awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
hp : p.length (r := r) ≠ 0
hl : p.eraseLast (r := r).toList (r := r).length = p.toList (r := r).dropLast.length
⊢ p.eraseLast (r := r).toList (r := r) = p.toList (r := r).dropLast
have h : ∀ (i : ℕ) (h₁ : i < p.eraseLast (r := r).toList (r := r).length) (h₂ : i < p.toList (r := r).dropLast.length),
  getElem (valid := fun (as : List α) (i : ℕ) ↦ i < as.length) (p.eraseLast (r := r).toList (r := r)) i h₁ =
    getElem (valid := fun (as : List α) (i : ℕ) ↦ i < as.length) p.toList (r := r).dropLast i h₂
Tactic: apply List.ext_getElem
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
h : p.length (r := r) ≠ 0
⊢ p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) ⋯ = p
have a : (p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) sorry).toList (r := r) = p.toList (r := r)
Tactic: apply toList_injective
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
h : p.length (r := r) ≠ 0
⊢ (p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) ⋯).toList (r := r) = p.toList (r := r)
have a : p.eraseLast (r := r).toList (r := r) ++ [p.last (r := r)] = p.toList (r := r)
Tactic: rw [toList_snoc, ← getLast_toList, toList_eraseLast _ h, List.dropLast_append_getLast]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
h : p.length (r := r) ≠ 0
a : p.eraseLast (r := r).toList (r := r) ++ [p.last (r := r)] = p.toList (r := r)
⊢ (p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) ⋯).toList (r := r) = p.toList (r := r)
have a₁ : p.eraseLast (r := r).toList (r := r) ++ [p.toList (r := r).getLast sorry] = p.toList (r := r)
Tactic: rw [toList_snoc, ← getLast_toList, toList_eraseLast _ h, List.dropLast_append_getLast]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
h : p.length (r := r) ≠ 0
a : p.eraseLast (r := r).toList (r := r) ++ [p.last (r := r)] = p.toList (r := r)
a₁ : p.eraseLast (r := r).toList (r := r) ++ [p.toList (r := r).getLast ⋯] = p.toList (r := r)
⊢ (p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) ⋯).toList (r := r) = p.toList (r := r)
have a₂ : p.toList (r := r).dropLast ++ [p.toList (r := r).getLast sorry] = p.toList (r := r)
Tactic: rw [toList_snoc, ← getLast_toList, toList_eraseLast _ h, List.dropLast_append_getLast]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p : RelSeries r
h : p.length (r := r) ≠ 0
a : p.eraseLast (r := r).toList (r := r) ++ [p.last (r := r)] = p.toList (r := r)
a₁ : p.eraseLast (r := r).toList (r := r) ++ [p.toList (r := r).getLast ⋯] = p.toList (r := r)
a₂ : p.toList (r := r).dropLast ++ [p.toList (r := r).getLast ⋯] = p.toList (r := r)
⊢ (p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) ⋯).toList (r := r) = p.toList (r := r)
have a₃ : p.toList (r := r) = p.toList (r := r)
Tactic: rw [toList_snoc, ← getLast_toList, toList_eraseLast _ h, List.dropLast_append_getLast]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
p : RelSeries r
heq : p.length (r := r) = 0
⊢ motive p
have h : p = RelSeries.singleton r (p.head (r := r))
Tactic: convert! singleton p.head
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
p : RelSeries r
heq : p.length (r := r) = 0
⊢ p = RelSeries.singleton r (p.head (r := r))
have h : p.length (r := r) = (RelSeries.singleton r (p.head (r := r))).length (r := r)
Tactic: ext n
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
p : RelSeries r
heq : p.length (r := r) = 0
h : p.length (r := r) = (RelSeries.singleton r (p.head (r := r))).length (r := r)
⊢ p = RelSeries.singleton r (p.head (r := r))
have h₁ : ∀ (n : Fin (p.length (r := r) + 1)),
  p.toFun (r := r) n = ((RelSeries.singleton r (p.head (r := r))).toFun (r := r) ∘ Fin.cast sorry) n
Tactic: ext n
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
d : ℕ
hd : (p : RelSeries r) → (heq : p.length (r := r) = d) → motive p
p : RelSeries r
heq : p.length (r := r) = d + 1
⊢ motive p
have succ : (ne0 : p.length (r := r) ≠ 0) → motive p
Tactic: have ne0 : p.length ≠ 0 := by simp [heq]
-- Lean 4 proof step from Mathlib.Order.RelSeries (hypothesis)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
d : ℕ
hd : (p : RelSeries r) → (heq : p.length (r := r) = d) → motive p
p : RelSeries r
heq : p.length (r := r) = d + 1
ne0 : p.length (r := r) ≠ 0
⊢ motive p
