→n(i0) + n), i→n(i0) + i→n(i1) = (i→n(i0) + n) % # ar + # ar, 0 ≤ (i0 + i1) % n→i(# ar), 0 ≤ i0, i1 ≤ n→i(# ar), i0 < n→i(# ar), inv(mk(ar, n→i(# ar), i0, i1)), i1 < n→i(# ar), inv(mk(ar[(i→n(i0) + n) % # ar, a], n→i(# ar), i0, i1 + 1)), 0 ≤ i0 + i1, n < i→n(i1), 0 < i1, (i→n(i0) + n) % # ar < # ar ⊦ ar[(i→n(i0) + n) % # ar] = a, # ar = 1, i1 = 0
Proof step 47:
# ar ≠ 0 ∧ # ar ≤ i→n(i0) + n, i→n(i0) + i→n(i1) = (i→n(i0) + n - # ar) % # ar + # ar, 0 ≤ (i0 + i1) % n→i(# ar), 0 ≤ i0, i1 ≤ n→i(# ar), i0 < n→i(# ar), inv(mk(ar, n→i(# ar), i0, i1)), i1 < n→i(# ar), inv(mk(ar[(i→n(i0) + n - # ar) % # ar, a], n→i(# ar), i0, i1 + 1)), 0 ≤ i0 + i1, n < i→n(i1), 0 < i1, (i→n(i0) + n - # ar) % # ar < # ar ⊦ ar[(i→n(i0) + n - # ar) % # ar] = a, # ar = 1, i1 = 0

Rule argument

[x0lemmaarg: spec-name: nat-div instance-name: lemma-name: mod-rec antecedent: n ≠ 0 succedent: n ≤ m → m % n = (m - n) % n substlist: [n, m] → [# ar, i→n(i0) + n] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 0, 1, 1, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (# ar, i→n(i0) + n)]]