declaration stringlengths 27 11.3k | file stringlengths 52 114 | context dict | tactic_states listlengths 1 1.24k |
|---|---|---|---|
example {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) [p.IsHomLift f φ] : f = f := by
subst_hom_lift p f φ
rename_i h
guard_hyp h : p.IsHomLift (p.map φ) φ
guard_target = p.map φ = p.map φ
trivial
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/SubstHomLift.lean | {
"open": [
"CategoryTheory Category"
],
"variables": [
"{𝒮 : Type u₁} {𝒳 : Type u₂} [Category.{v₁} 𝒳] [Category.{v₂} 𝒮] (p : 𝒳 ⥤ 𝒮)"
]
} | [
{
"line": "subst_hom_lift p f φ",
"before_state": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ f = f",
"after_state": "case map\n𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{... |
example {R S T : 𝒮} {a b c : 𝒳} (f : R ⟶ S) (g : S ⟶ T) (φ : a ⟶ b) (ψ : b ⟶ c)
[p.IsHomLift f (φ ≫ ψ)] : f = f := by
subst_hom_lift p f (φ ≫ ψ)
rename_i h
guard_hyp h : p.IsHomLift (p.map (φ ≫ ψ)) (φ ≫ ψ)
guard_target = p.map (φ ≫ ψ) = p.map (φ ≫ ψ)
trivial | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/SubstHomLift.lean | {
"open": [
"CategoryTheory Category"
],
"variables": [
"{𝒮 : Type u₁} {𝒳 : Type u₂} [Category.{v₁} 𝒳] [Category.{v₂} 𝒮] (p : 𝒳 ⥤ 𝒮)"
]
} | [
{
"line": "subst_hom_lift p f(φ ≫ ψ)",
"before_state": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S T : 𝒮\na b c : 𝒳\nf : R ⟶ S\ng : S ⟶ T\nφ : a ⟶ b\nψ : b ⟶ c\ninst✝ : p.IsHomLift f (φ ≫ ψ)\n⊢ f = f",
"after_state": "case map\n𝒮 : Type ... |
example (X₁ X₂ : C) :
((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫
(𝟙 (𝟙_ C) ⊗ (α_ (𝟙_ C) X₁ X₂).inv) =
𝟙 (𝟙_ C) ⊗ ((λ_ X₁).inv ⊗ 𝟙 X₂) := by
pure_coherence
-- This is just running:
-- change projectMap id _ _ (LiftHom.lift (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
-- ... | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "pure_coherence\n -- This is just running:\n -- change projectMap id _ _ (LiftHom.lift (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫\n -- (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫ (𝟙 (𝟙_ C) ⊗ (α_ (𝟙_ C) X₁ X₂).inv))) =\n -- projectMap id _ _ (LiftHom.lift (𝟙 (𝟙_ C) ⊗ ((λ_ X₁).inv ⊗ 𝟙 X₂)))\n ... |
example {Y Z : C} (f : Y ⟶ Z) (g) (w : false) : (λ_ _).hom ≫ f = g := by
liftable_prefixes
guard_target = (𝟙 _ ≫ (λ_ _).hom) ≫ f = (𝟙 _) ≫ g
cases w
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "liftable_prefixes",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nY Z : C\nf : Y ⟶ Z\ng : 𝟙_ C ⊗ Y ⟶ Z\nw : false = true\n⊢ (λ_ Y).hom ≫ f = g",
"after_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nY Z : C\nf : Y ⟶ Z\ng : ... |
example (f : 𝟙_ C ⟶ _) : f ≫ (λ_ (𝟙_ C)).hom = f ≫ (ρ_ (𝟙_ C)).hom := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf : 𝟙_ C ⟶ 𝟙_ C ⊗ 𝟙_ C\n⊢ f ≫ (λ_ (𝟙_ C)).hom = f ≫ (ρ_ (𝟙_ C)).hom",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n wh... |
