Theorem: List.IsZeckendorfRep.sum_fib_lt
File: Mathlib/Data/Nat/Fib/Zeckendorf.lean

Proof steps:
state_before: n a : ℕ
l : List ℕ
hl : (a :: l).IsZeckendorfRep
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
⊢ (map fib (a :: l)).sum < fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
hl : (∀ a' ∈ l ++ [0], a' + 2 ≤ a) ∧ Pairwise (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ (map fib (a :: l)).sum < fib n
tactic: simp only [IsZeckendorfRep, cons_append, chain'_iff_pairwise, pairwise_cons] at hl
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
hl : (∀ a' ∈ l ++ [0], a' + 2 ≤ a) ∧ Pairwise (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ (map fib (a :: l)).sum < fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
hl : (∀ a' ∈ l ++ [0], a' + 2 ≤ a) ∧ Pairwise (fun a b => b + 2 ≤ a) (l ++ [0])
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
⊢ (map fib (a :: l)).sum < fib n
tactic: have : ∀ b, b ∈ head? (l ++ [0]) → b < a - 1 :=
      fun b hb ↦ lt_tsub_iff_right.2 <| hl.1 _ <| mem_of_mem_head? hb
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
hl : (∀ a' ∈ l ++ [0], a' + 2 ≤ a) ∧ Pairwise (fun a b => b + 2 ≤ a) (l ++ [0])
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
⊢ (map fib (a :: l)).sum < fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ (map fib (a :: l)).sum < fib n
tactic: simp only [mem_append, mem_singleton, ← chain'_iff_pairwise, or_imp, forall_and, forall_eq,
      zero_add] at hl
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ (map fib (a :: l)).sum < fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib a + (map fib l).sum < fib n
tactic: simp only [map, List.sum_cons]
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib a + (map fib l).sum < fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib a + fib (a - 1) ≤ fib n
tactic: refine (add_lt_add_left (sum_fib_lt hl.2 this) _).trans_le ?_
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib a + fib (a - 1) ≤ fib n
state_after: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib (a + 1) ≤ fib n
tactic: rw [add_comm, ← fib_add_one (hl.1.2.trans_lt' zero_lt_two).ne']
state_before: n a : ℕ
l : List ℕ
hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n
this : ∀ b ∈ (l ++ [0]).head?, b < a - 1
hl : ((∀ x ∈ l, x + 2 ≤ a) ∧ 2 ≤ a) ∧ Chain' (fun a b => b + 2 ≤ a) (l ++ [0])
⊢ fib (a + 1) ≤ fib n
state_after: no goals
tactic: exact fib_mono (hn _ rfl)