declaration stringlengths 27 11.3k | file stringlengths 52 114 | context dict | tactic_states listlengths 1 1.24k |
|---|---|---|---|
theorem iff_existsUnique (aA : a0 ∈ A) (bB : b0 ∈ B) :
UniqueMul A B a0 b0 ↔ ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0 :=
⟨fun _ ↦ ⟨(a0, b0), ⟨Finset.mk_mem_product aA bB, rfl⟩, by simpa⟩,
fun h ↦ h.elim
(by
rintro ⟨x1, x2⟩ _ J x y hx hy l
rcases Prod.mk_inj.mp (J (a0, b0) ⟨Finset.mk_me... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/UniqueProds/Basic.lean | {
"open": [
"Finset"
],
"variables": [
"{G H : Type*} [Mul G] [Mul H] {A B : Finset G} {a0 b0 : G}"
]
} | [
{
"line": "simpa",
"before_state": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nx✝ : UniqueMul A B a0 b0\n⊢ ∀ (y : G × G), (fun ab => ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0) y → y = (a0, b0)",
"after_state": "No Goals!"
},
{
"line": "rintro ⟨x1, x2⟩ _ J x y hx ... |
theorem isUnit_iff_mulLeft_bijective {a : M} :
IsUnit a ↔ Function.Bijective (a * ·) :=
⟨fun h ↦ ⟨h.mul_right_injective, fun y ↦ ⟨h.unit⁻¹ * y, by simp [← mul_assoc]⟩⟩, fun h ↦
⟨⟨a, _, (h.2 1).choose_spec, h.1
(by simpa [mul_assoc] using congr_arg (· * a) (h.2 1).choose_spec)⟩, rfl⟩⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type u}",
"[Monoid α]",
"(b c : αˣ) {u : αˣ}",
"[Monoid α]",
"[LeftCancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[RightCancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[CancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[CommMonoid α]",... | [
{
"line": "simp [← mul_assoc]",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\na : M\nh : IsUnit a\ny : M\n⊢ (fun x => a * x) (↑h.unit⁻¹ * y) = y",
"after_state": "No Goals!"
},
{
"line": "simpa [mul_assoc] using congr_arg (· * a) (h.2 1).choose_spec",
"before_state": "M : Type u_1\... |
theorem isUnit_iff_mulRight_bijective {a : M} :
IsUnit a ↔ Function.Bijective (· * a) :=
⟨fun h ↦ ⟨h.mul_left_injective, fun y ↦ ⟨y * h.unit⁻¹, by simp [mul_assoc]⟩⟩,
fun h ↦ ⟨⟨a, _, h.1 (by simpa [mul_assoc] using congr_arg (a * ·) (h.2 1).choose_spec),
(h.2 1).choose_spec⟩, rfl⟩⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type u}",
"[Monoid α]",
"(b c : αˣ) {u : αˣ}",
"[Monoid α]",
"[LeftCancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[RightCancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[CancelMonoid α] [Subsingleton αˣ] {a b : α}",
"[CommMonoid α]",... | [
{
"line": "simp [mul_assoc]",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\na : M\nh : IsUnit a\ny : M\n⊢ (fun x => x * a) (y * ↑h.unit⁻¹) = y",
"after_state": "No Goals!"
},
{
"line": "simpa [mul_assoc] using congr_arg (a * ·) (h.2 1).choose_spec",
"before_state": "M : Type u_1\ni... |
theorem inv_mul_of_eq {a : α} (h : ↑u = a) : ↑u⁻¹ * a = 1 := by rw [← h, u.inv_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Defs.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type u}",
"[Monoid α]",
"(a b : αˣ) {u : αˣ}"
]
} | [
{
"line": "rw [← h, u.inv_mul]",
"before_state": "α : Type u\ninst✝ : Monoid α\nu : αˣ\na : α\nh : ↑u = a\n⊢ ↑u⁻¹ * a = 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [← h, u.inv_mul]",
"before_state": "α : Type u\ninst✝ : Monoid α\nu : αˣ\na : α\nh : ↑u = a\n⊢ ↑u⁻¹ * a = 1",
"afte... |
theorem mul_inv_of_eq {a : α} (h : ↑u = a) : a * ↑u⁻¹ = 1 := by rw [← h, u.mul_inv]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Defs.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type u}",
"[Monoid α]",
"(a b : αˣ) {u : αˣ}"
]
} | [
{
"line": "rw [← h, u.mul_inv]",
"before_state": "α : Type u\ninst✝ : Monoid α\nu : αˣ\na : α\nh : ↑u = a\n⊢ a * ↑u⁻¹ = 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [← h, u.mul_inv]",
"before_state": "α : Type u\ninst✝ : Monoid α\nu : αˣ\na : α\nh : ↑u = a\n⊢ a * ↑u⁻¹ = 1",
"afte... |
theorem mul_val_inv (h : IsUnit a) : a * ↑h.unit⁻¹ = 1 := by
rw [← h.unit.mul_inv]; congr
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Defs.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type u}",
"[Monoid α]",
"(a b : αˣ) {u : αˣ}",
"[DivisionMonoid α]",
"[Monoid α] {a : α}",
"{M : Type*} {N : Type*}",
"[Monoid M] {a b : M}",
"[Monoid M] {a : M}"
]
} | [
{
"line": "rw [← h.unit.mul_inv]",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\na : M\nh : IsUnit a\n⊢ a * ↑h.unit⁻¹ = 1",
"after_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\na : M\nh : IsUnit a\n⊢ a * ↑h.unit⁻¹ = ↑h.unit * ↑h.unit⁻¹"
},
{
"line": "rewrite [← h.unit.mul_inv]",
... |
theorem map [MonoidHomClass F M N] (f : F) {x : M} (h : IsUnit x) : IsUnit (f x) := by
rcases h with ⟨y, rfl⟩; exact (Units.map (f : M →* N) y).isUnit
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type*} {M : Type u} {N : Type v} {P : Type w} [Monoid M] [Monoid N] [Monoid P]",
"(M)",
"{M}",
"[DivisionMonoid α]",
"{F G M N : Type*} [FunLike F M N] [FunLike G N M]",
"[Monoid M] [Monoid N]"
]
} | [
{
"line": "rcases h with ⟨y, rfl⟩",
"before_state": "F : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝³ : FunLike F M N\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass F M N\nf : F\nx : M\nh : IsUnit x\n⊢ IsUnit (f x)",
"after_state": "case intro\nF : Type u_2\nM : Type u_4\nN : Type u_5\nin... |
theorem of_leftInverse [MonoidHomClass G N M] {f : F} {x : M} (g : G)
(hfg : Function.LeftInverse g f) (h : IsUnit (f x)) : IsUnit x := by
simpa only [hfg x] using h.map g
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type*} {M : Type u} {N : Type v} {P : Type w} [Monoid M] [Monoid N] [Monoid P]",
"(M)",
"{M}",
"[DivisionMonoid α]",
"{F G M N : Type*} [FunLike F M N] [FunLike G N M]",
"[Monoid M] [Monoid N]"
]
} | [
{
"line": "simpa only [hfg x] using h.map g",
"before_state": "F : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike G N M\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass G N M\nf : F\nx : M\ng : G\nhfg : LeftInverse ⇑g ⇑f\nh : IsUnit (f x)\n⊢ IsUnit ... |
theorem mul_liftRight_inv (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :
f x * ↑(IsUnit.liftRight f h x)⁻¹ = 1 := Units.mul_liftRight_inv (by intro; rfl) x
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type*} {M : Type u} {N : Type v} {P : Type w} [Monoid M] [Monoid N] [Monoid P]",
"(M)",
"{M}",
"[DivisionMonoid α]",
"{F G M N : Type*} [FunLike F M N] [FunLike G N M]",
"[Monoid M] [Monoid N]"
]
} | [
{
"line": "intro",
"before_state": "M : Type u_4\nN : Type u_5\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (f x)\nx : M\n⊢ ∀ (x : M), ↑⋯.unit = f x",
"after_state": "M : Type u_4\nN : Type u_5\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (f x)\nx x✝... |
theorem liftRight_inv_mul (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :
↑(IsUnit.liftRight f h x)⁻¹ * f x = 1 := Units.liftRight_inv_mul (by intro; rfl) x
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Units/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{α : Type*} {M : Type u} {N : Type v} {P : Type w} [Monoid M] [Monoid N] [Monoid P]",
"(M)",
"{M}",
"[DivisionMonoid α]",
"{F G M N : Type*} [FunLike F M N] [FunLike G N M]",
"[Monoid M] [Monoid N]"
]
} | [
{
"line": "intro",
"before_state": "M : Type u_4\nN : Type u_5\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (f x)\nx : M\n⊢ ∀ (x : M), ↑⋯.unit = f x",
"after_state": "M : Type u_4\nN : Type u_5\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (f x)\nx x✝... |
theorem map {M N : Type*} [Monoid M] [Monoid N] {F : Type*} [FunLike F M N] [MonoidHomClass F M N]
(f : F) {x y : M} (ha : Associated x y) : Associated (f x) (f y) := by
obtain ⟨u, ha⟩ := ha
exact ⟨Units.map f u, by rw [← ha, map_mul, Units.coe_map, MonoidHom.coe_coe]⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Associated.lean | {
"open": [],
"variables": [
"{M : Type*}"
]
} | [
{
"line": "obtain ⟨u, ha⟩ := ha",
"before_state": "M : Type u_2\nN : Type u_3\ninst✝³ : Monoid M\ninst✝² : Monoid N\nF : Type u_4\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\nx y : M\nha : Associated x y\n⊢ Associated (f x) (f y)",
"after_state": "case intro\nM : Type u_2\nN : Type u_3\... |
private theorem inv_eq_of_mul (h : a * b = 1) : a⁻¹ = b := by
rw [← inv_mul_cancel_left₀ (left_ne_zero_of_mul_eq_one h) b]
rw [h]
rw [mul_one]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{M₀ G₀ : Type*}",
"[MulZeroClass M₀] {a b : M₀}",
"[Mul M₀] [Zero M₀] [NoZeroDivisors M₀] {a b : M₀}",
"[MulZeroOneClass M₀]",
"[MulZeroOneClass M₀] [Nontrivial M₀] {a b : M₀}",
"[MonoidWithZero M₀] {a : M₀} {n : ℕ}",
"[NoZeroDivisors M₀]",... | [
{
"line": "rw [← inv_mul_cancel_left₀ (left_ne_zero_of_mul_eq_one h) b]",
"before_state": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na b : G₀\nh : a * b = 1\n⊢ a⁻¹ = b",
"after_state": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na b : G₀\nh : a * b = 1\n⊢ a⁻¹ = a⁻¹ * (a * b)"
},
{
"line": "rewrite ... |
lemma inverse_pow (r : M₀) : ∀ n : ℕ, Ring.inverse r ^ n = Ring.inverse (r ^ n)
| 0 => by rw [pow_zero, pow_zero, Ring.inverse_one]
| n + 1 => by
rw [pow_succ']
rw [pow_succ]
rw [Ring.mul_inverse_rev' ((Commute.refl r).pow_left n)]
rw [Ring.inverse_pow r n]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Commute.lean | {
"open": [],
"variables": [
"{M₀ G₀ : Type*}",
"[MonoidWithZero M₀]"
]
} | [
{
"line": "rw [pow_zero, pow_zero, Ring.inverse_one]",
"before_state": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\nr : M₀\n⊢ Ring.inverse r ^ 0 = Ring.inverse (r ^ 0)",
"after_state": "No Goals!"
