ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CochainComplex.HomComplex.Cochain.leftUnshift_neg
@[simp]
lemma leftUnshift_neg {n' a : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n') :
    (-γ).leftUnshift n hn = -γ.leftUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n')
Goal: (-γ).leftUnshift n hn = -γ.leftUnshift n hn
-- CochainComplex.HomComplex.Cochain.shift_neg
@[simp]
lemma shift_neg (a : ℤ) :
    (-γ).shift a = -γ.shift a 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a : ℤ)
Goal: (-γ).shift a = -γ.shift a
-- CochainComplex.HomComplex.Cochain.rightUnshift_add
@[simp]
lemma rightUnshift_add {n' a : ℤ} (γ₁ γ₂ : Cochain K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n) :
    (γ₁ + γ₂).rightUnshift n hn = γ₁.rightUnshift n hn + γ₂.rightUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ₁ γ₂ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n)
Goal: (γ₁ + γ₂).rightUnshift n hn = γ₁.rightUnshift n hn + γ₂.rightUnshift n hn
-- CochainComplex.HomComplex.Cochain.leftUnshift_add
@[simp]
lemma leftUnshift_add {n' a : ℤ} (γ₁ γ₂ : Cochain (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n') :
    (γ₁ + γ₂).leftUnshift n hn = γ₁.leftUnshift n hn + γ₂.leftUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ₁ γ₂ : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n')
Goal: (γ₁ + γ₂).leftUnshift n hn = γ₁.leftUnshift n hn + γ₂.leftUnshift n hn
-- CochainComplex.HomComplex.Cochain.rightShift_smul
@[simp]
lemma rightShift_smul (a n' : ℤ) (hn' : n' + a = n) (x : R) :
    (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n' + a = n) (x : R)
Goal: (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn'
-- CochainComplex.HomComplex.Cochain.leftShift_smul
@[simp]
lemma leftShift_smul (a n' : ℤ) (hn' : n + a = n') (x : R) :
    (x • γ).leftShift a n' hn' = x • γ.leftShift a n' hn' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n + a = n') (x : R)
Goal: (x • γ).leftShift a n' hn' = x • γ.leftShift a n' hn'
-- CochainComplex.HomComplex.Cochain.shift_smul
@[simp]
lemma shift_smul (a : ℤ) (x : R) :
    (x • γ).shift a = x • (γ.shift a) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a : ℤ) (x : R)
Goal: (x • γ).shift a = x • γ.shift a
-- CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv_apply
@[simps!]
def rightShiftLinearEquiv (n a n' : ℤ) (hn' : n' + a = n) :
    Cochain K L n ≃ₗ[R] Cochain K (L⟦a⟧) n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type u_1) [Ring R] [CategoryTheory.Linear R C] (K L : CochainComplex C ℤ) (n a n' : ℤ) (hn' : n' + a = n)
Goal: ∀ (a_1 : CochainComplex.HomComplex.Cochain K L n),
  (CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv R K L n a n' hn') a_1 = a_1.rightShift a n' hn'
-- CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv_symm_apply
@[simps!]
def rightShiftLinearEquiv (n a n' : ℤ) (hn' : n' + a = n) :
    Cochain K L n ≃ₗ[R] Cochain K (L⟦a⟧) n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type u_1) [Ring R] [CategoryTheory.Linear R C] (K L : CochainComplex C ℤ) (n a n' : ℤ) (hn' : n' + a = n)
Goal: ∀ (a_1 : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'),
  (CochainComplex.HomComplex.Cochain.rightShiftLinearEquiv R K L n a n' hn').symm a_1 = a_1.rightUnshift n hn'
-- CochainComplex.HomComplex.Cochain.leftShiftLinearEquiv_apply
@[simps!]
def leftShiftLinearEquiv (n a n' : ℤ) (hn : n + a = n') :
    Cochain K L n ≃ₗ[R] Cochain (K⟦a⟧) L n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type u_1) [Ring R] [CategoryTheory.Linear R C] (K L : CochainComplex C ℤ) (n a n' : ℤ) (hn : n + a = n')
Goal: ∀ (a_1 : CochainComplex.HomComplex.Cochain K L n),
  (CochainComplex.HomComplex.Cochain.leftShiftLinearEquiv R K L n a n' hn) a_1 = a_1.leftShift a n' hn
-- CochainComplex.HomComplex.Cochain.leftShiftLinearEquiv_symm_apply
@[simps!]