have len : p.eraseLast (r := r).length (r := r) = d
Tactic: have len : p.eraseLast.length = d := by simp [heq]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
d : ℕ
hd : (p : RelSeries r) → (heq : p.length (r := r) = d) → motive p
p : RelSeries r
heq : p.length (r := r) = d + 1
ne0 : p.length (r := r) ≠ 0
len : p.eraseLast (r := r).length (r := r) = d
⊢ motive p
have h : p = p.eraseLast (r := r).snoc (r := r) (p.last (r := r)) sorry
Tactic: convert! snoc p.eraseLast p.last (p.eraseLast_last_rel_last ne0) (hd _ len)
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
p : RelSeries r
n : ℕ
heq : p.length (r := r) = n
⊢ motive p
have zero : (p : RelSeries r) → (heq : p.length (r := r) = 0) → motive p
Tactic: induction n generalizing p with
    | zero =>
      convert! singleton p.head
      ext n
      · exact heq
      · simp [show n = 0 by lia, apply_zero]
    | succ d hd =>
      have ne0 : p.length ≠ 0 := by simp [heq]
      have len : p.eraseLast.length = d := by simp [heq]
      convert! snoc p.eraseLast p.last (p.eraseLast_last_rel_last ne0) (hd _ len)
      exact (p.snoc_self_eraseLast ne0).symm
Cleared hypotheses: p, n, heq
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
motive : RelSeries r → Sort u_3
singleton : (x : α) → motive (RelSeries.singleton r x)
snoc : (p : RelSeries r) → (x : α) → (hx : (p.last (r := r), x) ∈ r) → (hp : motive p) → motive (p.snoc (r := r) x hx)
p : RelSeries r
n : ℕ
heq : p.length (r := r) = n
zero : (p : RelSeries r) → (heq : p.length (r := r) = 0) → motive p
⊢ motive p
have succ : (d : ℕ) →
  (hd : (p : RelSeries r) → (heq : p.length (r := r) = d) → motive p) →
    (p : RelSeries r) → (heq : p.length (r := r) = d + 1) → motive p
Tactic: induction n generalizing p with
    | zero =>
      convert! singleton p.head
      ext n
      · exact heq
      · simp [show n = 0 by lia, apply_zero]
    | succ d hd =>
      have ne0 : p.length ≠ 0 := by simp [heq]
      have len : p.eraseLast.length = d := by simp [heq]
      convert! snoc p.eraseLast p.last (p.eraseLast_last_rel_last ne0) (hd _ len)
      exact (p.snoc_self_eraseLast ne0).symm
Cleared hypotheses: p, n, heq
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
⊢ ∀ (i : Fin (p.length (r := r) + q.length (r := r))),
    Membership.mem (α := α × α) r
      ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) i.castSucc,
          Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) i.succ) :
        α × α)
have left : ∀ (i : Fin (p.length (r := r))),
  Membership.mem (α := α × α) r
    ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castSucc (Fin.castAdd (q.length (r := r)) i)),
        Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.succ (Fin.castAdd (q.length (r := r)) i))) :
      α × α)
Tactic: apply Fin.addCases
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
left :
  ∀ (i : Fin (p.length (r := r))),
    Membership.mem (α := α × α) r
      ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castSucc (Fin.castAdd (q.length (r := r)) i)),
          Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.succ (Fin.castAdd (q.length (r := r)) i))) :
        α × α)
⊢ ∀ (i : Fin (p.length (r := r) + q.length (r := r))),
    Membership.mem (α := α × α) r
      ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) i.castSucc,
          Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
            (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) i.succ) :
        α × α)
have right : ∀ (i : Fin (q.length (r := r))),
  Membership.mem (α := α × α) r
    ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castSucc (Fin.natAdd (p.length (r := r)) i)),
        Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.succ (Fin.natAdd (p.length (r := r)) i))) :
      α × α)
Tactic: apply Fin.addCases
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
⊢ Membership.mem (α := α × α) r
    ((Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castSucc (Fin.castAdd (q.length (r := r)) i)),
        Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.succ (Fin.castAdd (q.length (r := r)) i))) :
      α × α)
have left : Membership.mem (α := α × α) r
  (((p.toFun (r := r) ∘ Fin.castSucc) i,
      Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
