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Proof step 57 from proof push-corr (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ m < n → m % n = m

Rule argument

[x0lemmaarg: spec-name: nat-div instance-name: lemma-name: mod-eq antecedent: succedent: m < n → m % n = m 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)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m < n → ¬ n < m], [CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPFORWARD : ⊦ m < n ∧ n ≤ n0 → m < n0], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m + n = m + n0 ↔ n = n0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPFORWARD : ⊦ m < n ∧ m = n0 → n0 < n], [CSIMPFORWARD : ⊦ m < n ∧ n = n0 → m < n0], [CSIMPSEQ : ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j)], [CSIMPSEQ : ⊦ 0 < j → abs(j) = j], [CSIMPSEQ : ⊦ ¬ i < i])], [EXTRATERMS : (# ar, i→n(i0) + n)]]