},
{
"line": "rewrite [pow_zero, pow_zero, Ring.inverse_one]",
"before_state": "M₀ : Type u_1\nin... |
lemma inverse_pow_mul_eq_iff_eq_mul {a : M₀} (b c : M₀) (ha : IsUnit a) {k : ℕ} :
Ring.inverse a ^ k * b = c ↔ b = a ^ k * c := by
rw [Ring.inverse_pow]
rw [Ring.inverse_mul_eq_iff_eq_mul _ _ _ (IsUnit.pow _ ha)]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Commute.lean | {
"open": [],
"variables": [
"{M₀ G₀ : Type*}",
"[MonoidWithZero M₀]"
]
} | [
{
"line": "rw [Ring.inverse_pow]",
"before_state": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\na b c : M₀\nha : IsUnit a\nk : ℕ\n⊢ Ring.inverse a ^ k * b = c ↔ b = a ^ k * c",
"after_state": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\na b c : M₀\nha : IsUnit a\nk : ℕ\n⊢ Ring.inverse (a ^ k) * b = c ↔ b = ... |
theorem div_left (hac : Commute a c) (hbc : Commute b c) : Commute (a / b) c := by
rw [div_eq_mul_inv]
exact hac.mul_left hbc.inv_left₀
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Commute.lean | {
"open": [],
"variables": [
"{M₀ G₀ : Type*}",
"[MonoidWithZero M₀]",
"[GroupWithZero G₀] {a b c : G₀}"
]
} | [
{
"line": "rw [div_eq_mul_inv]",
"before_state": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na b c : G₀\nhac : Commute a c\nhbc : Commute b c\n⊢ Commute (a / b) c",
"after_state": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na b c : G₀\nhac : Commute a c\nhbc : Commute b c\n⊢ Commute (a * b⁻¹) c"
},
{
... |
lemma of_injective {f : F} (hf : Injective f) [NeZero a] : NeZero (f a) :=
⟨by rw [← ZeroHomClass.map_zero f]; exact hf.ne NeZero.out⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{F α β : Type*} [Zero α] [Zero β] [FunLike F α β] [ZeroHomClass F α β] {a : α}"
]
} | [
{
"line": "rw [← ZeroHomClass.map_zero f]",
"before_state": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : FunLike F α β\ninst✝¹ : ZeroHomClass F α β\na : α\nf : F\nhf : Injective ⇑f\ninst✝ : NeZero a\n⊢ f a ≠ 0",
"after_state": "F : Type u_1\nα : Type u_2\nβ : Typ... |
theorem map_ite_zero_one {F : Type*} [FunLike F α β] [MonoidWithZeroHomClass F α β] (f : F)
(p : Prop) [Decidable p] :
f (ite p 0 1) = ite p 0 1 := by
split_ifs with h <;> simp [h]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{F α β : Type*} [Zero α] [Zero β] [FunLike F α β] [ZeroHomClass F α β] {a : α}",
"{F α β γ δ M₀ : Type*} [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ]",
"[FunLike F α β]"
]
} | [
{
"line": "focus\n split_ifs with h\n with_annotate_state\"<;>\" skip\n all_goals simp [h]",
"before_state": "α : Type u_5\nβ : Type u_6\ninst✝⁴ : MulZeroOneClass α\ninst✝³ : MulZeroOneClass β\nF : Type u_10\ninst✝² : FunLike F α β\ninst✝¹ : MonoidWithZeroHomClass F α β\nf : F\np : Prop\ninst✝ : Decidabl... |
theorem map_ite_one_zero {F : Type*} [FunLike F α β] [MonoidWithZeroHomClass F α β] (f : F)
(p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 := by
split_ifs with h <;> simp [h]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Hom.lean | {
"open": [
"Function"
],
"variables": [
"{F α β : Type*} [Zero α] [Zero β] [FunLike F α β] [ZeroHomClass F α β] {a : α}",
"{F α β γ δ M₀ : Type*} [MulZeroOneClass α] [MulZeroOneClass β] [MulZeroOneClass γ]",
"[FunLike F α β]"
]
} | [
{
"line": "focus\n split_ifs with h\n with_annotate_state\"<;>\" skip\n all_goals simp [h]",
"before_state": "α : Type u_5\nβ : Type u_6\ninst✝⁴ : MulZeroOneClass α\ninst✝³ : MulZeroOneClass β\nF : Type u_10\ninst✝² : FunLike F α β\ninst✝¹ : MonoidWithZeroHomClass F α β\nf : F\np : Prop\ninst✝ : Decidabl... |
lemma indicator_mul (s : Set ι) (f g : ι → M₀) :
indicator s (fun i ↦ f i * g i) = fun i ↦ indicator s f i * indicator s g i := by
funext
simp only [indicator]
split_ifs
· rfl
rw [mul_zero]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "funext",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns : Set ι\nf g : ι → M₀\n⊢ (s.indicator fun i => f i * g i) = fun i => s.indicator f i * s.indicator g i",
"after_state": "case h\nι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns : Set ι\nf g : ι → M₀\nx✝ ... |
lemma indicator_mul_left (s : Set ι) (f g : ι → M₀) :
indicator s (fun j ↦ f j * g j) i = indicator s f i * g i := by
simp only [indicator]
split_ifs
· rfl
· rw [zero_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "simp only [indicator]",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢ s.indicator (fun j => f j * g j) i = s.indicator f i * g i",
"after_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢... |
lemma indicator_mul_right (s : Set ι) (f g : ι → M₀) :
indicator s (fun j ↦ f j * g j) i = f i * indicator s g i := by
simp only [indicator]
split_ifs
· rfl
· rw [mul_zero]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "simp only [indicator]",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢ s.indicator (fun j => f j * g j) i = f i * s.indicator g i",
"after_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢... |
lemma indicator_mul_const (s : Set ι) (f : ι → M₀) (a : M₀) (i : ι) :
s.indicator (f · * a) i = s.indicator f i * a := by rw [indicator_mul_left]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "rw [indicator_mul_left]",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns : Set ι\nf : ι → M₀\na : M₀\ni : ι\n⊢ s.indicator (fun x => f x * a) i = s.indicator f i * a",
"after_state": "No Goals!"
},
{
"line": "rewrite [indicator_mul_left]",
"before_state":... |
lemma indicator_const_mul (s : Set ι) (f : ι → M₀) (a : M₀) (i : ι) :
s.indicator (a * f ·) i = a * s.indicator f i := by rw [indicator_mul_right]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "rw [indicator_mul_right]",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns : Set ι\nf : ι → M₀\na : M₀\ni : ι\n⊢ s.indicator (fun x => a * f x) i = a * s.indicator f i",
"after_state": "No Goals!"