example (f) : (λ_ (𝟙_ C)).hom ≫ f ≫ (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom ≫ f ≫ (ρ_ (𝟙_ C)).hom := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf : 𝟙_ C ⟶ 𝟙_ C ⊗ 𝟙_ C\n⊢ (λ_ (𝟙_ C)).hom ≫ f ≫ (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom ≫ f ≫ (ρ_ (𝟙_ C)).hom",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bica... |
example {U : C} (f : U ⟶ 𝟙_ C) : f ≫ (ρ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom = f := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU : C\nf : U ⟶ 𝟙_ C\n⊢ f ≫ (ρ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom = f",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n whisker... |
example (W X Y Z : C) (f) :
((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom ≫ (𝟙 W ⊗ (α_ X Y Z).hom) ≫ f ≫
(α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).hom =
(α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).hom ≫ f ≫
((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom ≫ (𝟙 W ⊗ (α_ X Y Z).hom) := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nW X Y Z : C\nf : W ⊗ X ⊗ Y ⊗ Z ⟶ ((W ⊗ X) ⊗ Y) ⊗ Z\n⊢ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫\n (α_ W (X ⊗ Y) Z).hom ≫ (𝟙 W ⊗ (α_ X Y Z).hom) ≫ f ≫ (α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).hom =\n (α_ (W ⊗ ... |
example {U V W X Y : C} (f : U ⟶ V ⊗ (W ⊗ X)) (g : (V ⊗ W) ⊗ X ⟶ Y) :
f ⊗≫ g = f ≫ (α_ _ _ _).inv ≫ g := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X Y : C\nf : U ⟶ V ⊗ W ⊗ X\ng : (V ⊗ W) ⊗ X ⟶ Y\n⊢ f ⊗≫ g = f ≫ (α_ V W X).inv ≫ g",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoid... |
example : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\n⊢ (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n whisker_simps -failIfUnchanged✝;\n m... |
example : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\n⊢ (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n whisker_simps -failIfUnchanged✝;\n m... |
example (X Y Z : C) : (α_ X Y Z).inv ≫ (α_ X Y Z).hom = 𝟙 (X ⊗ Y ⊗ Z) := by coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z : C\n⊢ (α_ X Y Z).inv ≫ (α_ X Y Z).hom = 𝟙 (X ⊗ Y ⊗ Z)",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n whisker_simps... |
example (X Y Z W : C) :
(𝟙 X ⊗ (α_ Y Z W).hom) ≫ (α_ X Y (Z ⊗ W)).inv ≫ (α_ (X ⊗ Y) Z W).inv =
(α_ X (Y ⊗ Z) W).inv ≫ ((α_ X Y Z).inv ⊗ 𝟙 W) := by
coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\n⊢ (𝟙 X ⊗ (α_ Y Z W).hom) ≫ (α_ X Y (Z ⊗ W)).inv ≫ (α_ (X ⊗ Y) Z W).inv = (α_ X (Y ⊗ Z) W).inv ≫ ((α_ X Y Z).inv ⊗ 𝟙 W)",
"after_state": "No Goals!"
},
{
"line": "(simp -fail... |
example (X Y : C) :
(𝟙 X ⊗ (λ_ Y).inv) ≫ (α_ X (𝟙_ C) Y).inv = (ρ_ X).inv ⊗ 𝟙 Y := by
coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y : C\n⊢ (𝟙 X ⊗ (λ_ Y).inv) ≫ (α_ X (𝟙_ C) Y).inv = (ρ_ X).inv ⊗ 𝟙 Y",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [bicategoricalComp✝, monoidalComp✝]);\n w... |
example (X Y : C) (f : 𝟙_ C ⟶ X) (g : X ⟶ Y) (_w : false) :
(λ_ (𝟙_ C)).hom ≫ f ≫ 𝟙 X ≫ g = (ρ_ (𝟙_ C)).hom ≫ f ≫ g := by
coherence
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y : C\nf : 𝟙_ C ⟶ X\ng : X ⟶ Y\n_w : false = true\n⊢ (λ_ (𝟙_ C)).hom ≫ f ≫ 𝟙 X ≫ g = (ρ_ (𝟙_ C)).hom ≫ f ≫ g",
"after_state": "No Goals!"
