awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  ∀ (hj : j < b.blocksFun ⟨i, hi⟩),
    a.sizeUpTo (b.sizeUpTo i + j) =
      (a.gather b).sizeUpTo i +
        sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < length (n := (a.gather b).blocksFun ⟨i, ⋯⟩) (a.sigmaCompositionAux b ⟨i, ⋯⟩)
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ a.sizeUpTo (b.sizeUpTo i + (j + 1)) =
    (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) (j + 1)
have succ : (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)) j +
    getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) a.blocks (b.sizeUpTo i + j) D =
  (a.gather b).sizeUpTo i +
    (sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)) j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
        (blocks (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)))
        j B)
Tactic: rw [this, sizeUpTo_succ _ D, IHj A, sizeUpTo_succ _ B]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  ∀ (hj : j < b.blocksFun ⟨i, hi⟩),
    a.sizeUpTo (b.sizeUpTo i + j) =
      (a.gather b).sizeUpTo i +
        sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < length (n := (a.gather b).blocksFun ⟨i, ⋯⟩) (a.sigmaCompositionAux b ⟨i, ⋯⟩)
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i +
        sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) a.blocks (b.sizeUpTo i + j) D =
    (a.gather b).sizeUpTo i +
      (sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j +
        getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
          (blocks (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
            (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)))
          j B)
have succ : (a.gather b).sizeUpTo i +
    (sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        {
          blocks :=
            getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i sorry,
          blocks_pos := sorry, blocks_sum := sorry : Composition ((a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length)) }
        j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) a.blocks (b.sizeUpTo i + j) D) =
  (a.gather b).sizeUpTo i +
    (sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        {
          blocks :=
            getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i sorry,
          blocks_pos := sorry, blocks_sum := sorry : Composition ((a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length)) }
        j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
        (getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i sorry) j sorry)
Tactic: simp only [sigmaCompositionAux, add_assoc]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  ∀ (hj : j < b.blocksFun ⟨i, hi⟩),
    a.sizeUpTo (b.sizeUpTo i + j) =
      (a.gather b).sizeUpTo i +
        sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < length (n := (a.gather b).blocksFun ⟨i, ⋯⟩) (a.sigmaCompositionAux b ⟨i, ⋯⟩)
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i +
      (sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          {
            blocks :=
              getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i ⋯,
            blocks_pos := ⋯, blocks_sum := ⋯ : Composition ((a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length)) }
          j +
        getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) a.blocks (b.sizeUpTo i + j) D) =
    (a.gather b).sizeUpTo i +
      (sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          {
            blocks :=
              getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i ⋯,
            blocks_pos := ⋯, blocks_sum := ⋯ : Composition ((a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length)) }
          j +
        getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
          (getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i ⋯) j
          ⋯)
have succ : (a.gather b).sizeUpTo i +
    (sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        {
          blocks :=
            getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i sorry,
          blocks_pos := sorry, blocks_sum := sorry : Composition ((a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length)) }
        j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) a.blocks (b.sizeUpTo i + j) D) =
  (a.gather b).sizeUpTo i +
    (sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        {
          blocks :=
            getElem (valid := fun (as : List (List ℕ)) (i : ℕ) ↦ i < as.length) (a.blocks.splitWrtComposition b) i sorry,
          blocks_pos := sorry, blocks_sum := sorry : Composition ((a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length)) }
        j +
      getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
        (drop (b.sizeUpTo i) (take (b.sizeUpTo (i + 1)) a.blocks)) j sorry)
Tactic: rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
n : ℕ
a : Composition n
b : Composition a.length
i j : ℕ
hi : i < b.length
hj : j < b.blocksFun ⟨i, hi⟩
⊢ a.sizeUpTo (b.sizeUpTo i + j) =
    (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j
have zero : ∀ (hj : 0 < b.blocksFun (⟨i, hi⟩ : Fin b.length)),
  a.sizeUpTo (b.sizeUpTo i + 0) =
    (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)) 0
Tactic: induction j with
  | zero =>
    change
      sum (take (b.blocks.take i).sum a.blocks) =
        sum (take i (map sum (splitWrtComposition a.blocks b)))
    induction i with
    | zero => rfl
    | succ i IH =>
      have A : i < b.length := Nat.lt_of_succ_lt hi
      have B : i < List.length (map List.sum (splitWrtComposition a.blocks b)) := by simp [A]
      have C : 0 < blocksFun b ⟨i, A⟩ := Composition.blocks_pos' _ _ _
      rw [sum_take_succ _ _ B, ← IH A C]
      have :
        take (sum (take i b.blocks)) a.blocks =
          take (sum (take i b.blocks)) (take (sum (take (i + 1) b.blocks)) a.blocks) := by
        rw [take_take, min_eq_left]
        apply monotone_sum_take _ (Nat.le_succ _)
      rw [this, getElem_map, getElem_splitWrtComposition, ←
        take_append_drop (sum (take i b.blocks)) (take (sum (take (Nat.succ i) b.blocks)) a.blocks),
        sum_append]
      congr
      rw [take_append_drop]
  | succ j IHj =>
    have A : j < blocksFun b ⟨i, hi⟩ := lt_trans (lt_add_one j) hj
    have B : j < length (sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ hi⟩) := by
      convert! A; rw [← length_sigmaCompositionAux]
    have C : sizeUpTo b i + j < sizeUpTo b (i + 1) := by
      simp only [sizeUpTo_succ b hi, add_lt_add_iff_left]
      exact A
    have D : sizeUpTo b i + j < length a := lt_of_lt_of_le C (b.sizeUpTo_le _)
    have : sizeUpTo b i + Nat.succ j = (sizeUpTo b i + j).succ := rfl
    rw [this, sizeUpTo_succ _ D, IHj A, sizeUpTo_succ _ B]
    simp only [sigmaCompositionAux, add_assoc]
    rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take]
Cleared hypotheses: j, hj
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
n : ℕ
a : Composition n
b : Composition a.length
i j : ℕ
hi : i < b.length
hj : j < b.blocksFun ⟨i, hi⟩
zero :
  ∀ (hj : 0 < b.blocksFun (⟨i, hi⟩ : Fin b.length)),
    a.sizeUpTo (b.sizeUpTo i + 0) =
      (a.gather b).sizeUpTo i +
        sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
          (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) 0
⊢ a.sizeUpTo (b.sizeUpTo i + j) =
    (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, ⋯⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, ⋯⟩ : Fin (a.gather b).length)) j
have succ : ∀ (j : ℕ)
  (IHj :
    ∀ (hj : j < b.blocksFun ⟨i, hi⟩),
      a.sizeUpTo (b.sizeUpTo i + j) =
        (a.gather b).sizeUpTo i +
          sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
            (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)) j)
  (hj : j + 1 < b.blocksFun ⟨i, hi⟩),
  a.sizeUpTo (b.sizeUpTo i + (j + 1)) =
    (a.gather b).sizeUpTo i +
      sizeUpTo (n := (a.gather b).blocksFun (⟨i, sorry⟩ : Fin (a.gather b).length))
        (a.sigmaCompositionAux b (⟨i, sorry⟩ : Fin (a.gather b).length)) (j + 1)
Tactic: induction j with
  | zero =>
    change
      sum (take (b.blocks.take i).sum a.blocks) =
        sum (take i (map sum (splitWrtComposition a.blocks b)))
    induction i with
    | zero => rfl
    | succ i IH =>
      have A : i < b.length := Nat.lt_of_succ_lt hi
      have B : i < List.length (map List.sum (splitWrtComposition a.blocks b)) := by simp [A]
      have C : 0 < blocksFun b ⟨i, A⟩ := Composition.blocks_pos' _ _ _
      rw [sum_take_succ _ _ B, ← IH A C]
      have :
        take (sum (take i b.blocks)) a.blocks =
          take (sum (take i b.blocks)) (take (sum (take (i + 1) b.blocks)) a.blocks) := by
        rw [take_take, min_eq_left]
        apply monotone_sum_take _ (Nat.le_succ _)
      rw [this, getElem_map, getElem_splitWrtComposition, ←
        take_append_drop (sum (take i b.blocks)) (take (sum (take (Nat.succ i) b.blocks)) a.blocks),
        sum_append]
      congr
      rw [take_append_drop]
  | succ j IHj =>
    have A : j < blocksFun b ⟨i, hi⟩ := lt_trans (lt_add_one j) hj
    have B : j < length (sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ hi⟩) := by
      convert! A; rw [← length_sigmaCompositionAux]
    have C : sizeUpTo b i + j < sizeUpTo b (i + 1) := by
      simp only [sizeUpTo_succ b hi, add_lt_add_iff_left]
      exact A
    have D : sizeUpTo b i + j < length a := lt_of_lt_of_le C (b.sizeUpTo_le _)
    have : sizeUpTo b i + Nat.succ j = (sizeUpTo b i + j).succ := rfl
    rw [this, sizeUpTo_succ _ D, IHj A, sizeUpTo_succ _ B]
    simp only [sigmaCompositionAux, add_assoc]
    rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take]
Cleared hypotheses: j, hj
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
⊢ ∀ {i_1 : ℕ},
    Membership.mem (γ := List ℕ)
        (flatten
          (ofFn (n := i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
            fun
              (j : Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
            (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).blocks))
        i_1 →
      0 < i_1
have simp : ∀ {i_1 : ℕ} (j : Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length),
  i_1 ∈ (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).blocks → 0 < i_1
Tactic: simp only [and_imp, List.mem_flatten, exists_imp, forall_mem_ofFn_iff]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
