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Proof step 40 from proof initial-ex-abstract-initial (previous step)

# apebs + 1 = # pebs, pebs[# apebs] = mkpeb(buf, m, true), n < # apebs, ubi-io-initial(apebs), ubi-io-initial(pebs), ∀ n0. n0 < # apebs → abs-peb(pebs[n0], apebs[n0]) ⊦ n = # apebs, abs-peb(pebs[n], copy(apebs, 0, mkapebs(# pebs), 0, # apebs)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ m0 < # apebs → copy(apebs0, n, apebs, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ apebs[m0];apebs0[m0 + n - n0])
Proof step 42:
# apebs + 1 = # pebs, pebs[# apebs] = mkpeb(buf, m, true), n < # apebs, n < # mkapebs(# pebs), ubi-io-initial(pebs), ubi-io-initial(apebs), ∀ n0. n0 < # apebs → abs-peb(pebs[n0], apebs[n0]) ⊦ n = # apebs, abs-peb(pebs[n], (n < 0 ∨ 0 + # apebs ≤ n ⊃ mkapebs(# pebs)[n];apebs[n + 0 - 0]))

Rule argument

spec: array-copy, inst: apebs name: at-copy seq: ⊦ m0 < # apebs → copy(apebs0, n, apebs, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ apebs[m0];apebs0[m0 + n - n0]) subst: [m0, m, n0, apebs, n, apebs0] → [n, # apebs, 0, mkapebs(# pebs), 0, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # mkapebs(n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%# apebs = m, # apebs0 = m ⊦ copy(apebs, 0, apebs0, 0, m) = apebs; (s) Used terms: n, # apebs, 0, mkapebs(# pebs), 0, apebs