        (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r))
        (if h : i.succ = Fin.last (p.length (r := r)) then Fin.natAdd (p.length (r := r)) 0
        else Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, sorry⟩)) :
    α × α)
Tactic: simp_rw [Fin.castSucc_castAdd, Fin.addCases_left, Fin.succ_castAdd]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
⊢ Membership.mem (α := α × α) r
    (((p.toFun (r := r) ∘ Fin.castSucc) i,
        Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
          (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r))
          (if h : i.succ = Fin.last (p.length (r := r)) then Fin.natAdd (p.length (r := r)) 0
          else Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, ⋯⟩)) :
      α × α)
have h : Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
    (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r))
    (if h : i.succ = Fin.last (p.length (r := r)) then Fin.natAdd (p.length (r := r)) 0
    else Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, sorry⟩) =
  p.toFun (r := r) i.succ
Tactic: convert! p.step i
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r))
      (if h : i.succ = Fin.last (p.length (r := r)) then Fin.natAdd (p.length (r := r)) 0
      else Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, ⋯⟩) =
    p.toFun (r := r) i.succ
have pos : ∀ (h : i.succ = Fin.last (p.length (r := r))),
  Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
pos :
  ∀ (h : i.succ = Fin.last (p.length (r := r))),
    Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
        (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
      p.toFun (r := r) i.succ
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r))
      (if h : i.succ = Fin.last (p.length (r := r)) then Fin.natAdd (p.length (r := r)) 0
      else Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, ⋯⟩) =
    p.toFun (r := r) i.succ
have neg : ∀ (h : ¬i.succ = Fin.last (p.length (r := r))),
  Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castAdd (q.length (r := r) + 1) ⟨(↑i : ℕ) + 1, sorry⟩) =
    p.toFun (r := r) i.succ
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
h : i.succ = Fin.last (p.length (r := r))
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
have pos : q.toFun (r := r) 0 = p.toFun (r := r) i.succ
Tactic: rw [Fin.addCases_right, h, ← last, connect, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
h : i.succ = Fin.last (p.length (r := r))
pos : q.toFun (r := r) 0 = p.toFun (r := r) i.succ
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
have pos₁ : q.toFun (r := r) 0 = p.toFun (r := r) (Fin.last (p.length (r := r)))
Tactic: rw [Fin.addCases_right, h, ← last, connect, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
h : i.succ = Fin.last (p.length (r := r))
pos : q.toFun (r := r) 0 = p.toFun (r := r) i.succ
pos₁ : q.toFun (r := r) 0 = p.toFun (r := r) (Fin.last (p.length (r := r)))
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
have pos₂ : q.toFun (r := r) 0 = p.last (r := r)
Tactic: rw [Fin.addCases_right, h, ← last, connect, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
h : i.succ = Fin.last (p.length (r := r))
pos : q.toFun (r := r) 0 = p.toFun (r := r) i.succ
pos₁ : q.toFun (r := r) 0 = p.toFun (r := r) (Fin.last (p.length (r := r)))
pos₂ : q.toFun (r := r) 0 = p.last (r := r)
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
have pos₃ : q.toFun (r := r) 0 = q.head (r := r)
Tactic: rw [Fin.addCases_right, h, ← last, connect, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
β : Type u_2
s : SetRel β β
p q : RelSeries r
connect : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r))
h : i.succ = Fin.last (p.length (r := r))
pos : q.toFun (r := r) 0 = p.toFun (r := r) i.succ
pos₁ : q.toFun (r := r) 0 = p.toFun (r := r) (Fin.last (p.length (r := r)))
pos₂ : q.toFun (r := r) 0 = p.last (r := r)
pos₃ : q.toFun (r := r) 0 = q.head (r := r)
⊢ Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
      (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) 0) =
    p.toFun (r := r) i.succ
have pos₄ : q.toFun (r := r) 0 = q.toFun (r := r) 0
Tactic: rw [Fin.addCases_right, h, ← last, connect, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r) + 1)
x : Fin (p.length (r := r))