},
{
"line": "rewrite [indicator_mul_right]",
"before_state... |
lemma inter_indicator_mul (f g : ι → M₀) (i : ι) :
(s ∩ t).indicator (fun j ↦ f j * g j) i = s.indicator f i * t.indicator g i := by
rw [← Set.indicator_indicator]
simp_rw [indicator]
split_ifs <;> simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "rw [← Set.indicator_indicator]",
"before_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns t : Set ι\nf g : ι → M₀\ni : ι\n⊢ (s ∩ t).indicator (fun j => f j * g j) i = s.indicator f i * t.indicator g i",
"after_state": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ns t ... |
lemma indicator_prod_one {t : Set κ} {j : κ} :
(s ×ˢ t).indicator (1 : ι × κ → M₀) (i, j) = s.indicator 1 i * t.indicator 1 j := by
simp_rw [indicator, mem_prod_eq]
split_ifs with h₀ <;> simp only [Pi.one_apply, mul_one, mul_zero] <;> tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Indicator.lean | {
"open": [
"Set"
],
"variables": [
"{ι κ G₀ M₀ R : Type*}",
"[MulZeroClass M₀] {s t : Set ι} {i : ι}",
"[MulZeroOneClass M₀] {s t : Set ι} {i : ι}"
]
} | [
{
"line": "simp_rw [indicator, mem_prod_eq]",
"before_state": "ι : Type u_1\nκ : Type u_2\nM₀ : Type u_4\ninst✝¹ : MulZeroClass M₀\ninst✝ : MulZeroOneClass M₀\ns : Set ι\ni : ι\nt : Set κ\nj : κ\n⊢ (s ×ˢ t).indicator 1 (i, j) = s.indicator 1 i * t.indicator 1 j",
"after_state": "ι : Type u_1\nκ : Type u... |
theorem ofAdd_zpow (a : ℤ) : (↑(ofAdd a) : ℤₘ₀) = ofAdd (1 : ℤ) ^ a := by
rw [← WithZero.coe_zpow]
rw [WithZero.coe_inj]
rw [← Int.ofAdd_mul]
rw [one_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Int.lean | {
"open": [
"Multiplicative"
],
"variables": []
} | [
{
"line": "rw [← WithZero.coe_zpow]",
"before_state": "a : ℤ\n⊢ ↑(ofAdd a) = ↑(ofAdd 1) ^ a",
"after_state": "a : ℤ\n⊢ ↑(ofAdd a) = ↑(ofAdd 1 ^ a)"
},
{
"line": "rewrite [← WithZero.coe_zpow]",
"before_state": "a : ℤ\n⊢ ↑(ofAdd a) = ↑(ofAdd 1) ^ a",
"after_state": "a : ℤ\n⊢ ↑(ofAdd a) = ... |
theorem zero_right [MulZeroClass G₀] (a : G₀) : SemiconjBy a 0 0 := by
simp only [SemiconjBy]
simp only [mul_zero]
simp only [zero_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}"
]
} | [
{
"line": "simp only [SemiconjBy]",
"before_state": "G₀ : Type u_1\ninst✝ : MulZeroClass G₀\na : G₀\n⊢ SemiconjBy a 0 0",
"after_state": "G₀ : Type u_1\ninst✝ : MulZeroClass G₀\na : G₀\n⊢ a * 0 = 0 * a"
},
{
"line": "simp only [mul_zero]",
"before_state": "G₀ : Type u_1\ninst✝ : MulZeroClass... |
theorem zero_left [MulZeroClass G₀] (x y : G₀) : SemiconjBy 0 x y := by
simp only [SemiconjBy]
simp only [mul_zero]
simp only [zero_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}"
]
} | [
{
"line": "simp only [SemiconjBy]",
"before_state": "G₀ : Type u_1\ninst✝ : MulZeroClass G₀\nx y : G₀\n⊢ SemiconjBy 0 x y",
"after_state": "G₀ : Type u_1\ninst✝ : MulZeroClass G₀\nx y : G₀\n⊢ 0 * x = y * 0"
},
{
"line": "simp only [mul_zero]",
"before_state": "G₀ : Type u_1\ninst✝ : MulZeroC... |
theorem inv_symm_left_iff₀ : SemiconjBy a⁻¹ x y ↔ SemiconjBy a y x :=
Classical.by_cases (fun ha : a = 0 => by simp only [ha, inv_zero, SemiconjBy.zero_left]) fun ha =>
@units_inv_symm_left_iff _ _ (Units.mk0 a ha) _ _
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}",
"[GroupWithZero G₀] {a x y x' y' : G₀}"
]
} | [
{
"line": "simp only [ha, inv_zero, SemiconjBy.zero_left]",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y : G₀\nha : a = 0\n⊢ SemiconjBy a⁻¹ x y ↔ SemiconjBy a y x",
"after_state": "No Goals!"
}
] |
theorem inv_right₀ (h : SemiconjBy a x y) : SemiconjBy a x⁻¹ y⁻¹ := by
by_cases ha : a = 0
· simp only [ha, zero_left]
by_cases hx : x = 0
· subst x
simp only [SemiconjBy] at h
simp only [mul_zero] at h
simp only [@eq_comm _ _ (y * a)] at h
simp only [mul_eq_zero] at h
simp [h.resolve_right ... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}",
"[GroupWithZero G₀] {a x y x' y' : G₀}"
]
} | [
{
"line": "by_cases ha : a = 0",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y : G₀\nh : SemiconjBy a x y\n⊢ SemiconjBy a x⁻¹ y⁻¹",
"after_state": "case pos\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y : G₀\nh : SemiconjBy a x y\nha : a = 0\n⊢ SemiconjBy a x⁻¹ y⁻¹\n---\ncase neg\nG₀... |
theorem div_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x / x') (y / y') := by
rw [div_eq_mul_inv]
rw [div_eq_mul_inv]
exact h.mul_right h'.inv_right₀
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}",
"[GroupWithZero G₀] {a x y x' y' : G₀}"
]
} | [
{
"line": "rw [div_eq_mul_inv]",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y x' y' : G₀\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n⊢ SemiconjBy a (x / x') (y / y')",
"after_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y x' y' : G₀\nh : SemiconjBy a x y\nh' : SemiconjBy ... |
lemma zpow_right₀ {a x y : G₀} (h : SemiconjBy a x y) : ∀ m : ℤ, SemiconjBy a (x ^ m) (y ^ m)
| (n : ℕ) => by simp [h.pow_right n]
| .negSucc n => by simp only [zpow_negSucc, (h.pow_right (n + 1)).inv_right₀]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Semiconj.lean | {
"open": [],
"variables": [
"{G₀ : Type*}",
"[GroupWithZero G₀] {a x y x' y' : G₀}"
]
} | [
{
"line": "simp [h.pow_right n]",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na x y : G₀\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ ↑n) (y ^ ↑n)",
"after_state": "No Goals!"
},
{
"line": "simp only [zpow_negSucc, (h.pow_right (n + 1)).inv_right₀]",
"before_state": "G₀ : ... |
theorem smul_zero_mk {α : Type*} [Monoid M] [AddMonoid α] [DistribMulAction M α] (a : M) (c : β) :
a • ((0 : α), c) = (0, a • c) := by rw [Prod.smul_mk, smul_zero]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Action/Prod.lean | {
"open": [],
"variables": [
"{M N α β : Type*}",
"[SMul M α] [SMul M β]"
]
} | [
{
"line": "rw [Prod.smul_mk, smul_zero]",
"before_state": "M : Type u_1\nβ : Type u_4\ninst✝³ : SMul M β\nα : Type u_5\ninst✝² : Monoid M\ninst✝¹ : AddMonoid α\ninst✝ : DistribMulAction M α\na : M\nc : β\n⊢ a • (0, c) = (0, a • c)",
"after_state": "No Goals!"
},
{
"line": "rewrite [Prod.smul_mk,... |
theorem smul_mk_zero {β : Type*} [Monoid M] [AddMonoid β] [DistribMulAction M β] (a : M) (b : α) :
a • (b, (0 : β)) = (a • b, 0) := by rw [Prod.smul_mk, smul_zero]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Action/Prod.lean | {
"open": [],
"variables": [
"{M N α β : Type*}",
"[SMul M α] [SMul M β]"
]
} | [
{
"line": "rw [Prod.smul_mk, smul_zero]",
"before_state": "M : Type u_1\nα : Type u_3\ninst✝³ : SMul M α\nβ : Type u_5\ninst✝² : Monoid M\ninst✝¹ : AddMonoid β\ninst✝ : DistribMulAction M β\na : M\nb : α\n⊢ a • (b, 0) = (a • b, 0)",
"after_state": "No Goals!"
