Theorem: Cycle.chain_of_pairwise
File: Mathlib/Data/List/Cycle.lean

Proof steps:
state_before: α : Type u_1
r : α → α → Prop
s : Cycle α
⊢ (∀ a ∈ s, ∀ b ∈ s, r a b) → Chain r s
state_after: case H0
α : Type u_1
r : α → α → Prop
s : Cycle α
⊢ (∀ a ∈ nil, ∀ b ∈ nil, r a b) → Chain r nil

case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
⊢ (∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b) → Chain r ↑(a :: l)
tactic: induction' s with a l _
state_before: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
⊢ (∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b) → Chain r ↑(a :: l)
state_after: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
⊢ Chain r ↑(a :: l)
tactic: intro hs
state_before: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
⊢ Chain r ↑(a :: l)
state_after: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
⊢ Chain r ↑(a :: l)
tactic: have Ha : a ∈ (a :: l : Cycle α) := by simp
state_before: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
⊢ Chain r ↑(a :: l)
state_after: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ Chain r ↑(a :: l)
tactic: have Hl : ∀ {b} (_hb : b ∈ l), b ∈ (a :: l : Cycle α) := @fun b hb => by simp [hb]
state_before: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ Chain r ↑(a :: l)
state_after: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ List.Chain r a (l ++ [a])
tactic: rw [Cycle.chain_coe_cons]
state_before: case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ List.Chain r a (l ++ [a])
state_after: case HI.p
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ List.Pairwise r (a :: (l ++ [a]))
tactic: apply Pairwise.chain
state_before: case HI.p
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ List.Pairwise r (a :: (l ++ [a]))
state_after: case HI.p
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ (∀ a' ∈ l ++ [a], r a a') ∧ List.Pairwise r (l ++ [a])
tactic: rw [pairwise_cons]
state_before: case HI.p
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
⊢ (∀ a' ∈ l ++ [a], r a a') ∧ List.Pairwise r (l ++ [a])
state_after: case HI.p.refine_1
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
b : α
hb : b ∈ l ++ [a]
⊢ r a b

case HI.p.refine_2
α : Type u_1
r : α → α → Prop
s : Cycle α
a : α
l : List α
a✝ : (∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b) → Chain r ↑l
hs : ∀ a_1 ∈ ↑(a :: l), ∀ b ∈ ↑(a :: l), r a_1 b
Ha : a ∈ ↑(a :: l)
Hl : ∀ {b : α}, b ∈ l → b ∈ ↑(a :: l)
b : α
hb : b ∈ l
c : α
hc : c ∈ [a]
⊢ r b c
tactic: refine
    ⟨fun b hb => ?_,
      pairwise_append.2
        ⟨pairwise_of_forall_mem_list fun b hb c hc => hs b (Hl hb) c (Hl hc),
          pairwise_singleton r a, fun b hb c hc => ?_⟩⟩