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Proof step 24 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, #(⌜buf0⌝ from n0) < ⌜m⌝ , n0 + m ≤ # buf0, n1 + m ≤ # buf1 ⊦ (m ≤ n3 ⊃ buf1[n3 + n1];buf0[n3 + n0]) = (⌜buf1⌝ from n1 + m)[n3 - (#((⌜buf0⌝ from n0) to m)).n], ⌜n3⌝ < #((⌜buf0⌝ from n0) to m)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, inst: buffer-inf name: fin-size-from seq: ⊦ ⌜n⌝ < # σ0 ∧ finp(σ0) → #(σ0 from n) = ⌜(# σ0).n - n⌝ subst: [n, σ0] → [n0, ⌜buf0⌝ ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜n⌝ < # σ0 ∧ finp(σ0) → #(σ0 from n) = ⌜(# σ0).n - n⌝ ; (s)% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m < n → ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf% ⊦ m < n → m ≤ n; (s)% ⊦ finp(⌜buf⌝ ); (ss)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ ¬ n∞ < ⌜0⌝ ; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ n + 0 = n; (s)% ⊦ n∞ + ⌜0⌝ = n∞; (s)%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf% ⊦ m - 0 = m; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ 0 ≤ m; (s)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s) Used terms: n0, ⌜buf0⌝