awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₄ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
  (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
hr₄ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ
      TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₅ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
  ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B)) ∘ₗ
    comul)
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
hr₄ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ
      TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
hr₅ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B)) ∘ₗ
      comul)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₆ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
  (TensorProduct.map (LinearMap.fst R A B ∘ₗ inr R A B) (LinearMap.fst R A B ∘ₗ inr R A B) ∘ₗ comul)
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
hr₄ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ
      TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
hr₅ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B)) ∘ₗ
      comul)
hr₆ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B ∘ₗ inr R A B) (LinearMap.fst R A B ∘ₗ inr R A B) ∘ₗ comul)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₇ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 (TensorProduct.map 0 0 ∘ₗ comul)
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
hr₄ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ
      TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
hr₅ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B)) ∘ₗ
      comul)
hr₆ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B ∘ₗ inr R A B) (LinearMap.fst R A B ∘ₗ inr R A B) ∘ₗ comul)
hr₇ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 (TensorProduct.map 0 0 ∘ₗ comul)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₈ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 (0 ∘ₗ comul)
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hr :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ LinearMap.fst R A B ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₁ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₂ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
hr₃ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul ∘ₗ inr R A B)
hr₄ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ
      TensorProduct.map (inr R A B) (inr R A B) ∘ₗ comul)
hr₅ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ TensorProduct.map (inr R A B) (inr R A B)) ∘ₗ
      comul)
hr₆ :
  Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0
    (TensorProduct.map (LinearMap.fst R A B ∘ₗ inr R A B) (LinearMap.fst R A B ∘ₗ inr R A B) ∘ₗ comul)
hr₇ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 (TensorProduct.map 0 0 ∘ₗ comul)
hr₈ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 (0 ∘ₗ comul)
⊢ Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) ((comul ∘ₗ LinearMap.fst R A B) ∘ₗ inr R A B)
    ((TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ comul) ∘ₗ inr R A B)
have hr₉ : Eq (α := LinearMap.{u, u, w, v} (RingHom.id R) B (A ⊗[R] A)) 0 0
Tactic: rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
      ← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
⊢ Eq (α := A × B →ₗ[R] B ⊗[R] B) (comul ∘ₗ LinearMap.snd R A B)
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul)
have hl : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
⊢ Eq (α := A × B →ₗ[R] B ⊗[R] B) (comul ∘ₗ LinearMap.snd R A B)
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul)
have hr : Eq (α := B →ₗ[R] B ⊗[R] B) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inr R A B)
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inr R A B)
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₁ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₁ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₂ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₁ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₂ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₃ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
  (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul ∘ₗ inl R A B)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₁ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₂ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₃ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul ∘ₗ inl R A B)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₄ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
  (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₁ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₂ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₃ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul ∘ₗ inl R A B)
hl₄ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ
      TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₅ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
  ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ TensorProduct.map (inl R A B) (inl R A B)) ∘ₗ
    comul)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
-- Lean 4 proof step from Mathlib.RingTheory.Coalgebra.Basic (goal)
Goal:
R : Type u
A : Type v
B : Type w
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : AddCommMonoid B
inst_3 : Module R A
inst_4 : Module R B
inst_5 : Coalgebra R A
inst_6 : Coalgebra R B
hl :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ LinearMap.snd R A B ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₁ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) (comul ∘ₗ 0)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₂ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
hl₃ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul ∘ₗ inl R A B)
hl₄ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    (TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ
      TensorProduct.map (inl R A B) (inl R A B) ∘ₗ comul)
hl₅ :
  Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ TensorProduct.map (inl R A B) (inl R A B)) ∘ₗ
      comul)
⊢ Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) ((comul ∘ₗ LinearMap.snd R A B) ∘ₗ inl R A B)
    ((TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ comul) ∘ₗ inl R A B)
have hl₆ : Eq (α := LinearMap.{u, u, v, w} (RingHom.id R) A (B ⊗[R] B)) 0
  (TensorProduct.map (LinearMap.snd R A B ∘ₗ inl R A B) (LinearMap.snd R A B ∘ₗ inl R A B) ∘ₗ comul)
Tactic: rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
      ← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]