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Proof step 4 from proof contractinf-tl (previous step)

⊦ #(⌜nf(0) '⌝ + ⌜λ n. nf(n + 1)⌝ ) = # ⌜nf⌝ ∧ (∀ n. ⌜n⌝ < #(⌜nf(0) '⌝ + ⌜λ n. nf(n + 1)⌝ ) → (⌜nf(0) '⌝ + ⌜λ n. nf(n + 1)⌝ )[n] = ⌜nf⌝ [n])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1;%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ;%size-inf: ⊦ # ⌜nf⌝ = ∞;%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2;%sel-inf: ⊦ ⌜nf⌝ [n] = nf(n);% ⊦ ⌜m⌝ < ∞; (s)%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf% ⊦ ∞ + ⌜m⌝ = ∞; (s)% ⊦ # a ' = 0 +1; (s)