},
{
"line": "(simp -failIfUnchanged✝ only [... |
example (X₁ X₂ : C) :
(α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫
(𝟙 (𝟙_ C) ⊗ (α_ (𝟙_ C) X₁ X₂).inv) ≫
(𝟙 (𝟙_ C) ⊗ (λ_ _).hom ≫ (ρ_ X₁).inv ⊗ 𝟙 X₂) ≫
(𝟙 (𝟙_ C) ⊗ (α_ X₁ (𝟙_ C) X₂).hom) ≫
(α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).inv ≫
((λ_ X₁).hom ≫ (ρ_ X₁).inv ⊗ 𝟙 (𝟙_ C ⊗ X₂)) ≫
... | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]"
]
} | [
{
"line": "coherence",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX₁ X₂ : C\n⊢ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫\n (𝟙 (𝟙_ C) ⊗ (α_ (𝟙_ C) X₁ X₂).inv) ≫\n (𝟙 (𝟙_ C) ⊗ (λ_ X₁).hom ≫ (ρ_ X₁).inv ⊗ 𝟙 X₂) ≫\n (𝟙 (𝟙_ C) ⊗ (α_ X₁ (𝟙_ C) X₂)... |
example {a : B} (f : a ⟶ a) : 𝟙 f ▷ f = 𝟙 (f ≫ f) := by whisker_simps
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory",
"scoped Bicategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{B : Type u} [Bicategory.{w, v} B] {a b c d e : B}"
]
} | [
{
"line": "whisker_simps",
"before_state": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nB : Type u\ninst✝ : Bicategory B\na✝ b c d e a : B\nf : a ⟶ a\n⊢ 𝟙 f ▷ f = 𝟙 (f ≫ f)",
"after_state": "No Goals!"
},
{
"line": "simp only [Category.assoc✝, Bicategory.comp_whiskerLe... |
example (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
(α_ f g h).inv ≫ (α_ f g h).hom = 𝟙 (f ≫ g ≫ h) := by
bicategory_coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory",
"scoped Bicategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{B : Type u} [Bicategory.{w, v} B] {a b c d e : B}"
]
} | [
{
"line": "bicategory_coherence",
"before_state": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nB : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\n⊢ (α_ f g h).inv ≫ (α_ f g h).hom = 𝟙 (f ≫ g ≫ h)",
"after_state": "No Goals!"
},
{
"line": "... |
example (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i := by
bicategory_coherence | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Coherence.lean | {
"open": [
"CategoryTheory",
"scoped MonoidalCategory",
"scoped Bicategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{B : Type u} [Bicategory.{w, v} B] {a b c d e : B}"
]
} | [
{
"line": "bicategory_coherence",
"before_state": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nB : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv = (α_ f (g ≫ h) i).inv ≫ (α_... |
example (hyp : f₁ ≫ g₁ = f₂ ≫ g₂) : f₁ ≫ g₁ ≫ h ≫ l = (f₂ ≫ g₂) ≫ (h ≫ l) := by
conv =>
rhs
slice 2 3
show f₁ ≫ g₁ ≫ h ≫ l = f₂ ≫ (g₂ ≫ h) ≫ l
conv =>
lhs
slice 1 2
rw [hyp]
show ((f₂ ≫ g₂) ≫ h) ≫ l = f₂ ≫ (g₂ ≫ h) ≫ l
conv =>
lhs
slice 2 3
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Slice.lean | {
"open": [
"CategoryTheory"
],
"variables": [
"(C : Type) [Category C] (X Y Z W U : C)",
"(f₁ f₂ : X ⟶ Y) (g g₁ g₂ : Y ⟶ Z) (h : Z ⟶ W) (l : W ⟶ U)"
]
} | [
{
"line": "conv =>\n rhs\n slice 2 3",
"before_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ f₂ : X ⟶ Y\ng g₁ g₂ : Y ⟶ Z\nh : Z ⟶ W\nl : W ⟶ U\nhyp : f₁ ≫ g₁ = f₂ ≫ g₂\n⊢ f₁ ≫ g₁ ≫ h ≫ l = (f₂ ≫ g₂) ≫ h ≫ l",
"after_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ ... |
example (hyp : f₁ ≫ g₁ = f₂ ≫ g₂) : f₁ ≫ g₁ ≫ h ≫ l = (f₂ ≫ g₂) ≫ (h ≫ l) := by
slice_lhs 1 2 => { rw [hyp] }; slice_rhs 1 2 => skip
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Slice.lean | {
"open": [
"CategoryTheory"
],
"variables": [
"(C : Type) [Category C] (X Y Z W U : C)",
"(f₁ f₂ : X ⟶ Y) (g g₁ g₂ : Y ⟶ Z) (h : Z ⟶ W) (l : W ⟶ U)"
]
} | [
{
"line": "slice_lhs 1 2 => {rw [hyp]}",
"before_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ f₂ : X ⟶ Y\ng g₁ g₂ : Y ⟶ Z\nh : Z ⟶ W\nl : W ⟶ U\nhyp : f₁ ≫ g₁ = f₂ ≫ g₂\n⊢ f₁ ≫ g₁ ≫ h ≫ l = (f₂ ≫ g₂) ≫ h ≫ l",
"after_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ ... |
example (h₁ : f₁ = f₂) : f₁ ≫ g ≫ h ≫ l = ((f₂ ≫ g) ≫ h) ≫ l := by
slice_lhs 1 1 => rw [h₁]
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Slice.lean | {
"open": [
"CategoryTheory"
],
"variables": [
"(C : Type) [Category C] (X Y Z W U : C)",
"(f₁ f₂ : X ⟶ Y) (g g₁ g₂ : Y ⟶ Z) (h : Z ⟶ W) (l : W ⟶ U)"
]
} | [
{
"line": "slice_lhs 1 1 => rw [h₁]",
"before_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ f₂ : X ⟶ Y\ng g₁ g₂ : Y ⟶ Z\nh : Z ⟶ W\nl : W ⟶ U\nh₁ : f₁ = f₂\n⊢ f₁ ≫ g ≫ h ≫ l = ((f₂ ≫ g) ≫ h) ≫ l",
"after_state": "No Goals!"
