internlm/Lean-Github huggingface dataset:
Theorem: concat_aux_eqs
Proof state before:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
ww : p + zero < l := Eq.mpr (id (congrArg (fun _a => _a < l) (Nat.add_zero p))) w
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) p w = seq1 p pw
Tactic:
let hyp2 := (hyp zero).right (Nat.zero_lt_succ _) ww
Proof state after:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
ww : p + zero < l := Eq.mpr (id (congrArg (fun _a => _a < l) (Nat.add_zero p))) w
hyp2 : concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + zero) ww =
  (FinSeq.last seq1+|seq2) zero (_ : 0 < succ m) :=
  (hyp zero).right (zero_lt_succ m) ww
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) p w = seq1 p pw
Theorem: concat_aux_eqs
Proof state before:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
ww : p + zero < l := Eq.mpr (id (congrArg (fun _a => _a < l) (Nat.add_zero p))) w
hyp2 : concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + zero) ww =
  (FinSeq.last seq1+|seq2) zero (_ : 0 < succ m) :=
  (hyp zero).right (zero_lt_succ m) ww
lem2 : concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + zero) ww =
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) p w :=
  match p + zero, Nat.add_zero p, ww with
  | .(p), Eq.refl p, ww => rfl
⊢ (FinSeq.last seq1+|seq2) zero (_ : 0 < succ m) = seq1 p pw
Tactic:
rfl
Proof state after:
no goals
Theorem: concat_aux_eqs
Proof state before:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
⊢ p + zero < l
Tactic:
rw [(Nat.add_zero p)]
Proof state after:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
⊢ p < l
Theorem: concat_aux_eqs
Proof state before:
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
eql : succ k = p + 1
pw : p < p + 1
w : p < l
⊢ p < l
Tactic:
assumption
Proof state after:
no goals
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww = seq2 k kw
Tactic:
let hypw := (hyp (1 + k)).right
Proof state after:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww = seq2 k kw
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww = seq2 k kw
Tactic:
let kww : 1 + k < m + 1 := by
  rw [comm]
  apply Nat.succ_lt_succ
  assumption
Proof state after:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
kww : 1 + k < m + 1 := Eq.mpr (id (congrArg (fun _a => _a < m + 1) comm)) (succ_lt_succ kw)
⊢ concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww = seq2 k kw
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
kww : 1 + k < m + 1 := Eq.mpr (id (congrArg (fun _a => _a < m + 1) comm)) (succ_lt_succ kw)
resol : concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww =
  (FinSeq.last seq1+|seq2) (1 + k) kww :=
  hypw kww ww
⊢ (FinSeq.last seq1+|seq2) (1 + k) kww = seq2 k kw
Tactic:
match 1 + k, comm, kww with
| .(k + 1), rfl, kwww => rfl
Proof state after:
no goals
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ 1 + k < m + 1
Tactic:
rw [comm]
Proof state after:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ k + 1 < m + 1
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ k + 1 < m + 1
Tactic:
apply Nat.succ_lt_succ
Proof state after:
case a
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ k < m
Theorem: concat_aux_eqs
Proof state before:
case a
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
⊢ k < m
Tactic:
assumption
Proof state after:
no goals
Theorem: concat_aux_eqs
Proof state before:
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
ass : p + 1 + k = p + (1 + k) := Nat.add_assoc p 1 k
comm : 1 + k = k + 1 := Nat.add_comm 1 k
kw : k < m
w : p + 1 + k < l
ww : p + (1 + k) < l
hypw : ∀ (kw : 1 + k < m + 1) (w : p + (1 + k) < l),
  concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) w = (FinSeq.last seq1+|seq2) (1 + k) kw :=
  (hyp (1 + k)).right
kww : 1 + k < m + 1 := Eq.mpr (id (congrArg (fun _a => _a < m + 1) comm)) (succ_lt_succ kw)
resol : concatSeqAux ss (FinSeq.init seq1) (FinSeq.last seq1+|seq2) (p + (1 + k)) ww =
  (FinSeq.last seq1+|seq2) (1 + k) kww :=
  hypw kww ww
kwww : k + 1 < m + 1
⊢ (FinSeq.last seq1+|seq2) (k + 1) kwww = seq2 k kw
Tactic:
rfl
Proof state after:
no goals
Theorem: concat_empty_seq_id
Proof state before:
α : Type
n : ℕ
⊢ ∀ (seq : FinSeq n α), seq++|FinSeq.empty = seq
Tactic:
intro seq
Proof state after:
α : Type
n : ℕ
seq : FinSeq n α
⊢ seq++|FinSeq.empty = seq
Theorem: concat_empty_seq_id
Proof state before:
α : Type
n : ℕ
seq : FinSeq n α
⊢ seq++|FinSeq.empty = seq
Tactic:
apply funext
Proof state after:
case h
α : Type
n : ℕ
seq : FinSeq n α
⊢ ∀ (x : ℕ), (seq++|FinSeq.empty) x = seq x
Theorem: concat_empty_seq_id
Proof state before:
case h
α : Type
n : ℕ
seq : FinSeq n α
⊢ ∀ (x : ℕ), (seq++|FinSeq.empty) x = seq x
Tactic:
intro k
Proof state after:
case h
α : Type
n : ℕ
seq : FinSeq n α
k : ℕ
⊢ (seq++|FinSeq.empty) k = seq k