},
{
"line": "rewrite [Prod.smul_mk,... |
lemma zero_mem_smul_set_iff (ha : a ≠ 0) : (0 : β) ∈ a • t ↔ (0 : β) ∈ t := by
refine ⟨?_, zero_mem_smul_set⟩
rintro ⟨b, hb, h⟩
rwa [(eq_zero_or_eq_zero_of_smul_eq_zero h).resolve_left ha] at hb
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Action/Pointwise/Set.lean | {
"open": [
"Function",
"scoped Pointwise"
],
"variables": [
"{α β : Type*}",
"[Zero β] [SMulZeroClass α β] {s : Set α} {t : Set β} {a : α}",
"[Zero α] [NoZeroSMulDivisors α β]"
]
} | [
{
"line": "refine ⟨?_, zero_mem_smul_set⟩",
"before_state": "α : Type u_1\nβ : Type u_2\ninst✝³ : Zero β\ninst✝² : SMulZeroClass α β\nt : Set β\na : α\ninst✝¹ : Zero α\ninst✝ : NoZeroSMulDivisors α β\nha : a ≠ 0\n⊢ 0 ∈ a • t ↔ 0 ∈ t",
"after_state": "α : Type u_1\nβ : Type u_2\ninst✝³ : Zero β\ninst✝² :... |
theorem mul_left_eq_zero (u : M₀ˣ) {a : M₀} : a * u = 0 ↔ a = 0 :=
⟨fun h => by simpa using mul_eq_zero_of_left h ↑u⁻¹, fun h => mul_eq_zero_of_left h u⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Units/Basic.lean | {
"open": [],
"variables": [
"{α M₀ G₀ : Type*}",
"[MonoidWithZero M₀]"
]
} | [
{
"line": "simpa using mul_eq_zero_of_left h ↑u⁻¹",
"before_state": "M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : a * ↑u = 0\n⊢ a = 0",
"after_state": "No Goals!"
}
] |
theorem mul_right_eq_zero (u : M₀ˣ) {a : M₀} : ↑u * a = 0 ↔ a = 0 :=
⟨fun h => by simpa using mul_eq_zero_of_right (↑u⁻¹) h, mul_eq_zero_of_right (u : M₀)⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Units/Basic.lean | {
"open": [],
"variables": [
"{α M₀ G₀ : Type*}",
"[MonoidWithZero M₀]"
]
} | [
{
"line": "simpa using mul_eq_zero_of_right (↑u⁻¹) h",
"before_state": "M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : ↑u * a = 0\n⊢ a = 0",
"after_state": "No Goals!"
}
] |
theorem mk0_inj {a b : G₀} (ha : a ≠ 0) (hb : b ≠ 0) : Units.mk0 a ha = Units.mk0 b hb ↔ a = b :=
⟨fun h => by injection h, fun h => Units.ext h⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Units/Basic.lean | {
"open": [
"Classical in"
],
"variables": [
"{α M₀ G₀ : Type*}",
"[MonoidWithZero M₀]",
"(M₀)",
"{M₀}",
"[GroupWithZero G₀]"
]
} | [
{
"line": "injection h",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\na b : G₀\nha : a ≠ 0\nhb : b ≠ 0\nh : Units.mk0 a ha = Units.mk0 b hb\n⊢ a = b",
"after_state": "No Goals!"
}
] |
theorem exists_iff_ne_zero {p : G₀ → Prop} : (∃ u : G₀ˣ, p u) ↔ ∃ x ≠ 0, p x := by
simp [exists0]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/GroupWithZero/Units/Basic.lean | {
"open": [
"Classical in"
],
"variables": [
"{α M₀ G₀ : Type*}",
"[MonoidWithZero M₀]",
"(M₀)",
"{M₀}",
"[GroupWithZero G₀]"
]
} | [
{
"line": "simp [exists0]",
"before_state": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\np : G₀ → Prop\n⊢ (∃ u, p ↑u) ↔ ∃ x, x ≠ 0 ∧ p x",
"after_state": "No Goals!"
}
] |
lemma next_eq_self' (c : ComplexShape ι) (j : ι) (hj : ∀ k, ¬ c.Rel j k) :
c.next j = j :=
dif_neg (by simpa using hj)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/ComplexShape.lean | {
"open": [
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}"
]
} | [
{
"line": "simpa using hj",
"before_state": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ¬∃ j_1, c.Rel j j_1",
"after_state": "No Goals!"
}
] |
lemma prev_eq_self' (c : ComplexShape ι) (j : ι) (hj : ∀ i, ¬ c.Rel i j) :
c.prev j = j :=
dif_neg (by simpa using hj)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/ComplexShape.lean | {
"open": [
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}"
]
} | [
{
"line": "simpa using hj",
"before_state": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (i : ι), ¬c.Rel i j\n⊢ ¬∃ i, c.Rel i j",
"after_state": "No Goals!"
}
] |
lemma next_eq_self (c : ComplexShape ι) (j : ι) (hj : ¬ c.Rel j (c.next j)) :
c.next j = j :=
c.next_eq_self' j (fun k hk' => hj (by simpa only [c.next_eq' hk'] using hk'))
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/ComplexShape.lean | {
"open": [
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}"
]
} | [
{
"line": "simpa only [c.next_eq' hk'] using hk'",
"before_state": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ¬c.Rel j (c.next j)\nk : ι\nhk' : c.Rel j k\n⊢ c.Rel j (c.next j)",
"after_state": "No Goals!"
}
] |
lemma prev_eq_self (c : ComplexShape ι) (j : ι) (hj : ¬ c.Rel (c.prev j) j) :
c.prev j = j :=
c.prev_eq_self' j (fun k hk' => hj (by simpa only [c.prev_eq' hk'] using hk'))
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/ComplexShape.lean | {
"open": [
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}"
]
} | [
{
"line": "simpa only [c.prev_eq' hk'] using hk'",
"before_state": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ¬c.Rel (c.prev j) j\nk : ι\nhk' : c.Rel k j\n⊢ c.Rel (c.prev j) j",
"after_state": "No Goals!"
}
] |
lemma shape_f (K : HomologicalComplex₂ C c₁ c₂) (i₁ i₁' : I₁) (h : ¬ c₁.Rel i₁ i₁') (i₂ : I₂) :
(K.d i₁ i₁').f i₂ = 0 := by
rw [K.shape _ _ h]
rw [zero_f]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalBicomplex.lean | {
"open": [
"CategoryTheory Limits",
"HomologicalComplex"
],
"variables": [
"(C : Type*) [Category C] [HasZeroMorphisms C]",
"{C c₁ c₂}",
"(C c₁ c₂) in",
"(c₁ c₂)",
"(X : GradedObject (I₁ × I₂) C)"
]
} | [
{
"line": "rw [K.shape _ _ h]",
"before_state": "I₁ : Type u_2\nI₂ : Type u_3\nC : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\ninst✝ : HasZeroMorphisms C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni₁ i₁' : I₁\nh : ¬c₁.Rel i₁ i₁'\ni₂ : I₂\n⊢ (K.d i₁ i₁').f i₂ = 0",
"after... |
lemma d_f_comp_d_f (K : HomologicalComplex₂ C c₁ c₂)
(i₁ i₁' i₁'' : I₁) (i₂ : I₂) :
(K.d i₁ i₁').f i₂ ≫ (K.d i₁' i₁'').f i₂ = 0 := by
rw [← comp_f]
rw [d_comp_d]
rw [zero_f]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalBicomplex.lean | {
"open": [
"CategoryTheory Limits",
"HomologicalComplex"
],
"variables": [
"(C : Type*) [Category C] [HasZeroMorphisms C]",
"{C c₁ c₂}",
"(C c₁ c₂) in",
"(c₁ c₂)",
"(X : GradedObject (I₁ × I₂) C)"
]
} | [
{
"line": "rw [← comp_f]",
"before_state": "I₁ : Type u_2\nI₂ : Type u_3\nC : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\ninst✝ : HasZeroMorphisms C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni₁ i₁' i₁'' : I₁\ni₂ : I₂\n⊢ (K.d i₁ i₁').f i₂ ≫ (K.d i₁' i₁'').f i₂ = 0",
"aft... |
lemma d_comm (K : HomologicalComplex₂ C c₁ c₂) (i₁ i₁' : I₁) (i₂ i₂' : I₂) :
(K.d i₁ i₁').f i₂ ≫ (K.X i₁').d i₂ i₂' = (K.X i₁).d i₂ i₂' ≫ (K.d i₁ i₁').f i₂' := by
simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalBicomplex.lean | {
"open": [
"CategoryTheory Limits",
"HomologicalComplex"
],
"variables": [
"(C : Type*) [Category C] [HasZeroMorphisms C]",
"{C c₁ c₂}",
"(C c₁ c₂) in",
"(c₁ c₂)",
"(X : GradedObject (I₁ × I₂) C)"
]
} | [
{
"line": "simp",
"before_state": "I₁ : Type u_2\nI₂ : Type u_3\nC : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\ninst✝ : HasZeroMorphisms C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\n⊢ (K.d i₁ i₁').f i₂ ≫ (K.X i₁').d i₂ i₂' = (K.X i₁).d i₂ i₂' ≫ (K.... |
lemma mem_quasiIso_iff (f : K ⟶ L) : quasiIso C c f ↔ QuasiIso f := by rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/QuasiIso.lean | {
"open": [
"CategoryTheory Limits",
"HomologicalComplex"
],
"variables": [
"{ι : Type*} {C : Type u} [Category.{v} C] [HasZeroMorphisms C]",
"{C₁ C₂ : Type*} [Category C₁] [Category C₂] [Preadditive C₁] [Preadditive C₂]",
"(i : ι) [K.HasHomology i] [L.HasHomology i]",
"[∀ i, K.HasHomology... | [
{
"line": "rfl",
"before_state": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nc : ComplexShape ?m.713\ninst✝ : CategoryWithHomology C\nι✝ : Type u_4\nK L : HomologicalComplex C c\nf : K ⟶ L\n⊢ quasiIso C c f ↔ QuasiIso f",
"after_state": "No Goals!"