def leftShiftLinearEquiv (n a n' : ℤ) (hn : n + a = n') :
    Cochain K L n ≃ₗ[R] Cochain (K⟦a⟧) L n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type u_1) [Ring R] [CategoryTheory.Linear R C] (K L : CochainComplex C ℤ) (n a n' : ℤ) (hn : n + a = n')
Goal: ∀ (a_1 : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'),
  (CochainComplex.HomComplex.Cochain.leftShiftLinearEquiv R K L n a n' hn).symm a_1 = a_1.leftUnshift n hn
-- CochainComplex.HomComplex.Cochain.shiftLinearMap_apply
@[simps!]
def shiftLinearMap (n a : ℤ) :
    Cochain K L n →ₗ[R] Cochain (K⟦a⟧) (L⟦a⟧) n 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type u_1) [Ring R] [CategoryTheory.Linear R C] (K L : CochainComplex C ℤ) (n a : ℤ)
Goal: ∀ (a_1 : CochainComplex.HomComplex.Cochain K L n),
  (CochainComplex.HomComplex.Cochain.shiftLinearMap R K L n a) a_1 = a_1.shift a
-- CochainComplex.HomComplex.Cochain.rightShift_units_smul
@[simp]
lemma rightShift_units_smul (a n' : ℤ) (hn' : n' + a = n) (x : Rˣ) :
    (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n' + a = n) (x : Rˣ)
Goal: (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn'
-- CochainComplex.HomComplex.Cochain.leftShift_units_smul
@[simp]
lemma leftShift_units_smul (a n' : ℤ) (hn' : n + a = n') (x : Rˣ) :
    (x • γ).leftShift a n' hn' = x • γ.leftShift a n' hn' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n + a = n') (x : Rˣ)
Goal: (x • γ).leftShift a n' hn' = x • γ.leftShift a n' hn'
-- CochainComplex.HomComplex.Cochain.shift_units_smul
@[simp]
lemma shift_units_smul (a : ℤ) (x : Rˣ) :
    (x • γ).shift a = x • (γ.shift a) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a : ℤ) (x : Rˣ)
Goal: (x • γ).shift a = x • γ.shift a
-- CochainComplex.HomComplex.Cochain.rightUnshift_smul
@[simp]
lemma rightUnshift_smul {n' a : ℤ} (γ : Cochain K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n) (x : R) :
    (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n) (x : R)
Goal: (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn
-- CochainComplex.HomComplex.Cochain.rightUnshift_units_smul
@[simp]
lemma rightUnshift_units_smul {n' a : ℤ} (γ : Cochain K (L⟦a⟧) n') (n : ℤ)
    (hn : n' + a = n) (x : Rˣ) :
    (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n) (x : Rˣ)
Goal: (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn
-- CochainComplex.HomComplex.Cochain.leftUnshift_smul
@[simp]
lemma leftUnshift_smul {n' a : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n') (x : R) :
    (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n') (x : R)
Goal: (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn
-- CochainComplex.HomComplex.Cochain.leftUnshift_units_smul
@[simp]
lemma leftUnshift_units_smul {n' a : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ)
    (hn : n + a = n') (x : Rˣ) :
    (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n') (x : Rˣ)
Goal: (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn
-- CochainComplex.HomComplex.Cochain.rightUnshift_comp
lemma rightUnshift_comp {m : ℤ} {a : ℤ} (γ' : Cochain L (M⟦a⟧) m) {nm : ℤ} (hnm : n + m = nm)
    (nm' : ℤ) (hnm' : nm + a = nm') (m' : ℤ) (hm' : m + a = m') :
    (γ.comp γ' hnm).rightUnshift nm' hnm' =
      γ.comp (γ'.rightUnshift m' hm') (by omega) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L M : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) {m a : ℤ} (γ' : CochainComplex.HomComplex.Cochain L ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj M) m) {nm : ℤ} (hnm : n + m = nm) (nm' : ℤ) (hnm' : nm + a = nm') (m' : ℤ) (hm' : m + a = m')
Goal: (γ.comp γ' hnm).rightUnshift nm' hnm' = γ.comp (γ'.rightUnshift m' hm') ⋯
-- CochainComplex.HomComplex.Cochain.leftShift_comp
lemma leftShift_comp (a n' : ℤ) (hn' : n + a = n') {m t t' : ℤ} (γ' : Cochain L M m)
    (h : n + m = t) (ht' : t + a = t') :
    (γ.comp γ' h).leftShift a t' ht' = (a * m).negOnePow • (γ.leftShift a n' hn').comp γ'
      (by rw [← ht', ← h, ← hn', add_assoc, add_comm a, add_assoc]) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L M : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n + a = n') {m t t' : ℤ} (γ' : CochainComplex.HomComplex.Cochain L M m) (h : n + m = t) (ht' : t + a = t')
Goal: (γ.comp γ' h).leftShift a t' ht' = (a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯
-- CochainComplex.HomComplex.Cochain.leftShift_comp_zero_cochain
@[simp]
lemma leftShift_comp_zero_cochain (a n' : ℤ) (hn' : n + a = n') (γ' : Cochain L M 0) :
    (γ.comp γ' (add_zero n)).leftShift a n' hn' =
      (γ.leftShift a n' hn').comp γ' (add_zero n') 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L M : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n + a = n') (γ' : CochainComplex.HomComplex.Cochain L M 0)
Goal: (γ.comp γ' ⋯).leftShift a n' hn' = (γ.leftShift a n' hn').comp γ' ⋯
-- CochainComplex.HomComplex.Cochain.δ_rightShift
lemma δ_rightShift (a n' m' : ℤ) (hn' : n' + a = n) (m : ℤ) (hm' : m' + a = m) :
    δ n' m' (γ.rightShift a n' hn') = a.negOnePow • (δ n m γ).rightShift a m' hm' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' m' : ℤ) (hn' : n' + a = n) (m : ℤ) (hm' : m' + a = m)
Goal: CochainComplex.HomComplex.δ n' m' (γ.rightShift a n' hn') =
  a.negOnePow • (CochainComplex.HomComplex.δ n m γ).rightShift a m' hm'
-- CochainComplex.HomComplex.Cochain.δ_rightUnshift
lemma δ_rightUnshift {a n' : ℤ} (γ : Cochain K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n)
    (m m' : ℤ) (hm' : m' + a = m) :
    δ n m (γ.rightUnshift n hn) = a.negOnePow • (δ n' m' γ).rightUnshift m hm' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {a n' : ℤ} (γ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n) (m m' : ℤ) (hm' : m' + a = m)
Goal: CochainComplex.HomComplex.δ n m (γ.rightUnshift n hn) =
  a.negOnePow • (CochainComplex.HomComplex.δ n' m' γ).rightUnshift m hm'
-- CochainComplex.HomComplex.Cochain.δ_leftShift
lemma δ_leftShift (a n' m' : ℤ) (hn' : n + a = n') (m : ℤ) (hm' : m + a = m') :
    δ n' m' (γ.leftShift a n' hn') = a.negOnePow • (δ n m γ).leftShift a m' hm' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' m' : ℤ) (hn' : n + a = n') (m : ℤ) (hm' : m + a = m')
Goal: CochainComplex.HomComplex.δ n' m' (γ.leftShift a n' hn') =
  a.negOnePow • (CochainComplex.HomComplex.δ n m γ).leftShift a m' hm'
-- CochainComplex.HomComplex.Cochain.δ_leftUnshift
lemma δ_leftUnshift {a n' : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n')
    (m m' : ℤ) (hm' : m + a = m') :
    δ n m (γ.leftUnshift n hn) = a.negOnePow • (δ n' m' γ).leftUnshift m hm' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {a n' : ℤ} (γ : CochainComplex.HomComplex.Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n') (m m' : ℤ) (hm' : m + a = m')
Goal: CochainComplex.HomComplex.δ n m (γ.leftUnshift n hn) =
  a.negOnePow • (CochainComplex.HomComplex.δ n' m' γ).leftUnshift m hm'
-- CochainComplex.HomComplex.Cochain.δ_shift
@[simp]
lemma δ_shift (a m : ℤ) :
    δ n m (γ.shift a) = a.negOnePow • (δ n m γ).shift a 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a m : ℤ)
Goal: CochainComplex.HomComplex.δ n m (γ.shift a) = a.negOnePow • (CochainComplex.HomComplex.δ n m γ).shift a
-- CochainComplex.HomComplex.Cochain.leftShift_rightShift
lemma leftShift_rightShift (a n' : ℤ) (hn' : n' + a = n) :
    (γ.rightShift a n' hn').leftShift a n hn' =
      (a * n + (a * (a - 1)) / 2).negOnePow • γ.shift a 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n' + a = n)
Goal: (γ.rightShift a n' hn').leftShift a n hn' = (a * n + a * (a - 1) / 2).negOnePow • γ.shift a
-- CochainComplex.HomComplex.Cochain.rightShift_leftShift
lemma rightShift_leftShift (a n' : ℤ) (hn' : n + a = n') :
    (γ.leftShift a n' hn').rightShift a n hn' =
      (a * n' + (a * (a - 1)) / 2).negOnePow • γ.shift a 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' : ℤ) (hn' : n + a = n')
Goal: (γ.leftShift a n' hn').rightShift a n hn' = (a * n' + a * (a - 1) / 2).negOnePow • γ.shift a
-- CochainComplex.HomComplex.Cochain.leftShift_rightShift_eq_negOnePow_rightShift_leftShift
lemma leftShift_rightShift_eq_negOnePow_rightShift_leftShift
    (a n' n'' : ℤ) (hn' : n' + a = n) (hn'' : n + a = n'') :
    (γ.rightShift a n' hn').leftShift a n hn' =
      a.negOnePow • (γ.leftShift a n'' hn'').rightShift a n hn'' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cochain K L n) (a n' n'' : ℤ) (hn' : n' + a = n) (hn'' : n + a = n'')
Goal: (γ.rightShift a n' hn').leftShift a n hn' = a.negOnePow • (γ.leftShift a n'' hn'').rightShift a n hn''
-- CochainComplex.HomComplex.Cocycle.rightShift_coe
@[simps!]
