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Proof step 13 from proof copy-stream-append (previous step)

# buf1 = n2 + n1 + m, ⌜# buf1⌝ = ⌜n1⌝ + #((⌜buf0⌝ from n0) to m) + #(⌜buf1⌝ from n1 + m), n < # buf1, n0 + m ≤ # buf0, n1 + m ≤ # buf1 ⊦ (n < n1 ∨ n1 + m ≤ n ⊃ buf1[n];buf0[n + n0 - n1]) = ((⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m))[n]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 15:
# buf1 = n2 + n1 + m, ⌜# buf1⌝ = ⌜n1⌝ + #((⌜buf0⌝ from n0) to m) + #(⌜buf1⌝ from n1 + m), n < # buf1, n1 + m ≤ # buf1, n0 + m ≤ # buf0 ⊦ (n < n1 ∨ n1 + m ≤ n ⊃ buf1[n];buf0[n + n0 - n1]) = ((⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m))[n], ⌜n⌝ < #(⌜buf1⌝ to n1)

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [n, ((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m), ⌜buf1⌝ to n1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + m0 < n → ¬ n + n0 < m; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)%m < n, ¬ # σ0 < ⌜n⌝ ⊦ (σ0 to n)[m] = σ0[m]; (s)% ⊦ n < # buf → ⌜buf⌝ [n] = buf[n]; (s) Used terms: n, ((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m), ⌜buf1⌝ to n1