},
{
"line": "exact I... |
theorem NatTrans.mapHomologicalComplex_naturality {c : ComplexShape ι} {F G : W₁ ⥤ W₂}
[F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms]
(α : F ⟶ G) {C D : HomologicalComplex W₁ c} (f : C ⟶ D) :
(F.mapHomologicalComplex c).map f ≫ (NatTrans.mapHomologicalComplex α c).app D =
(NatTrans.mapHomologic... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Additive.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits HomologicalComplex"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{W : Type*} [Category W] [Preadditive W]",
"{W₁ W₂ : Type*} [Category W₁] [Category W₂] [HasZeroMorphisms W₁] [HasZer... | [
{
"line": "simp",
"before_state": "ι : Type u_1\nW₁ : Type u_3\nW₂ : Type u_4\ninst✝⁵ : Category.{u_5, u_3} W₁\ninst✝⁴ : Category.{u_6, u_4} W₂\ninst✝³ : HasZeroMorphisms W₁\ninst✝² : HasZeroMorphisms W₂\nc : ComplexShape ι\nF G : W₁ ⥤ W₂\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : G.PreservesZeroMorphisms\... |
theorem chainComplex_d_succ_succ_zero (C : ChainComplex V ℕ) (i : ℕ) : C.d (i + 2) 0 = 0 := by
rw [C.shape]
exact i.succ_succ_ne_one.symm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Augment.lean | {
"open": [
"CategoryTheory Limits HomologicalComplex"
],
"variables": [
"{V : Type u} [Category.{v} V]",
"[HasZeroMorphisms V]"
]
} | [
{
"line": "rw [C.shape]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : ChainComplex V ℕ\ni : ℕ\n⊢ C.d (i + 2) 0 = 0",
"after_state": "case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : ChainComplex V ℕ\ni : ℕ\n⊢ ¬(ComplexShape.down... |
theorem cochainComplex_d_succ_succ_zero (C : CochainComplex V ℕ) (i : ℕ) : C.d 0 (i + 2) = 0 := by
rw [C.shape]
simp only [ComplexShape.up_Rel]
simp only [zero_add]
exact (Nat.one_lt_succ_succ _).ne
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Augment.lean | {
"open": [
"CategoryTheory Limits HomologicalComplex"
],
"variables": [
"{V : Type u} [Category.{v} V]",
"[HasZeroMorphisms V]",
"[HasZeroMorphisms V]"
]
} | [
{
"line": "rw [C.shape]",
"before_state": "V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ inst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ C.d 0 (i + 2) = 0",
"after_state": "case a\nV : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ inst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\ni : ℕ\n⊢ ¬... |
theorem objEqToHom_d {x y : β} (h : x = y) :
X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom (by cases h; rfl) := by cases h; dsimp; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DifferentialObject.lean | {
"open": [
"CategoryTheory CategoryTheory.Limits"
],
"variables": [
"{β : Type*} [AddCommGroup β] {b : β}",
"{V : Type*} [Category V] [HasZeroMorphisms V]",
"(X : DifferentialObject ℤ (GradedObjectWithShift b V))"
]
} | [
{
"line": "cases h",
"before_state": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{?u.578, u_2} V\ninst✝ : HasZeroMorphisms V\nX : DifferentialObject ℤ (GradedObjectWithShift b V)\nx y : β\nh : x = y\n⊢ (fun b_1 => b_1 + { as := 1 }.as • b) x = (fun b_1 => b_1 + { as := 1 }... |
theorem eqToHom_f' {X Y : DifferentialObject ℤ (GradedObjectWithShift b V)} (f : X ⟶ Y) {x y : β}
(h : x = y) : X.objEqToHom h ≫ f.f y = f.f x ≫ Y.objEqToHom h := by cases h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DifferentialObject.lean | {
"open": [
"CategoryTheory CategoryTheory.Limits"
],
"variables": [
"{β : Type*} [AddCommGroup β] {b : β}",
"{V : Type*} [Category V] [HasZeroMorphisms V]",
"(X : DifferentialObject ℤ (GradedObjectWithShift b V))"
]
} | [
{
"line": "cases h",
"before_state": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{u_3, u_2} V\ninst✝ : HasZeroMorphisms V\nX Y : DifferentialObject ℤ (GradedObjectWithShift b V)\nf : X ⟶ Y\nx y : β\nh : x = y\n⊢ X.objEqToHom h ≫ f.f y = f.f x ≫ Y.objEqToHom h",
"after_... |
theorem d_eqToHom (X : HomologicalComplex V (ComplexShape.up' b)) {x y z : β} (h : y = z) :
X.d x y ≫ eqToHom (congr_arg X.X h) = X.d x z := by cases h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DifferentialObject.lean | {
"open": [
"CategoryTheory CategoryTheory.Limits",
"CategoryTheory.DifferentialObject"
],
"variables": [
"{β : Type*} [AddCommGroup β] {b : β}",
"{V : Type*} [Category V] [HasZeroMorphisms V]",
"(X : DifferentialObject ℤ (GradedObjectWithShift b V))",
"{β : Type*} [AddCommGroup β] (b : β)... | [
{
"line": "cases h",
"before_state": "β : Type u_3\ninst✝² : AddCommGroup β\nb : β\nV : Type u_4\ninst✝¹ : Category.{u_5, u_4} V\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex V (ComplexShape.up' b)\nx y z : β\nh : y = z\n⊢ X.d x y ≫ eqToHom ⋯ = X.d x z",
"after_state": "case refl\nβ : Type u_3\nin... |
lemma double_d_eq_zero₀ (a b : ι) (ha : a ≠ i₀) :
(double f hi₀₁).d a b = 0 :=
dif_neg (by tauto)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Double.lean | {
"open": [
"CategoryTheory Category Limits ZeroObject Opposite",
"Classical in"
],
"variables": [
"{C : Type*} [Category C] [HasZeroMorphisms C] [HasZeroObject C]",
"{X₀ X₁ : C} (f : X₀ ⟶ X₁) {ι : Type*} {c : ComplexShape ι}"
]
} | [
{
"line": "tauto",
"before_state": "ι : Type u_1\ni₀ : ι\nx✝ : Sort u_2\ndouble : x✝\na b : ι\nha : a ≠ i₀\n⊢ ¬?m.1519",
"after_state": "No Goals!"
},
{
"line": "contradiction",
"before_state": "ι : Type u_1\ni₀ : ι\nx✝ : Sort u_2\ndouble : x✝\na b : ι\nha : a ≠ i₀\n⊢ ¬?m.1519",
"after_s... |
lemma double_d_eq_zero₁ (a b : ι) (hb : b ≠ i₁) :
(double f hi₀₁).d a b = 0 :=
dif_neg (by tauto)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Double.lean | {
"open": [
"CategoryTheory Category Limits ZeroObject Opposite",
"Classical in"
],
"variables": [
"{C : Type*} [Category C] [HasZeroMorphisms C] [HasZeroObject C]",
"{X₀ X₁ : C} (f : X₀ ⟶ X₁) {ι : Type*} {c : ComplexShape ι}"
]
} | [
{
"line": "tauto",
"before_state": "ι : Type u_1\ni₁ : ι\nx✝ : Sort u_2\ndouble : x✝\na b : ι\nhb : b ≠ i₁\n⊢ ¬?m.1519",
"after_state": "No Goals!"