},
{
"line": "conv => lhs; slice 1 1; (rw [h₁])",
... |
example (h₁ : f₁ = f₂) : ((f₂ ≫ g) ≫ h) ≫ l = f₁ ≫ g ≫ h ≫ l := by
slice_rhs 1 1 => rw [h₁] | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Slice.lean | {
"open": [
"CategoryTheory"
],
"variables": [
"(C : Type) [Category C] (X Y Z W U : C)",
"(f₁ f₂ : X ⟶ Y) (g g₁ g₂ : Y ⟶ Z) (h : Z ⟶ W) (l : W ⟶ U)"
]
} | [
{
"line": "slice_rhs 1 1 => rw [h₁]",
"before_state": "C : Type\ninst✝ : Category.{u_1, 0} C\nX Y Z W U : C\nf₁ f₂ : X ⟶ Y\ng g₁ g₂ : Y ⟶ Z\nh : Z ⟶ W\nl : W ⟶ U\nh₁ : f₁ = f₂\n⊢ ((f₂ ≫ g) ≫ h) ≫ l = f₁ ≫ g ≫ h ≫ l",
"after_state": "No Goals!"
},
{
"line": "conv => rhs; slice 1 1; (rw [h₁])",
... |
example {f j : a ⟶ d} {g : a ⟶ b} {h : b ⟶ c} {i : c ⟶ d}
(η : f ⟶ g ≫ (h ≫ i)) (θ : (g ≫ h) ≫ i ⟶ j) :
η ⊗≫ θ = η ≫ (α_ _ _ _).inv ≫ θ := by
bicategory
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Bicategory/Basic.lean | {
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"Bicategory"
],
"variables": [
"{B : Type u} [Bicategory.{w, v} B]",
"{a b c d : B}"
]
} | [
{
"line": "bicategory",
"before_state": "B : Type u\ninst✝ : Bicategory B\na b c d : B\nf j : a ⟶ d\ng : a ⟶ b\nh : b ⟶ c\ni : c ⟶ d\nη : f ⟶ g ≫ h ≫ i\nθ : (g ≫ h) ≫ i ⟶ j\n⊢ η ⊗≫ θ = η ≫ (α_ g h i).inv ≫ θ",
"after_state": "No Goals!"
}
] |
example {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d} (η : h ⟶ i) :
(f ≫ g) ◁ η = (α_ _ _ _).hom ≫ f ◁ g ◁ η ≫ (α_ _ _ _).inv := by
bicategory
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Bicategory/Basic.lean | {
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"Bicategory"
],
"variables": [
"{B : Type u} [Bicategory.{w, v} B]",
"{a b c d : B}"
]
} | [
{
"line": "bicategory",
"before_state": "B : Type u\ninst✝ : Bicategory B\na b c d : B\nf : a ⟶ b\ng : b ⟶ c\nh i : c ⟶ d\nη : h ⟶ i\n⊢ (f ≫ g) ◁ η = (α_ f g h).hom ≫ f ◁ g ◁ η ≫ (α_ f g i).inv",
"after_state": "No Goals!"
}
] |
example {f g h : a ⟶ b} {η : f ⟶ g} {θ : g ⟶ h} : η ≫ θ = η ≫ θ := by
bicategory | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Bicategory/Basic.lean | {
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"Bicategory"
],
"variables": [
"{B : Type u} [Bicategory.{w, v} B]",
"{a b c d : B}"
]
} | [
{
"line": "bicategory",
"before_state": "B : Type u\ninst✝ : Bicategory B\na b c d : B\nf g h : a ⟶ b\nη : f ⟶ g\nθ : g ⟶ h\n⊢ η ≫ θ = η ≫ θ",
"after_state": "No Goals!"
}
] |
example (X : Type u) [Group X] : ⇑(𝟙 (of X)) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u\ninst✝ : Group X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Group X] [Group Y] (f : X →* Y) :
⇑(ofHom f) = ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Group X\ninst✝ : Group Y\nf : X →* Y\n⊢ ⇑(ConcreteCategory.hom (ofHom f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Group X] [Group Y] (f : X →* Y)
(x : X) : (ofHom f) x = f x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Group X\ninst✝ : Group Y\nf : X →* Y\nx : X\n⊢ (ConcreteCategory.hom (ofHom f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : Grp} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Grp\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Type u} [Group X] [Group Y] [Group Z]
(f : X →* Y) (g : Y →* Z) :
⇑(ofHom f ≫ ofHom g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : Group Y\ninst✝ : Group Z\nf : X →* Y\ng : Y →* Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ ofHom g)) = ⇑g ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Group X] [Group Y] {Z : Grp}
(f : X →* Y) (g : of Y ⟶ Z) :
⇑(ofHom f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Group X\ninst✝ : Group Y\nZ : Grp\nf : X →* Y\ng : of Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Grp} {Z : Type u} [Group Z] (f : X ⟶ Y) (g : Y ⟶ of Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Grp\nZ : Type u\ninst✝ : Group Z\nf : X ⟶ Y\ng : Y ⟶ of Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {Y Z : Grp} {X : Type u} [Group X] (f : of X ⟶ Y) (g : Y ⟶ Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "Y Z : Grp\nX : Type u\ninst✝ : Group X\nf : of X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Grp} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Grp\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ (ConcreteCategory.hom (f ≫ g)) x = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) x)",
"after_state": "No Goals!"
}
] |
example {X Y : Grp} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Grp\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : Grp} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Grp\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : Grp) : ⇑(𝟙 X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Grp\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {X : Type*} [Group X] : ⇑(MonoidHom.id X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u_1\ninst✝ : Group X\n⊢ ⇑(MonoidHom.id X) = id",
"after_state": "No Goals!"
}
] |
example {M N : Grp} (f : M ⟶ N) (x y : M) : f (x * y) = f x * f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : Grp\nf : M ⟶ N\nx y : ↑M\n⊢ (ConcreteCategory.hom f) (x * y) = (ConcreteCategory.hom f) x * (ConcreteCategory.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : Grp} (f : M ⟶ N) : f 1 = 1 := by
simp | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Grp.lean | {
"open": [
"CategoryTheory Grp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : Grp\nf : M ⟶ N\n⊢ (ConcreteCategory.hom f) 1 = 1",
"after_state": "No Goals!"
}
] |
example (X : Type v) [AddCommGroup X] [Module R X] : ⇑(𝟙 (of R X)) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nX : Type v\ninst✝¹ : AddCommGroup X\ninst✝ : Module R X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of R X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X →ₗ[R] Y) :
⇑(ModuleCat.ofHom f) = ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : AddCommGroup X\ninst✝² : Module R X\ninst✝¹ : AddCommGroup Y\ninst✝ : Module R Y\nf : X →ₗ[R] Y\n⊢ ⇑(ConcreteCategory.hom (↟f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X →ₗ[R] Y)
(x : X) : (ModuleCat.ofHom f) x = f x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : AddCommGroup X\ninst✝² : Module R X\ninst✝¹ : AddCommGroup Y\ninst✝ : Module R Y\nf : X →ₗ[R] Y\nx : X\n⊢ (ConcreteCategory.hom (↟f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : ModuleCat R} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Ring Z]
[Algebra R Z] (f : X →ₗ[R] Y) (g : Y →ₗ[R] Z) :
⇑(ModuleCat.ofHom f ≫ ModuleCat.ofHom g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁶ : CommRing R\nX Y Z : Type v\ninst✝⁵ : AddCommGroup X\ninst✝⁴ : Module R X\ninst✝³ : AddCommGroup Y\ninst✝² : Module R Y\ninst✝¹ : Ring Z\ninst✝ : Algebra R Z\nf : X →ₗ[R] Y\ng : Y →ₗ[R] Z\n⊢ ⇑(ConcreteCategory.hom (↟f ≫ ↟g)) = ⇑g ∘ ⇑f",
"after_state... |
example {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {Z : ModuleCat R}
(f : X →ₗ[R] Y) (g : of R Y ⟶ Z) :
⇑(ModuleCat.ofHom f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : AddCommGroup X\ninst✝² : Module R X\ninst✝¹ : AddCommGroup Y\ninst✝ : Module R Y\nZ : ModuleCat R\nf : X →ₗ[R] Y\ng : of R Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (↟f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state"... |
example {X Y : ModuleCat R} {Z : Type v} [Ring Z] [Algebra R Z] (f : X ⟶ Y) (g : Y ⟶ of R Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nX Y : ModuleCat R\nZ : Type v\ninst✝¹ : Ring Z\ninst✝ : Algebra R Z\nf : X ⟶ Y\ng : Y ⟶ of R Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {Y Z : ModuleCat R} {X : Type v} [AddCommGroup X] [Module R X] (f : of R X ⟶ Y) (g : Y ⟶ Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nY Z : ModuleCat R\nX : Type v\ninst✝¹ : AddCommGroup X\ninst✝ : Module R X\nf : of R X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : ModuleCat R} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y Z : ModuleCat R\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ (ConcreteCategory.hom (f ≫ g)) x = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) x)",
"after_state": "No Goals!"
