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Proof step 7 from proof at-copy (previous step)

m0 < # ar1, ∀ m0, ar1, ar0, n, n0. m0 < # ar1 → copy(ar0, n, ar1, n0, m1)[m0] = (m0 < n0 ∨ n0 + m1 ≤ m0 ⊃ ar1[m0];ar0[m0 + n - n0]) ⊦ copy(ar0, n, ar1, n0, m1)[n0 + m1, ar0[n + m1]][m0] = (m0 < n0 ∨ n0 + m1 < m0 ⊃ ar1[m0];ar0[m0 + n - n0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ m ≠ n → ar[n, a][m] = ar[m]
Proof step 9:
m0 < # ar1, ∀ m0, ar1, ar0, n, n0. m0 < # ar1 → copy(ar0, n, ar1, n0, m1)[m0] = (m0 < n0 ∨ n0 + m1 ≤ m0 ⊃ ar1[m0];ar0[m0 + n - n0]) ⊦ copy(ar0, n, ar1, n0, m1)[m0] = (m0 < n0 ∨ n0 + m1 < m0 ⊃ ar1[m0];ar0[m0 + n - n0]), m0 = n0 + m1

Rule argument

spec: array name: Put-other seq: ⊦ m ≠ n → ar[n, a][m] = ar[m] subst: [m, a, n, ar] → [m0, ar0[n + m1], n0 + m1, copy(ar0, n, ar1, n0, m1)] flags: <all paths>

Additional information

Used simplifier rules: size-copy: ⊦ # copy(ar, m, ar0, m0, n) = # ar0;% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%m < # ar ⊦ ar[m, a][m] = a; (s)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n < n; (s)% ⊦ m + n - m = n; (s) Used terms: m0, ar0[n + m1], n0 + m1, copy(ar0, n, ar1, n0, m1)