},
{
"line": "contradiction",
"before_state": "ι : Type u_1\ni₁ : ι\nx✝ : Sort u_2\ndouble : x✝\na b : ι\nhb : b ≠ i₁\n⊢ ¬?m.1519",
"after_s... |
lemma not_rel_of_eq {j' : ι } (h : j = j') : ¬ c.Rel j j' := by
subst h
exact c.not_rel_self j
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HasNoLoop.lean | {
"open": [],
"variables": [
"{ι : Type*}",
"(c : ComplexShape ι) [c.HasNoLoop] (j : ι)",
"{j} in"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj j' : ι\nh : j = j'\n⊢ ¬c.Rel j j'",
"after_state": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj : ι\n⊢ ¬c.Rel j j"
},
{
"line": "exact c.not_rel_self j",
"before_state": "ι : Type u_1\... |
lemma exists_distinct_prev_or :
(∃ (k : ι), c.Rel j k ∧ j ≠ k) ∨ ∀ (k : ι), ¬ c.Rel j k := by
by_cases h : ∃ (k : ι), c.Rel j k
· obtain ⟨k, hk⟩ := h
exact Or.inl ⟨k, hk, fun hjk ↦ c.not_rel_of_eq hjk hk⟩
· exact Or.inr (by simpa using h)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HasNoLoop.lean | {
"open": [],
"variables": [
"{ι : Type*}",
"(c : ComplexShape ι) [c.HasNoLoop] (j : ι)",
"{j} in"
]
} | [
{
"line": "by_cases h : ∃ (k : ι), c.Rel j k",
"before_state": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj : ι\n⊢ (∃ k, c.Rel j k ∧ j ≠ k) ∨ ∀ (k : ι), ¬c.Rel j k",
"after_state": "case pos\nι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj : ι\nh : ∃ k, c.Rel j k\n⊢ (∃ k, c.Rel j k... |
lemma exists_distinct_next_or :
(∃ (i : ι), c.Rel i j ∧ i ≠ j) ∨ ∀ (i : ι), ¬ c.Rel i j := by
by_cases h : ∃ (i : ι), c.Rel i j
· obtain ⟨i, hi⟩ := h
exact Or.inl ⟨i, hi, fun hij ↦ c.not_rel_of_eq hij hi⟩
· exact Or.inr (by simpa using h)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HasNoLoop.lean | {
"open": [],
"variables": [
"{ι : Type*}",
"(c : ComplexShape ι) [c.HasNoLoop] (j : ι)",
"{j} in"
]
} | [
{
"line": "by_cases h : ∃ (i : ι), c.Rel i j",
"before_state": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj : ι\n⊢ (∃ i, c.Rel i j ∧ i ≠ j) ∨ ∀ (i : ι), ¬c.Rel i j",
"after_state": "case pos\nι : Type u_1\nc : ComplexShape ι\ninst✝ : c.HasNoLoop\nj : ι\nh : ∃ i, c.Rel i j\n⊢ (∃ i, c.Rel i j... |
theorem d_comp_d (C : HomologicalComplex V c) (i j k : ι) : C.d i j ≫ C.d j k = 0 := by
by_cases hij : c.Rel i j
· by_cases hjk : c.Rel j k
· exact C.d_comp_d' i j k hij hjk
· rw [C.shape j k hjk, comp_zero]
· rw [C.shape i j hij, zero_comp]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "by_cases hij : c.Rel i j",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j k : ι\n⊢ C.d i j ≫ C.d j k = 0",
"after_state": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\nin... |
lemma XIsoOfEq_hom_comp_d (K : HomologicalComplex V c) {p₁ p₂ : ι} (h : p₁ = p₂) (p₃ : ι) :
(K.XIsoOfEq h).hom ≫ K.d p₂ p₃ = K.d p₁ p₃ := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₁ p₂ : ι\nh : p₁ = p₂\np₃ : ι\n⊢ (K.XIsoOfEq h).hom ≫ K.d p₂ p₃ = K.d p₁ p₃",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Catego... |
lemma XIsoOfEq_inv_comp_d (K : HomologicalComplex V c) {p₂ p₁ : ι} (h : p₂ = p₁) (p₃ : ι) :
(K.XIsoOfEq h).inv ≫ K.d p₂ p₃ = K.d p₁ p₃ := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₂ p₁ : ι\nh : p₂ = p₁\np₃ : ι\n⊢ (K.XIsoOfEq h).inv ≫ K.d p₂ p₃ = K.d p₁ p₃",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Catego... |
lemma d_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₂ = p₃) (p₁ : ι) :
K.d p₁ p₂ ≫ (K.XIsoOfEq h).hom = K.d p₁ p₃ := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₂ p₃ : ι\nh : p₂ = p₃\np₁ : ι\n⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq h).hom = K.d p₁ p₃",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Catego... |
lemma d_comp_XIsoOfEq_inv (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₃ = p₂) (p₁ : ι) :
K.d p₁ p₂ ≫ (K.XIsoOfEq h).inv = K.d p₁ p₃ := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK : HomologicalComplex V c\np₂ p₃ : ι\nh : p₃ = p₂\np₁ : ι\n⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq h).inv = K.d p₁ p₃",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Catego... |
theorem next_nat_zero : (ComplexShape.down ℕ).next 0 = 0 := by
classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "classical\nrefine dif_neg ?_\npush_neg\nintro\napply Nat.noConfusion",
"before_state": "⊢ (ComplexShape.down ℕ).next 0 = 0",
"after_state": "No Goals!"
},
{
"line": "refine dif_neg ?_",
"before_state": "⊢ (ComplexShape.down ℕ).next 0 = 0",
"after_state": "⊢ ¬∃ j, (ComplexShape... |
theorem prev_nat_zero : (ComplexShape.up ℕ).prev 0 = 0 := by
classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}"
]
} | [
{
"line": "classical\nrefine dif_neg ?_\npush_neg\nintro\napply Nat.noConfusion",
"before_state": "⊢ (ComplexShape.up ℕ).prev 0 = 0",
"after_state": "No Goals!"
},
{
"line": "refine dif_neg ?_",
"before_state": "⊢ (ComplexShape.up ℕ).prev 0 = 0",
"after_state": "⊢ ¬∃ i, (ComplexShape.up ... |
theorem Hom.comm {A B : HomologicalComplex V c} (f : A.Hom B) (i j : ι) :
f.f i ≫ B.d i j = A.d i j ≫ f.f j := by
by_cases hij : c.Rel i j
· exact f.comm' i j hij
· rw [A.shape i j hij, B.shape i j hij, comp_zero, zero_comp]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)"
]
} | [
{
"line": "by_cases hij : c.Rel i j",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nA B : HomologicalComplex V c\nf : A.Hom B\ni j : ι\n⊢ f.f i ≫ B.d i j = A.d i j ≫ f.f j",
"after_state": "case pos\nι : Type u_1\nV : Type u\ninst✝... |
lemma hom_ext {C D : HomologicalComplex V c} (f g : C ⟶ D)
(h : ∀ i, f.f i = g.f i) : f = g := by
apply Hom.ext
funext
apply h
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)"
]
} | [
{
"line": "apply Hom.ext",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf g : C ⟶ D\nh : ∀ (i : ι), f.f i = g.f i\n⊢ f = g",
"after_state": "No Goals!"
}
] |
theorem eqToHom_f {C₁ C₂ : HomologicalComplex V c} (h : C₁ = C₂) (n : ι) :
HomologicalComplex.Hom.f (eqToHom h) n =
eqToHom (congr_fun (congr_arg HomologicalComplex.X h) n) := by
subst h
rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC₁ C₂ : HomologicalComplex V c\nh : C₁ = C₂\nn : ι\n⊢ (eqToHom h).f n = eqToHom ⋯",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Ha... |
lemma XIsoOfEq_hom_naturality {K L : HomologicalComplex V c} (φ : K ⟶ L) {n n' : ι} (h : n = n') :
φ.f n ≫ (L.XIsoOfEq h).hom = (K.XIsoOfEq h).hom ≫ φ.f n' := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK L : HomologicalComplex V c\nφ : K ⟶ L\nn n' : ι\nh : n = n'\n⊢ φ.f n ≫ (L.XIsoOfEq h).hom = (K.XIsoOfEq h).hom ≫ φ.f n'",
"after_state": "ι : Type u_1\nV : Type u... |
lemma XIsoOfEq_inv_naturality {K L : HomologicalComplex V c} (φ : K ⟶ L) {n n' : ι} (h : n = n') :
φ.f n' ≫ (L.XIsoOfEq h).inv = (K.XIsoOfEq h).inv ≫ φ.f n := by subst h; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "subst h",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nK L : HomologicalComplex V c\nφ : K ⟶ L\nn n' : ι\nh : n = n'\n⊢ φ.f n' ≫ (L.XIsoOfEq h).inv = (K.XIsoOfEq h).inv ≫ φ.f n",
"after_state": "ι : Type u_1\nV : Type u... |
theorem d_comp_eqToHom {i j j' : ι} (rij : c.Rel i j) (rij' : c.Rel i j') :
C.d i j' ≫ eqToHom (congr_arg C.X (c.next_eq rij' rij)) = C.d i j := by
obtain rfl := c.next_eq rij rij'
simp only [eqToHom_refl]
simp only [comp_id]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "obtain rfl := c.next_eq rij rij'",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j j' : ι\nrij : c.Rel i j\nrij' : c.Rel i j'\n⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j",
"after_state": "ι : Type u_1... |
theorem eqToHom_comp_d {i i' j : ι} (rij : c.Rel i j) (rij' : c.Rel i' j) :
eqToHom (congr_arg C.X (c.prev_eq rij rij')) ≫ C.d i' j = C.d i j := by
obtain rfl := c.prev_eq rij rij'
simp only [eqToHom_refl]
simp only [id_comp]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "obtain rfl := c.prev_eq rij rij'",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni i' j : ι\nrij : c.Rel i j\nrij' : c.Rel i' j\n⊢ eqToHom ⋯ ≫ C.d i' j = C.d i j",
"after_state": "ι : Type u_1... |
theorem dTo_eq {i j : ι} (r : c.Rel i j) : C.dTo j = (C.xPrevIso r).hom ≫ C.d i j := by
obtain rfl := c.prev_eq' r
exact (Category.id_comp _).symm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "obtain rfl := c.prev_eq' r",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nr : c.Rel i j\n⊢ C.dTo j = (C.xPrevIso r).hom ≫ C.d i j",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹ :... |
theorem dFrom_eq {i j : ι} (r : c.Rel i j) : C.dFrom i = C.d i j ≫ (C.xNextIso r).inv := by
obtain rfl := c.next_eq' r
exact (Category.comp_id _).symm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "obtain rfl := c.next_eq' r",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nr : c.Rel i j\n⊢ C.dFrom i = C.d i j ≫ (C.xNextIso r).inv",
"after_state": "ι : Type u_1\nV : Type u\ninst✝¹... |
theorem xPrevIso_comp_dTo {i j : ι} (r : c.Rel i j) : (C.xPrevIso r).inv ≫ C.dTo j = C.d i j := by
simp [C.dTo_eq r]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "simp [C.dTo_eq r]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nr : c.Rel i j\n⊢ (C.xPrevIso r).inv ≫ C.dTo j = C.d i j",
"after_state": "No Goals!"
