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Proof step 140 from proof infinite-append-butlast (previous step)

finp((σ from m0 + 1) to # y) ∧ #((σ from m0 + 1) to # y) = # ⌜y⌝ , lf(2 + n1) = a ' + x1, ii(n1 + 1) = m0 + 1, ii(0) = 0, ii(2 + n1) = 2 + m0 + # y, (σ from m0 + 1) to # y = ⌜y⌝ ∧ ⌜(σ from m0 + 1)[# y] '⌝ = ⌜a '⌝ , ⌜# y⌝ + ⌜m0⌝ + ⌜1⌝ < # σ, ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).rest) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ σ[m0 + # y + 1] = a, # σ < ⌜2⌝ + ⌜m0⌝ + ⌜# y⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 141:
(σ from m0 + 1) to # y = ⌜y⌝ , ii(2 + n1) = 2 + m0 + # y, ii(0) = 0, ii(n1 + 1) = m0 + 1, lf(2 + n1) = (σ from m0 + 1)[# y] ' + x1, ⌜# y⌝ + ⌜m0⌝ + ⌜1⌝ < # σ, (σ from m0 + 1)[# y] ∈ lf(2 + n1), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).rest) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n ≠ 0 → ii(n) ≠ 0 ⊦ σ[m0 + # y + 1] = (σ from m0 + 1)[# y], # σ < ⌜2⌝ + ⌜m0⌝ + ⌜# y⌝

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list%in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%fin-to: ⊦ finp(σ to m); from specification stream%c-04: ⊦ a ' = b ' ↔ a = b; from specification list%inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream