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Proof step 9 from proof from-to-rest (previous step)

# buf = n + m, #(⌜buf⌝ from n) < ⌜m⌝ ⊦ #((⌜buf⌝ from n) to m) = #(⌜buf⌝ from n) ∧ (⌜n0⌝ < #((⌜buf⌝ from n) to m) → ((⌜buf⌝ from n) to m)[n0] = (⌜buf⌝ from n)[n0])

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] → [n, ⌜buf⌝ ] flags: <all paths>

Additional information

Used simplifier rules: tostream-size: ⊦ # ⌜buf⌝ = ⌜# buf⌝ ;%finp-buf: ⊦ finp(⌜buf⌝ );%tostream-size: ⊦ # ⌜buf⌝ = ⌜# buf⌝ ;% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%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%stream-fin-forward: ⊦ finp(σ0) → (∃ dx. ⌜dx⌝ = σ0); (forward) from specification stream-basic[buffer-inf]% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ ¬ n∞ < ⌜0⌝ ; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ # σ0 = ⌜n⌝ → σ0 to n = σ0; (s)% ⊦ n + 0 = n; (s)% ⊦ # σ0 = ⌜0⌝ ↔ σ0 = []; (s) Used terms: n, ⌜buf⌝