}
] |
theorem xPrevIsoSelf_comp_dTo {j : ι} (h : ¬c.Rel (c.prev j) j) :
(C.xPrevIsoSelf h).inv ≫ C.dTo j = 0 := by simp [h]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "simp [h]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\nj : ι\nh : ¬c.Rel (c.prev j) j\n⊢ (C.xPrevIsoSelf h).inv ≫ C.dTo j = 0",
"after_state": "No Goals!"
}
] |
theorem dFrom_comp_xNextIso {i j : ι} (r : c.Rel i j) :
C.dFrom i ≫ (C.xNextIso r).hom = C.d i j := by
simp [C.dFrom_eq r]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "simp [C.dFrom_eq r]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j : ι\nr : c.Rel i j\n⊢ C.dFrom i ≫ (C.xNextIso r).hom = C.d i j",
"after_state": "No Goals!"
}
] |
theorem dFrom_comp_xNextIsoSelf {i : ι} (h : ¬c.Rel i (c.next i)) :
C.dFrom i ≫ (C.xNextIsoSelf h).hom = 0 := by simp [h]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "simp [h]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni : ι\nh : ¬c.Rel i (c.next i)\n⊢ C.dFrom i ≫ (C.xNextIsoSelf h).hom = 0",
"after_state": "No Goals!"
}
] |
theorem kernel_from_eq_kernel [HasKernels V] {i j : ι} (r : c.Rel i j) :
kernelSubobject (C.dFrom i) = kernelSubobject (C.d i j) := by
rw [C.dFrom_eq r]
apply kernelSubobject_comp_mono
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "rw [C.dFrom_eq r]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\ni j : ι\nr : c.Rel i j\n⊢ kernelSubobject (C.dFrom i) = kernelSubobject (C.d i j)",
"after_state": "ι :... |
theorem image_to_eq_image [HasImages V] [HasEqualizers V] {i j : ι} (r : c.Rel i j) :
imageSubobject (C.dTo j) = imageSubobject (C.d i j) := by
rw [C.dTo_eq r]
apply imageSubobject_iso_comp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomologicalComplex.lean | {
"open": [
"CategoryTheory CategoryTheory.Category CategoryTheory.Limits",
"ZeroObject"
],
"variables": [
"{ι : Type*}",
"(V : Type u) [Category.{v} V] [HasZeroMorphisms V]",
"{V} {c : ComplexShape ι}",
"{V}",
"{c : ComplexShape ι} (C : HomologicalComplex V c)",
"(V c)"
]
} | [
{
"line": "rw [C.dTo_eq r]",
"before_state": "ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject (C.dTo j) = imageSubobject (C.d i j)",
... |
theorem prevD_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
prevD j f = f j (j + 1) ≫ Q.d _ _ := by
dsimp [prevD]
have : (ComplexShape.down ℕ).prev j = j + 1 := ChainComplex.prev ℕ j
congr 2
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [prevD]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\nj : ℕ\n⊢ (prevD j) f = f j (j + 1) ≫ Q.d (j + 1) j",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : Chai... |
theorem dNext_succ_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
dNext (i + 1) f = P.d _ _ ≫ f i (i + 1) := by
dsimp [dNext]
have : (ComplexShape.down ℕ).next (i + 1) = i := ChainComplex.next_nat_succ _
congr 2
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [dNext]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\ni : ℕ\n⊢ (dNext (i + 1)) f = P.d (i + 1) i ≫ f i (i + 1)",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q ... |
theorem dNext_zero_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : dNext 0 f = 0 := by
dsimp [dNext]
rw [P.shape]
rw [zero_comp]
rw [ChainComplex.next_nat_zero]; dsimp; decide
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [dNext]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ (dNext 0) f = 0",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X... |
theorem dNext_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
dNext j f = P.d _ _ ≫ f (j + 1) j := by
dsimp [dNext]
have : (ComplexShape.up ℕ).next j = j + 1 := CochainComplex.next ℕ j
congr 2
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [dNext]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\nj : ℕ\n⊢ (dNext j) f = P.d j (j + 1) ≫ f (j + 1) j",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : Co... |
theorem prevD_succ_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
prevD (i + 1) f = f (i + 1) _ ≫ Q.d i (i + 1) := by
dsimp [prevD]
have : (ComplexShape.up ℕ).prev (i + 1) = i := CochainComplex.prev_nat_succ i
congr 2
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [prevD]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\ni : ℕ\n⊢ (prevD (i + 1)) f = f (i + 1) i ≫ Q.d i (i + 1)",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP ... |
theorem prevD_zero_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : prevD 0 f = 0 := by
dsimp [prevD]
rw [Q.shape]
rw [comp_zero]
rw [CochainComplex.prev_nat_zero]; dsimp; decide
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Homotopy.lean | {
"open": [
"CategoryTheory Category Limits HomologicalComplex",
"Classical in",
"Classical in"
],
"variables": [
"{ι : Type*}",
"{V : Type u} [Category.{v} V] [Preadditive V]",
"{c : ComplexShape ι} {C D E : HomologicalComplex V c}",
"(f g : C ⟶ D) (h k : D ⟶ E) (i : ι)",
"{f g}",... | [
{
"line": "dsimp [prevD]",
"before_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ (prevD 0) f = 0",
"after_state": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) →... |
lemma descSigma_ext_iff {φ : F ⟶ G} {K : HomologicalComplex C c}
(x y : Σ (α : G ⟶ K), Homotopy (φ ≫ α) 0) :
x = y ↔ x.1 = y.1 ∧ (∀ (i j : ι) (_ : c.Rel j i), x.2.hom i j = y.2.hom i j) := by
constructor
· rintro rfl
tauto
· obtain ⟨x₁, x₂⟩ := x
obtain ⟨y₁, y₂⟩ := y
rintro ⟨rfl, h⟩
simp on... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/HomotopyCofiber.lean | {
"open": [
"CategoryTheory Category Limits Preadditive"
],
"variables": [
"{C : Type*} [Category C] [Preadditive C]",
"{ι : Type*} {c : ComplexShape ι} {F G K : HomologicalComplex C c} (φ : F ⟶ G)",
"[HasHomotopyCofiber φ] [DecidableRel c.Rel]",
"(hc : ∀ j, ∃ i, c.Rel i j)",
"(α : G ⟶ K) ... | [
{
"line": "constructor",
"before_state": "C : Type u_1\ninst✝¹ : Category.{u_3, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\nK : HomologicalComplex C c\nx y : (α : G ⟶ K) × Homotopy (φ ≫ α) 0\n⊢ x = y ↔ x.fst = y.fst ∧ ∀ (i j : ι), c.Rel j i → x.s... |
lemma opcyclesOpIso_hom_toCycles_op :
(K.opcyclesOpIso i).hom ≫ (K.toCycles j i).op = K.op.fromOpcycles i j := by
by_cases hij : c.Rel j i
· obtain rfl := c.prev_eq' hij
exact (K.sc i).opcyclesOpIso_hom_toCycles_op
· rw [K.toCycles_eq_zero hij, K.op.fromOpcycles_eq_zero hij, op_zero, comp_zero]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Opposite.lean | {
"open": [
"Opposite CategoryTheory CategoryTheory.Limits"
],
"variables": [
"{V : Type*} [Category V] [Abelian V]",
"{ι V : Type*} [Category V] {c : ComplexShape ι}",
"[HasZeroMorphisms V]",
"(V c)",
"{V c}",
"(K : HomologicalComplex V c) (i : ι) [K.HasHomology i]",
"(j : ι)"
]... | [
{
"line": "by_cases hij : c.Rel j i",
"before_state": "ι : Type u_2\nV : Type u_3\ninst✝² : Category.{u_4, u_3} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex V c\ni : ι\ninst✝ : K.HasHomology i\nj : ι\n⊢ (K.opcyclesOpIso i).hom ≫ (K.toCycles j i).op = K.op.fromOpcycles i j",
... |
lemma fromOpcycles_op_cyclesOpIso_inv :
(K.fromOpcycles i j).op ≫ (K.cyclesOpIso i).inv = K.op.toCycles j i := by
by_cases hij : c.Rel i j
· obtain rfl := c.next_eq' hij
exact (K.sc i).fromOpcycles_op_cyclesOpIso_inv
· rw [K.op.toCycles_eq_zero hij, K.fromOpcycles_eq_zero hij, op_zero, zero_comp]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Opposite.lean | {
"open": [
"Opposite CategoryTheory CategoryTheory.Limits"
],
"variables": [
"{V : Type*} [Category V] [Abelian V]",
"{ι V : Type*} [Category V] {c : ComplexShape ι}",
"[HasZeroMorphisms V]",
"(V c)",
"{V c}",
"(K : HomologicalComplex V c) (i : ι) [K.HasHomology i]",
"(j : ι)"
]... | [