}
] |
example {X Y : ModuleCat R} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y : ModuleCat R\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : ModuleCat R} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y : ModuleCat R\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : ModuleCat R) : ⇑(𝟙 X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX : ModuleCat R\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {M N : ModuleCat.{v} R} (f : M ⟶ N) (x y : M) : f (x + y) = f x + f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : ModuleCat R\nf : M ⟶ N\nx y : ↑M\n⊢ (ConcreteCategory.hom f) (x + y) = (ConcreteCategory.hom f) x + (ConcreteCategory.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : ModuleCat.{v} R} (f : M ⟶ N) : f 0 = 0 := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : ModuleCat R\nf : M ⟶ N\n⊢ (ConcreteCategory.hom f) 0 = 0",
"after_state": "No Goals!"
}
] |
example {M N : ModuleCat.{v} R} (f : M ⟶ N) (r : R) (m : M) : f (r • m) = r • f m := by
simp | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ModuleCat.lean | {
"open": [
"CategoryTheory ModuleCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
} | [
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : ModuleCat R\nf : M ⟶ N\nr : R\nm : ↑M\n⊢ (ConcreteCategory.hom f) (r • m) = r • (ConcreteCategory.hom f) m",
"after_state": "No Goals!"
}
] |
example (X : Type u) [Monoid X] : ⇑(𝟙 (of X)) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u\ninst✝ : Monoid X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Monoid X] [Monoid Y] (f : X →* Y) :
⇑(ofHom f) = ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Monoid X\ninst✝ : Monoid Y\nf : X →* Y\n⊢ ⇑(ConcreteCategory.hom (ofHom f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Monoid X] [Monoid Y] (f : X →* Y)
(x : X) : (ofHom f) x = f x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Monoid X\ninst✝ : Monoid Y\nf : X →* Y\nx : X\n⊢ (ConcreteCategory.hom (ofHom f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : MonCat} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : MonCat\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Type u} [Monoid X] [Monoid Y] [Monoid Z]
(f : X →* Y) (g : Y →* Z) :
⇑(ofHom f ≫ ofHom g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Monoid X\ninst✝¹ : Monoid Y\ninst✝ : Monoid Z\nf : X →* Y\ng : Y →* Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ ofHom g)) = ⇑g ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Monoid X] [Monoid Y] {Z : MonCat}
(f : X →* Y) (g : of Y ⟶ Z) :
⇑(ofHom f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Monoid X\ninst✝ : Monoid Y\nZ : MonCat\nf : X →* Y\ng : of Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : MonCat} {Z : Type u} [Monoid Z] (f : X ⟶ Y) (g : Y ⟶ of Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : MonCat\nZ : Type u\ninst✝ : Monoid Z\nf : X ⟶ Y\ng : Y ⟶ of Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {Y Z : MonCat} {X : Type u} [Monoid X] (f : of X ⟶ Y) (g : Y ⟶ Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "Y Z : MonCat\nX : Type u\ninst✝ : Monoid X\nf : of X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : MonCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : MonCat\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ (ConcreteCategory.hom (f ≫ g)) x = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) x)",
"after_state": "No Goals!"
