url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.ext | [18, 1] | [21, 22] | rw [mk.injEq] | α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
⊢ a = b | α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
⊢ a.mem = b.mem | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
⊢ a = b
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.ext | [18, 1] | [21, 22] | funext x | α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
⊢ a.mem = b.mem | case h
α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
x : α
⊢ mem a x = mem b x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
⊢ a.mem = b.mem
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.ext | [18, 1] | [21, 22] | exact propext (h x) | case h
α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
x : α
⊢ mem a x = mem b x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
a b : Set α
h : ∀ (x : α), x ∈ a ↔ x ∈ b
x : α
⊢ mem a x = mem b x
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_id | [83, 1] | [86, 7] | ext | α : Type u_1
s : Set α
⊢ image id s = s | case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ image id s ↔ x✝ ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Set α
⊢ image id s = s
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_id | [83, 1] | [86, 7] | rw [image] | case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ image id s ↔ x✝ ∈ s | case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ id a = x} ↔ x✝ ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ image id s ↔ x✝ ∈ s
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_id | [83, 1] | [86, 7] | simp | case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ id a = x} ↔ x✝ ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
s : Set α
x✝ : α
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ id a = x} ↔ x✝ ∈ s
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_comp | [89, 1] | [93, 8] | ext | β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
⊢ image (g ∘ f) s = image g (image f s) | case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ image (g ∘ f) s ↔ x✝ ∈ image g (image f s) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
⊢ image (g ∘ f) s = image g (image f s)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_comp | [89, 1] | [93, 8] | unfold image | case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ image (g ∘ f) s ↔ x✝ ∈ image g (image f s) | case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ (g ∘ f) a = x} ↔ x✝ ∈ {x | ∃ a, a ∈ {x | ∃ a, a ∈ s ∧ f a = x} ∧ g a = x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ image (g ∘ f) s ↔ x✝ ∈ image g (image f s)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_comp | [89, 1] | [93, 8] | aesop | case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ (g ∘ f) a = x} ↔ x✝ ∈ {x | ∃ a, a ∈ {x | ∃ a, a ∈ s ∧ f a = x} ∧ g a = x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
β : Type u_1
γ : Type u_2
α : Type u_3
g : β → γ
f : α → β
s : Set α
x✝ : γ
⊢ x✝ ∈ {x | ∃ a, a ∈ s ∧ (g ∘ f) a = x} ↔ x✝ ∈ {x | ∃ a, a ∈ {x | ∃ a, a ∈ s ∧ f a = x} ∧ g a = x}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | ext x | α : Type u_1
β : Type u_2
f : α → β
x : α
⊢ image f {x} = {f x} | case h
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} ↔ x ∈ {f x✝} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
x : α
⊢ image f {x} = {f x}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | constructor | case h
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} ↔ x ∈ {f x✝} | case h.mp
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} → x ∈ {f x✝}
case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ {f x✝} → x ∈ image f {x✝} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} ↔ x ∈ {f x✝}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | rintro ⟨_, rfl, rfl⟩ | case h.mp
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} → x ∈ {f x✝} | case h.mp.intro.intro
α : Type u_1
β : Type u_2
f : α → β
w✝ : α
⊢ f w✝ ∈ {f w✝} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ image f {x✝} → x ∈ {f x✝}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | rfl | case h.mp.intro.intro
α : Type u_1
β : Type u_2
f : α → β
w✝ : α
⊢ f w✝ ∈ {f w✝} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
α : Type u_1
β : Type u_2
f : α → β
w✝ : α
⊢ f w✝ ∈ {f w✝}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | rintro rfl | case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ {f x✝} → x ∈ image f {x✝} | case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x : α
⊢ f x ∈ image f {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x✝ : α
