Theorem: concat_aux_eqs
Proof state before:
case left
α : Type
n m l p : ℕ
s : p + 1 + m = l
seq1 : FinSeq (p + 1) α
seq2 : FinSeq m α
k : ℕ
ss : p + (m + 1) = l :=
  Eq.mpr (id (congrArg (fun _a => p + (m + 1) = _a) s.symm))
    (Eq.mpr (id (congrArg (fun _a => p + _a = p + 1 + m) (Nat.add_comm m 1)))
      (Eq.mpr (id (congrArg (fun _a => p + (1 + m) = _a) (Nat.add_assoc p 1 m))) (Eq.refl (p + (1 + m)))))
resolve : concatSeqAux s seq1 seq2 = concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) := rfl
hyp : ∀ (k : ℕ),
  (∀ (kw : k < p) (w : k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw) ∧
    ∀ (kw : k < m + 1) (w : p + k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + k) w = (FinSeq.last seq1+|seq2) k kw :=
  concat_aux_eqs ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2)
hyp1 : ∀ (kw : k < p) (w : k < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw :=
  (hyp k).left
⊢ ∀ (kw : k < p + 1) (w : k < l), concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = seq1 k kw

case right
α : Type
n m l p : ℕ
s : p + 1 + m = l
seq1 : FinSeq (p + 1) α
seq2 : FinSeq m α
k : ℕ
ss : p + (m + 1) = l :=
  Eq.mpr (id (congrArg (fun _a => p + (m + 1) = _a) s.symm))
    (Eq.mpr (id (congrArg (fun _a => p + _a = p + 1 + m) (Nat.add_comm m 1)))
      (Eq.mpr (id (congrArg (fun _a => p + (1 + m) = _a) (Nat.add_assoc p 1 m))) (Eq.refl (p + (1 + m)))))
resolve : concatSeqAux s seq1 seq2 = concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) := rfl
hyp : ∀ (k : ℕ),
  (∀ (kw : k < p) (w : k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw) ∧
    ∀ (kw : k < m + 1) (w : p + k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + k) w = (FinSeq.last seq1+|seq2) k kw :=
  concat_aux_eqs ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2)
hyp1 : ∀ (kw : k < p) (w : k < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw :=
  (hyp k).left
⊢ ∀ (kw : k < m) (w : p + 1 + k < l),
    concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + 1 + k) w = seq2 k kw
Tactic:
focus
    intro kw w
    match Nat.eq_or_lt_of_le kw with
    | Or.inr kww => exact hyp1 (le_of_succ_le_succ kww) w
    | Or.inl eql =>
        match k, eql, kw, w with
        | .(p), rfl, pw, w =>
          let ww : p + zero < l := by
            rw [(Nat.add_zero p)]
            assumption
          let hyp2 := (hyp zero).right (Nat.zero_lt_succ _) ww
          let lem2 : concatSeqAux ss (seq1.init) (seq1.last+|seq2) (p + zero) ww =
                concatSeqAux ss (seq1.init) (seq1.last+|seq2) p w :=
                match (p + zero), Nat.add_zero p, ww with
                | .(p), rfl, ww => rfl
          rw [← lem2]
          rw [hyp2]
          rfl
Proof state after:
case right
α : Type
n m l p : ℕ
s : p + 1 + m = l
seq1 : FinSeq (p + 1) α
seq2 : FinSeq m α
k : ℕ
ss : p + (m + 1) = l :=
  Eq.mpr (id (congrArg (fun _a => p + (m + 1) = _a) s.symm))
    (Eq.mpr (id (congrArg (fun _a => p + _a = p + 1 + m) (Nat.add_comm m 1)))
      (Eq.mpr (id (congrArg (fun _a => p + (1 + m) = _a) (Nat.add_assoc p 1 m))) (Eq.refl (p + (1 + m)))))
resolve : concatSeqAux s seq1 seq2 = concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) := rfl
hyp : ∀ (k : ℕ),
  (∀ (kw : k < p) (w : k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw) ∧
    ∀ (kw : k < m + 1) (w : p + k < l),
      concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + k) w = (FinSeq.last seq1+|seq2) k kw :=
  concat_aux_eqs ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2)
hyp1 : ∀ (kw : k < p) (w : k < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) k w = FinSeq.init seq1 k kw :=
  (hyp k).left
⊢ ∀ (kw : k < m) (w : p + 1 + k < l),
    concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + 1 + k) w = seq2 k kw