def rightShift (γ : Cocycle K L n) (a n' : ℤ) (hn' : n' + a = n) :
    Cocycle K (L⟦a⟧) n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cocycle K L n) (a n' : ℤ) (hn' : n' + a = n)
Goal: ↑(γ.rightShift a n' hn') = (↑γ).rightShift a n' hn'
-- CochainComplex.HomComplex.Cocycle.rightUnshift_coe
@[simps!]
def rightUnshift {n' a : ℤ} (γ : Cocycle K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n) :
    Cocycle K L n 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cocycle K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n)
Goal: ↑(γ.rightUnshift n hn) = (↑γ).rightUnshift n hn
-- CochainComplex.HomComplex.Cocycle.leftShift_coe
@[simps!]
def leftShift (γ : Cocycle K L n) (a n' : ℤ) (hn' : n + a = n') :
    Cocycle (K⟦a⟧) L n' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cocycle K L n) (a n' : ℤ) (hn' : n + a = n')
Goal: ↑(γ.leftShift a n' hn') = (↑γ).leftShift a n' hn'
-- CochainComplex.HomComplex.Cocycle.leftUnshift_coe
@[simps!]
def leftUnshift {n' a : ℤ} (γ : Cocycle (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n') :
    Cocycle K L n 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cocycle ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n') (n : ℤ) (hn : n + a = n')
Goal: ↑(γ.leftUnshift n hn) = (↑γ).leftUnshift n hn
-- CochainComplex.HomComplex.Cocycle.shift_coe
@[simps!]
def shift (γ : Cocycle K L n) (a : ℤ) :
    Cocycle (K⟦a⟧) (L⟦a⟧) n 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {K L : CochainComplex C ℤ} {n : ℤ} (γ : CochainComplex.HomComplex.Cocycle K L n) (a : ℤ)
Goal: ↑(γ.shift a) = (↑γ).shift a
-- CategoryTheory.ShortComplex.LeftHomologyData.map_i
@[simps]
noncomputable def map : (S.map F).LeftHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).i = F.map h.i
-- CategoryTheory.ShortComplex.LeftHomologyData.map_K
@[simps]
noncomputable def map : (S.map F).LeftHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).K = F.obj h.K
-- CategoryTheory.ShortComplex.LeftHomologyData.map_H
@[simps]
noncomputable def map : (S.map F).LeftHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).H = F.obj h.H
-- CategoryTheory.ShortComplex.LeftHomologyData.map_π
@[simps]
noncomputable def map : (S.map F).LeftHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).π = F.map h.π
-- CategoryTheory.ShortComplex.LeftHomologyData.map_f'
@[simp]
lemma map_f' : (h.map F).f' = F.map h.f' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).f' = F.map h.f'
-- CategoryTheory.ShortComplex.LeftHomologyMapData.map_φH
@[simps]
def LeftHomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData}
    {h₂ : S₂.LeftHomologyData} (ψ : LeftHomologyMapData φ h₁ h₂) (F : C ⥤ D)
    [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F] :
    LeftHomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (ψ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F]
Goal: (ψ.map F).φH = F.map ψ.φH
-- CategoryTheory.ShortComplex.LeftHomologyMapData.map_φK
@[simps]
def LeftHomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData}
    {h₂ : S₂.LeftHomologyData} (ψ : LeftHomologyMapData φ h₁ h₂) (F : C ⥤ D)
    [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F] :
    LeftHomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (ψ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F]
Goal: (ψ.map F).φK = F.map ψ.φK
-- CategoryTheory.ShortComplex.RightHomologyData.map_H
@[simps]
noncomputable def map : (S.map F).RightHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).H = F.obj h.H
-- CategoryTheory.ShortComplex.RightHomologyData.map_Q
@[simps]
noncomputable def map : (S.map F).RightHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).Q = F.obj h.Q