⊢ (p.smash (r := r) q h).toFun (r := r) (Fin.castLE ⋯ x.castSucc) = p.toFun (r := r) x.castSucc
have dsimp : Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
    (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.castLE sorry x.castSucc) =
  p.toFun (r := r) x.castSucc
Tactic: dsimp only [smash]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r) + 1)
⊢ (p.smash (r := r) q h).toFun (r := r) (Fin.castLE ⋯ i) = p.toFun (r := r) i
have refine : (p.smash (r := r) q h).toFun (r := r) (Fin.castLE sorry (Fin.last (p.length (r := r)))) =
  p.toFun (r := r) (Fin.last (p.length (r := r)))
Tactic: refine i.lastCases ?_ fun _ ↦ by dsimp only [smash]; apply Fin.addCases_left
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
i : Fin (p.length (r := r) + 1)
⊢ (p.smash (r := r) q h).toFun (r := r) (Fin.castLE ⋯ (Fin.last (p.length (r := r)))) =
    p.toFun (r := r) (Fin.last (p.length (r := r)))
have change : (p.smash (r := r) q h).toFun (r := r) (Fin.natAdd (p.length (r := r)) 0) =
  p.toFun (r := r) (Fin.last (p.length (r := r)))
Tactic: change p.smash q h (Fin.natAdd p.length (0 : Fin (q.length + 1))) = _
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
i : Fin (q.length (r := r))
⊢ (p.smash (r := r) q h).toFun (r := r) (Fin.castSucc (Fin.natAdd (p.length (r := r)) i)) = q.toFun (r := r) i.castSucc
have dsimp : Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
    (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) i.castSucc) =
  q.toFun (r := r) i.castSucc
Tactic: dsimp only [smash, Fin.castSucc_natAdd]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
i : Fin (q.length (r := r))
⊢ (p.smash (r := r) q h).toFun (r := r) (Fin.succ (Fin.natAdd (p.length (r := r)) i)) = q.toFun (r := r) i.succ
have dsimp : Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
    (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.natAdd (p.length (r := r)) i.succ) =
  q.toFun (r := r) i.succ
Tactic: dsimp only [smash, Fin.succ_natAdd]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
⊢ (p.smash (r := r) q h).head (r := r) = p.head (r := r)
have zero : ∀ (toFun : Fin (HAdd.hAdd (α := ℕ) 0 1) → α) (step : ∀ (i : Fin 0), (toFun i.castSucc, toFun i.succ) ∈ r)
  (h : (mk (r := r) 0 toFun step).last (r := r) = q.head (r := r)),
  ((mk (r := r) 0 toFun step).smash (r := r) q h).head (r := r) = (mk (r := r) 0 toFun step).head (r := r)
Tactic: obtain ⟨_ | _, _⟩ := p
Cleared hypotheses: p, h
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
zero :
  ∀ (toFun : Fin (HAdd.hAdd (α := ℕ) 0 1) → α) (step : ∀ (i : Fin 0), (toFun i.castSucc, toFun i.succ) ∈ r)
    (h : (mk (r := r) 0 toFun step).last (r := r) = q.head (r := r)),
    ((mk (r := r) 0 toFun step).smash (r := r) q h).head (r := r) = (mk (r := r) 0 toFun step).head (r := r)
⊢ (p.smash (r := r) q h).head (r := r) = p.head (r := r)
have succ : ∀ (n : ℕ) (toFun : Fin (n + 1 + 1) → α) (step : ∀ (i : Fin (n + 1)), (toFun i.castSucc, toFun i.succ) ∈ r)
  (h : (mk (r := r) (n + 1) toFun step).last (r := r) = q.head (r := r)),
  ((mk (r := r) (n + 1) toFun step).smash (r := r) q h).head (r := r) = (mk (r := r) (n + 1) toFun step).head (r := r)
Tactic: obtain ⟨_ | _, _⟩ := p
Cleared hypotheses: p, h
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
q : RelSeries r
n : ℕ
toFun : Fin (n + 1 + 1) → α
step : ∀ (i : Fin (n + 1)), (toFun i.castSucc, toFun i.succ) ∈ r
h : (mk (r := r) (n + 1) toFun step).last (r := r) = q.head (r := r)
⊢ ((mk (r := r) (n + 1) toFun step).smash (r := r) q h).head (r := r) = (mk (r := r) (n + 1) toFun step).head (r := r)
have succ : Fin.addCases (motive := fun (x : Fin (n + 1 + (q.length (r := r) + 1))) ↦ α) (toFun ∘ Fin.castSucc) (q.toFun (r := r))
    0 =
  toFun 0
Tactic: dsimp only [smash, head]
-- Lean 4 proof step from Mathlib.Order.RelSeries (goal)
Goal:
α : Type u_1
r : SetRel α α
p q : RelSeries r
h : p.last (r := r) = q.head (r := r)
⊢ (p.smash (r := r) q h).last (r := r) = q.last (r := r)
have dsimp : Fin.addCases (motive := fun (x : Fin (p.length (r := r) + (q.length (r := r) + 1))) ↦ α)
    (p.toFun (r := r) ∘ Fin.castSucc) (q.toFun (r := r)) (Fin.last (p.length (r := r) + q.length (r := r))) =
  q.toFun (r := r) (Fin.last (q.length (r := r)))
Tactic: dsimp only [smash, last]