k : ℕ
hk :
  Membership.mem (γ := List ℕ)
    (ofFn (n := i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
      fun (j : Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
      (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).length)
    k
⊢ 0 < k
have refine : ∀ (j : Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length),
  0 < (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).length
Tactic: refine ((forall_mem_ofFn_iff (P := fun i => 0 < i)).2 fun j => ?_) k hk
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
⊢ sum
      (ofFn (n := i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
        fun (j : Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
        (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).length) =
    {
        blocks :=
          flatten
            (ofFn (n := i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length)
              fun
                (j :
                  Fin i.fst (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)).length) ↦
              (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.length) → Composition (c.blocksFun i)) j).blocks),
        blocks_pos := ⋯, blocks_sum := ⋯ : Composition n }.length
have dsimp : sum
    (ofFn (n :=
      i.fst (β := fun (c : Composition n) ↦ (i : Fin c.blocks.length) → Composition (c.blocksFun i)).blocks.length)
      fun
        (j :
          Fin
            i.fst (β := fun (c : Composition n) ↦
                  (i : Fin c.blocks.length) → Composition (c.blocksFun i)).blocks.length) ↦
      (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.blocks.length) → Composition (c.blocksFun i))
            j).blocks.length) =
  List.length (α := ℕ)
    (flatten
      (ofFn (n :=
        i.fst (β := fun (c : Composition n) ↦ (i : Fin c.blocks.length) → Composition (c.blocksFun i)).blocks.length)
        fun
          (j :
            Fin
              i.fst (β := fun (c : Composition n) ↦
                    (i : Fin c.blocks.length) → Composition (c.blocksFun i)).blocks.length) ↦
        (i.snd (β := fun (c : Composition n) ↦ (i : Fin c.blocks.length) → Composition (c.blocksFun i)) j).blocks))
Tactic: dsimp only [Composition.length]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
⊢ flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
      a.blocks ∧
    (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).length) = b.blocks
have left : flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
  a.blocks
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
left :
  flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
    a.blocks
⊢ flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
      a.blocks ∧
    (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).length) = b.blocks
have right : (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).length) = b.blocks
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
| a.blocks
have rw : (ofFn (a.blocks.splitWrtComposition b).get).flatten
Tactic: rw [← flatten_splitWrtComposition a.blocks b, ← ofFn_get (splitWrtComposition a.blocks b)]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
⊢ flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
    a.blocks
have left : flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
  (ofFn (a.blocks.splitWrtComposition b).get).flatten
Tactic: conv_rhs =>
        rw [← flatten_splitWrtComposition a.blocks b, ← ofFn_get (splitWrtComposition a.blocks b)]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
⊢ flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
    (ofFn (a.blocks.splitWrtComposition b).get).flatten
have A : (a.gather b).length = (a.blocks.splitWrtComposition b).length
Tactic: have A : length (gather a b) = List.length (splitWrtComposition a.blocks b) := by
        simp only [length, gather, length_map, length_splitWrtComposition]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
A : (a.gather b).length = (a.blocks.splitWrtComposition b).length
⊢ flatten (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks) =
    (ofFn (a.blocks.splitWrtComposition b).get).flatten
have left : HEq (α := Fin (a.gather b).length → List ℕ) (fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).blocks)
  (a.blocks.splitWrtComposition b).get
Tactic: congr! 2
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
⊢ (ofFn (n := (a.gather b).length) fun (j : Fin (a.gather b).length) ↦ (a.sigmaCompositionAux b j).length) = b.blocks
have B : (a.gather b).length = b.blocks.length
Tactic: have B : Composition.length (Composition.gather a b) = List.length b.blocks :=
        Composition.length_gather _ _
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Composition (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst : NontriviallyNormedField 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : NormedAddCommGroup H
inst_8 : NormedSpace 𝕜 H
n_1 n : ℕ
a : Composition n
b : Composition a.length
B : (a.gather b).length = b.blocks.length
| b.blocks
have rw : ofFn (n := b.blocks.length) fun (i : Fin b.blocks.length) ↦
  getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) b.blocks (↑i : ℕ) sorry
Tactic: rw [← ofFn_getElem (xs := b.blocks)]