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Proof step 18 from proof copy-stream-append (previous step)

# buf1 = n2 + n1 + m, ⌜# buf1⌝ = ⌜n1⌝ + #((⌜buf0⌝ from n0) to m) + #(⌜buf1⌝ from n1 + m), n3 < n2 + m, n1 < # buf1, n0 + m ≤ # buf0, n1 + m ≤ # buf1 ⊦ (m ≤ n3 ⊃ buf1[n3 + n1];buf0[n3 + n0]) = (((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m))[n3]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 20:
# buf1 = n2 + n1 + m, ⌜# buf1⌝ = ⌜n1⌝ + #((⌜buf0⌝ from n0) to m) + #(⌜buf1⌝ from n1 + m), n1 < # buf1, n3 < n2 + m, n1 + m ≤ # buf1, n0 + m ≤ # buf0 ⊦ (m ≤ n3 ⊃ buf1[n3 + n1];buf0[n3 + n0]) = (((⌜buf0⌝ from n0) to m) + (⌜buf1⌝ from n1 + m))[n3], ⌜n3⌝ < #((⌜buf0⌝ from n0) to m)
Proof step 34:
# buf1 = n2 + n1 + m, ⌜# buf1⌝ = ⌜n1⌝ + #((⌜buf0⌝ from n0) to m) + #(⌜buf1⌝ from n1 + m), n1 < # buf1, n3 < n2 + m, ⌜n3⌝ < #((⌜buf0⌝ from n0) to m), n1 + m ≤ # buf1, n0 + m ≤ # buf0 ⊦ (m ≤ n3 ⊃ buf1[n3 + n1];buf0[n3 + n0]) = ((⌜buf0⌝ from n0) to m)[n3]

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [n3, ⌜buf1⌝ from n1 + m, (⌜buf0⌝ from n0) to m] flags: <all paths>

Additional information

Used terms: n3, ⌜buf1⌝ from n1 + m, (⌜buf0⌝ from n0) to m