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Proof step 15 from proof from-to-rest (previous step)

# buf = n + m ⊦ ⌜m⌝ = #(⌜buf⌝ from n), #(⌜buf⌝ from n) < ⌜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] → [n, ⌜buf⌝ ] flags: <all paths>

Additional information

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