x : β
⊢ x ∈ {f x✝} → x ∈ image f {x✝}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Set.lean | Set.image_singleton | [96, 1] | [102, 24] | exact ⟨x, rfl, rfl⟩ | case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x : α
⊢ f x ∈ image f {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
β : Type u_2
f : α → β
x : α
⊢ f x ∈ image f {x}
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΦΨ | [50, 1] | [52, 7] | unfold Φ Ψ | α : Type u_1
β : Type ?u.4710
γ : Type ?u.4713
δ : Type ?u.4716
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
x : Functor.obj F A
⊢ Φ (Ψ x) = x | α : Type u_1
β : Type ?u.4710
γ : Type ?u.4713
δ : Type ?u.4716
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
x : Functor.obj F A
⊢ NatTrans.app
{ app := fun B f => Functor.map F f x,
naturality :=
(_ :
∀ {B C : α} (f : B ⟶ C),
(fun B f => Functor.map F f x) C ∘ Functor.map (CoHom A) f =
Functor.map F f ∘ (fun B f => Functor.map F f x) B) }
A (𝟙 A) =
x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.4710
γ : Type ?u.4713
δ : Type ?u.4716
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
x : Functor.obj F A
⊢ Φ (Ψ x) = x
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΦΨ | [50, 1] | [52, 7] | simp | α : Type u_1
β : Type ?u.4710
γ : Type ?u.4713
δ : Type ?u.4716
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
x : Functor.obj F A
⊢ NatTrans.app
{ app := fun B f => Functor.map F f x,
naturality :=
(_ :
∀ {B C : α} (f : B ⟶ C),
(fun B f => Functor.map F f x) C ∘ Functor.map (CoHom A) f =
Functor.map F f ∘ (fun B f => Functor.map F f x) B) }
A (𝟙 A) =
x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.4710
γ : Type ?u.4713
δ : Type ?u.4716
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
x : Functor.obj F A
⊢ NatTrans.app
{ app := fun B f => Functor.map F f x,
naturality :=
(_ :
∀ {B C : α} (f : B ⟶ C),
(fun B f => Functor.map F f x) C ∘ Functor.map (CoHom A) f =
Functor.map F f ∘ (fun B f => Functor.map F f x) B) }
A (𝟙 A) =
x
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΨΦ | [55, 1] | [60, 18] | ext B | α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
⊢ Ψ (Φ η) = η | case app.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
⊢ NatTrans.app (Ψ (Φ η)) B = NatTrans.app η B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
⊢ Ψ (Φ η) = η
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΨΦ | [55, 1] | [60, 18] | funext f | case app.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
⊢ NatTrans.app (Ψ (Φ η)) B = NatTrans.app η B | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
⊢ NatTrans.app (Ψ (Φ η)) B = NatTrans.app η B
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΨΦ | [55, 1] | [60, 18] | have := congrFun (η.naturality f) (𝟙 A) | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : (NatTrans.app η B ∘ Functor.map (CoHom A) f) (𝟙 A) = (Functor.map F f ∘ NatTrans.app η A) (𝟙 A)
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΨΦ | [55, 1] | [60, 18] | simp at this | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : (NatTrans.app η B ∘ Functor.map (CoHom A) f) (𝟙 A) = (Functor.map F f ∘ NatTrans.app η A) (𝟙 A)
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : NatTrans.app η B f = Functor.map F f (NatTrans.app η A (𝟙 A))
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : (NatTrans.app η B ∘ Functor.map (CoHom A) f) (𝟙 A) = (Functor.map F f ∘ NatTrans.app η A) (𝟙 A)
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Yoneda.lean | Category.ΨΦ | [55, 1] | [60, 18] | exact this.symm | case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : NatTrans.app η B f = Functor.map F f (NatTrans.app η A (𝟙 A))
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h.h
α : Type u_1
β : Type ?u.4979
γ : Type ?u.4982
δ : Type ?u.4985
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ Type u_2
η : NatTrans (CoHom A) F
B : α
f : Functor.obj (CoHom A) B
this : NatTrans.app η B f = Functor.map F f (NatTrans.app η A (𝟙 A))
⊢ NatTrans.app (Ψ (Φ η)) B f = NatTrans.app η B f
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_strictly_positive | [11, 1] | [12, 8] | aesop | x y : ℤ
⊢ 0 < x ∧ 0 < y → 0 < x * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℤ
⊢ 0 < x ∧ 0 < y → 0 < x * y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_nonzero | [22, 1] | [23, 8] | aesop | x y : ℤ
⊢ x * y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℤ
⊢ x * y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_is_associative | [32, 1] | [33, 21] | rw [Int.mul_assoc] | x y z : ℤ
⊢ x * (y * z) = x * y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℤ
