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Proof step 3 from proof from-empty (previous step)

⊦ #(⌜buf⌝ from # buf) = # [] ∧ (∀ n. ⌜n⌝ < #(⌜buf⌝ from # buf) → (⌜buf⌝ from # buf)[n] = ([])[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ¬ # σ0 < ⌜n⌝ ∧ finp(σ0) → #(σ0 from n) = ⌜(# σ0).n - n⌝
Proof step 5:
⊦ # ⌜buf⌝ = ⌜# buf⌝
Proof step 6:
⊦ finp(⌜buf⌝ )
Proof step 7:
⊦ # ⌜buf⌝ = ⌜# buf⌝

Rule argument

spec: stream, inst: buffer-inf name: fin-size-from2 seq: ⊦ ¬ # σ0 < ⌜n⌝ ∧ finp(σ0) → #(σ0 from n) = ⌜(# σ0).n - n⌝ subst: [n, σ0] → [# buf, ⌜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 ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ m; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ m - m = 0; (s)% ⊦ # [] = ⌜0⌝ ; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s)% ⊦ ¬ m < 0; (s)% ⊦ ¬ n < n; (s) Used terms: # buf, ⌜buf⌝