}
] |
example {X Y : MonCat} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : MonCat\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : MonCat} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : MonCat\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : MonCat) : ⇑(𝟙 X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : MonCat\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {X : Type*} [Monoid X] : ⇑(MonoidHom.id X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u_1\ninst✝ : Monoid X\n⊢ ⇑(MonoidHom.id X) = id",
"after_state": "No Goals!"
}
] |
example {M N : MonCat} (f : M ⟶ N) (x y : M) : f (x * y) = f x * f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : MonCat\nf : M ⟶ N\nx y : ↑M\n⊢ (ConcreteCategory.hom f) (x * y) = (ConcreteCategory.hom f) x * (ConcreteCategory.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : MonCat} (f : M ⟶ N) : f 1 = 1 := by
simp | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/MonCat.lean | {
"open": [
"CategoryTheory MonCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : MonCat\nf : M ⟶ N\n⊢ (ConcreteCategory.hom f) 1 = 1",
"after_state": "No Goals!"
}
] |
example {U V W : ProfiniteGrp} (f : U ⟶ V) (g : V ⟶ W) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nU V W : ProfiniteGrp.{u_1}\nf : U ⟶ V\ng : V ⟶ W\n⊢ ⇑(Hom.hom (f ≫ g)) = ⇑(Hom.hom g) ∘ ⇑(Hom.hom f)",
"after_state": "No Goals!"
}
] |
example {U V W : ProfiniteGrp} (f : U ⟶ V) (g : V ⟶ W) (u : U) : (f ≫ g) u = g (f u) := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nU V W : ProfiniteGrp.{u_1}\nf : U ⟶ V\ng : V ⟶ W\nu : ↑U.toProfinite.toTop\n⊢ (Hom.hom (f ≫ g)) u = (Hom.hom g) ((Hom.hom f) u)",
"after_state": "No Goals!"
}
] |
example {U V : ProfiniteGrp} (e : U ≅ V) (u : U) : e.inv (e.hom u) = u := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nU V : ProfiniteGrp.{u_1}\ne : U ≅ V\nu : ↑U.toProfinite.toTop\n⊢ (Hom.hom e.inv) ((Hom.hom e.hom) u) = u",
"after_state": "No Goals!"
}
] |
example {U V : ProfiniteGrp} (e : U ≅ V) (v : V) : e.hom (e.inv v) = v := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nU V : ProfiniteGrp.{u_1}\ne : U ≅ V\nv : ↑V.toProfinite.toTop\n⊢ (Hom.hom e.hom) ((Hom.hom e.inv) v) = v",
"after_state": "No Goals!"
}
] |
example (U : ProfiniteGrp) : ⇑(𝟙 U) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nU : ProfiniteGrp.{u_1}\n⊢ ⇑(Hom.hom (𝟙 U)) = id",
"after_state": "No Goals!"
}
] |
example {M N : ProfiniteGrp.{u}} (f : M ⟶ N) (x y : M) : f (x * y) = f x * f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nM N : ProfiniteGrp.{u}\nf : M ⟶ N\nx y : ↑M.toProfinite.toTop\n⊢ (Hom.hom f) (x * y) = (Hom.hom f) x * (Hom.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : ProfiniteGrp.{u}} (f : M ⟶ N) : f 1 = 1 := by
simp | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/ProfiniteGrp.lean | {
"open": [
"CategoryTheory ProfiniteGrp"
],
"variables": [
"{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]"
]
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Group X\ninst✝¹ : TopologicalSpace X\ninst✝ : IsTopologicalGroup X\nM N : ProfiniteGrp.{u}\nf : M ⟶ N\n⊢ (Hom.hom f) 1 = 1",
"after_state": "No Goals!"