{
"line": "by_cases hij : c.Rel i j",
"before_state": "ι : Type u_2\nV : Type u_3\ninst✝² : Category.{u_4, u_3} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex V c\ni : ι\ninst✝ : K.HasHomology i\nj : ι\n⊢ (K.fromOpcycles i j).op ≫ (K.cyclesOpIso i).inv = K.op.toCycles j i",
"... |
lemma isSplitMono_from_singleFunctor_obj_of_injective
{I : C} [Injective I] {L : CochainComplex C ℤ} {i : ℤ}
(ι : (CochainComplex.singleFunctor C i).obj I ⟶ L) [L.IsStrictlyGE i] [QuasiIsoAt ι i] :
IsSplitMono ι := by
let e := L.pOpcyclesIso (i - 1) i (by simp)
((L.isZero_of_isStrictlyGE i (i - 1) (by... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughInjectives.lean | {
"open": [
"CategoryTheory Category",
"HomologicalComplex"
],
"variables": [
"{C : Type u} [Category.{v} C] [Abelian C]"
]
} | [
{
"line": "let e := L.pOpcyclesIso (i - 1) i (by simp) ((L.isZero_of_isStrictlyGE i (i - 1) (by simp)).eq_of_src _ _)",
"before_state": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nI : C\ninst✝² : Injective I\nL : CochainComplex C ℤ\ni : ℤ\nι : (CochainComplex.singleFunctor C i).obj I ⟶ L\ni... |
lemma hasExt_of_enoughInjectives [LocallySmall.{w} C] [EnoughInjectives C] :
HasExt.{w} C := by
letI := HasDerivedCategory.standard C
have := hasExt_of_hasDerivedCategory C
rw [hasExt_iff_small_ext.{w}]
intro X Y n
induction n generalizing X Y with
| zero =>
rw [small_congr Ext.homEquiv₀... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughInjectives.lean | {
"open": [
"CategoryTheory Category",
"HomologicalComplex",
"Limits",
"DerivedCategory",
"Abelian"
],
"variables": [
"{C : Type u} [Category.{v} C] [Abelian C]",
"[HasDerivedCategory.{w} C]",
"{C : Type u} [Category.{v} C] [Abelian C]",
"(C)"
]
} | [
{
"line": "letI := HasDerivedCategory.standard C",
"before_state": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : EnoughInjectives C\n⊢ HasExt C",
"after_state": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : LocallySmall.{w... |
lemma isSplitEpi_to_singleFunctor_obj_of_projective
{P : C} [Projective P] {K : CochainComplex C ℤ} {i : ℤ}
(π : K ⟶ (CochainComplex.singleFunctor C i).obj P) [K.IsStrictlyLE i] [QuasiIsoAt π i] :
IsSplitEpi π := by
let e := K.iCyclesIso i (i + 1) (by simp)
((K.isZero_of_isStrictlyLE i (i + 1) (by sim... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughProjectives.lean | {
"open": [
"CategoryTheory Category",
"HomologicalComplex"
],
"variables": [
"{C : Type u} [Category.{v} C] [Abelian C]"
]
} | [
{
"line": "let e := K.iCyclesIso i (i + 1) (by simp) ((K.isZero_of_isStrictlyLE i (i + 1) (by simp)).eq_of_tgt _ _)",
"before_state": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nP : C\ninst✝² : Projective P\nK : CochainComplex C ℤ\ni : ℤ\nπ : K ⟶ (CochainComplex.singleFunctor C i).obj P\nin... |
lemma hasExt_of_enoughProjectives [LocallySmall.{w} C] [EnoughProjectives C] :
HasExt.{w} C := by
letI := HasDerivedCategory.standard C
have := hasExt_of_hasDerivedCategory C
rw [hasExt_iff_small_ext.{w}]
intro X Y n
induction n generalizing X Y with
| zero =>
rw [small_congr Ext.homEqui... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughProjectives.lean | {
"open": [
"CategoryTheory Category",
"HomologicalComplex",
"Limits",
"DerivedCategory",
"Abelian"
],
"variables": [
"{C : Type u} [Category.{v} C] [Abelian C]",
"[HasDerivedCategory.{w} C]",
"{C : Type u} [Category.{v} C] [Abelian C]",
"(C)"
]
} | [
{
"line": "letI := HasDerivedCategory.standard C",
"before_state": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : EnoughProjectives C\n⊢ HasExt C",
"after_state": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : LocallySmall.{... |
lemma isZero_of_isStrictlyGE (n i : ℤ) (hi : i < n) [K.IsStrictlyGE n] :
IsZero (K.X i) :=
isZero_X_of_isStrictlySupported K (embeddingUpIntGE n) i
(by simpa only [not_mem_range_embeddingUpIntGE_iff] using hi)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "simpa only [not_mem_range_embeddingUpIntGE_iff] using hi",
"before_state": "C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\nn i : ℤ\nhi : i < n\ninst✝ : K.IsStrictlyGE n\n⊢ ∀ (i_1 : ℕ... |
lemma isZero_of_isStrictlyLE (n i : ℤ) (hi : n < i) [K.IsStrictlyLE n] :
IsZero (K.X i) :=
isZero_X_of_isStrictlySupported K (embeddingUpIntLE n) i
(by simpa only [not_mem_range_embeddingUpIntLE_iff] using hi)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "simpa only [not_mem_range_embeddingUpIntLE_iff] using hi",
"before_state": "C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\nn i : ℤ\nhi : n < i\ninst✝ : K.IsStrictlyLE n\n⊢ ∀ (i_1 : ℕ... |
lemma exactAt_of_isGE (n i : ℤ) (hi : i < n) [K.IsGE n] :
K.ExactAt i :=
exactAt_of_isSupported K (embeddingUpIntGE n) i
(by simpa only [not_mem_range_embeddingUpIntGE_iff] using hi)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "simpa only [not_mem_range_embeddingUpIntGE_iff] using hi",
"before_state": "C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\nn i : ℤ\nhi : i < n\ninst✝ : K.IsGE n\n⊢ ∀ (i_1 : ℕ), (embe... |
lemma exactAt_of_isLE (n i : ℤ) (hi : n < i) [K.IsLE n] :
K.ExactAt i :=
exactAt_of_isSupported K (embeddingUpIntLE n) i
(by simpa only [not_mem_range_embeddingUpIntLE_iff] using hi)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "simpa only [not_mem_range_embeddingUpIntLE_iff] using hi",
"before_state": "C : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\nn i : ℤ\nhi : n < i\ninst✝ : K.IsLE n\n⊢ ∀ (i_1 : ℕ), (embe... |
lemma isStrictlyGE_iff (n : ℤ) :
K.IsStrictlyGE n ↔ ∀ (i : ℤ) (_ : i < n), IsZero (K.X i) := by
constructor
· intro _ i hi
exact K.isZero_of_isStrictlyGE n i hi
· intro h
refine IsStrictlySupported.mk (fun i hi ↦ ?_)
rw [not_mem_range_embeddingUpIntGE_iff] at hi
exact h i hi
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "constructor",
"before_state": "C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝¹ : HasZeroObject C\ninst✝ : ∀ (i : ℤ), HasHomology K i\nn : ℤ\n⊢ K.IsStrictlyGE n ↔ ∀ i < n, IsZero (K.X i)",
"after_state": "case mp\nC : Type u_1\ninst✝³ :... |
lemma isStrictlyLE_iff (n : ℤ) :
K.IsStrictlyLE n ↔ ∀ (i : ℤ) (_ : n < i), IsZero (K.X i) := by
constructor
· intro _ i hi
exact K.isZero_of_isStrictlyLE n i hi
· intro h
refine IsStrictlySupported.mk (fun i hi ↦ ?_)
rw [not_mem_range_embeddingUpIntLE_iff] at hi
exact h i hi
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "constructor",
"before_state": "C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝¹ : HasZeroObject C\ninst✝ : ∀ (i : ℤ), HasHomology K i\nn : ℤ\n⊢ K.IsStrictlyLE n ↔ ∀ (i : ℤ), n < i → IsZero (K.X i)",
"after_state": "case mp\nC : Type u_1... |
lemma isGE_iff (n : ℤ) :
K.IsGE n ↔ ∀ (i : ℤ) (_ : i < n), K.ExactAt i := by
constructor
· intro _ i hi
exact K.exactAt_of_isGE n i hi
· intro h
refine IsSupported.mk (fun i hi ↦ ?_)
rw [not_mem_range_embeddingUpIntGE_iff] at hi
exact h i hi
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Homology/Embedding/CochainComplex.lean | {
"open": [
"CategoryTheory Category Limits ComplexShape ZeroObject",
"HomologicalComplex"
],
"variables": [
"{C : Type*} [Category C]",
"[HasZeroMorphisms C] (K L : CochainComplex C ℤ) (φ : K ⟶ L) (e : K ≅ L)",
"[HasZeroObject C] [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]",
"{K L}"
]... | [
{
"line": "constructor",
"before_state": "C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝¹ : HasZeroObject C\ninst✝ : ∀ (i : ℤ), HasHomology K i\nn : ℤ\n⊢ K.IsGE n ↔ ∀ i < n, ExactAt K i",
"after_state": "case mp\nC : Type u_1\ninst✝³ : Category.{... |
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