}
] |
example (X : Type u) [Semiring X] : ⇑(𝟙 (of X)) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u\ninst✝ : Semiring X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semiring X] [Semiring Y] (f : X →+* Y) :
⇑(ofHom f) = ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semiring X\ninst✝ : Semiring Y\nf : X →+* Y\n⊢ ⇑(ConcreteCategory.hom (ofHom f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semiring X] [Semiring Y] (f : X →+* Y)
(x : X) : (ofHom f) x = f x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semiring X\ninst✝ : Semiring Y\nf : X →+* Y\nx : X\n⊢ (ConcreteCategory.hom (ofHom f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : SemiRingCat} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : SemiRingCat\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Type u} [Semiring X] [Semiring Y] [Semiring Z]
(f : X →+* Y) (g : Y →+* Z) :
⇑(ofHom f ≫ ofHom g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Semiring X\ninst✝¹ : Semiring Y\ninst✝ : Semiring Z\nf : X →+* Y\ng : Y →+* Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ ofHom g)) = ⇑g ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semiring X] [Semiring Y] {Z : SemiRingCat}
(f : X →+* Y) (g : of Y ⟶ Z) :
⇑(ofHom f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semiring X\ninst✝ : Semiring Y\nZ : SemiRingCat\nf : X →+* Y\ng : of Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : SemiRingCat} {Z : Type u} [Semiring Z] (f : X ⟶ Y) (g : Y ⟶ of Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : SemiRingCat\nZ : Type u\ninst✝ : Semiring Z\nf : X ⟶ Y\ng : Y ⟶ of Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {Y Z : SemiRingCat} {X : Type u} [Semiring X] (f : of X ⟶ Y) (g : Y ⟶ Z) :
⇑(f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "Y Z : SemiRingCat\nX : Type u\ninst✝ : Semiring X\nf : of X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : SemiRingCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : SemiRingCat\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ (ConcreteCategory.hom (f ≫ g)) x = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) x)",
"after_state": "No Goals!"
}
] |
example {X Y : SemiRingCat} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : SemiRingCat\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : SemiRingCat} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : SemiRingCat\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : SemiRingCat) : ⇑(𝟙 X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : SemiRingCat\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {X : Type*} [Semiring X] : ⇑(RingHom.id X) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u_1\ninst✝ : Semiring X\n⊢ ⇑(RingHom.id X) = id",
"after_state": "No Goals!"
}
] |
example {M N : SemiRingCat} (f : M ⟶ N) (x y : M) : f (x + y) = f x + f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : SemiRingCat\nf : M ⟶ N\nx y : ↑M\n⊢ (ConcreteCategory.hom f) (x + y) = (ConcreteCategory.hom f) x + (ConcreteCategory.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : SemiRingCat} (f : M ⟶ N) (x y : M) : f (x * y) = f x * f y := by
simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : SemiRingCat\nf : M ⟶ N\nx y : ↑M\n⊢ (ConcreteCategory.hom f) (x * y) = (ConcreteCategory.hom f) x * (ConcreteCategory.hom f) y",
"after_state": "No Goals!"
}
] |
example {M N : SemiRingCat} (f : M ⟶ N) : f 0 = 0 := by
simp | /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Ring.lean | {
"open": [
"CategoryTheory SemiRingCat"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "M N : SemiRingCat\nf : M ⟶ N\n⊢ (ConcreteCategory.hom f) 0 = 0",
"after_state": "No Goals!"
}
] |
example (X : Type u) [Semigroup X] : ⇑(𝟙 (of X)) = id := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X : Type u\ninst✝ : Semigroup X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semigroup X] [Semigroup Y] (f : X →ₙ* Y) :
⇑(ofHom f) = ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semigroup X\ninst✝ : Semigroup Y\nf : X →ₙ* Y\n⊢ ⇑(ConcreteCategory.hom (ofHom f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semigroup X] [Semigroup Y] (f : X →ₙ* Y)
(x : X) : (ofHom f) x = f x := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semigroup X\ninst✝ : Semigroup Y\nf : X →ₙ* Y\nx : X\n⊢ (ConcreteCategory.hom (ofHom f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : Semigrp} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Semigrp\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom f)",
"after_state": "No Goals!"
}
] |
example {X Y Z : Type u} [Semigroup X] [Semigroup Y] [Semigroup Z]
(f : X →ₙ* Y) (g : Y →ₙ* Z) :
⇑(ofHom f ≫ ofHom g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y Z : Type u\ninst✝² : Semigroup X\ninst✝¹ : Semigroup Y\ninst✝ : Semigroup Z\nf : X →ₙ* Y\ng : Y →ₙ* Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ ofHom g)) = ⇑g ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type u} [Semigroup X] [Semigroup Y] {Z : Semigrp}
(f : X →ₙ* Y) (g : of Y ⟶ Z) :
⇑(ofHom f ≫ g) = g ∘ f := by simp
| /root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean | {
"open": [
"CategoryTheory Semigrp"
],
"variables": []
} | [
{
"line": "simp",
"before_state": "X Y : Type u\ninst✝¹ : Semigroup X\ninst✝ : Semigroup Y\nZ : Semigrp\nf : X →ₙ* Y\ng : of Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state": "No Goals!"
}
] |
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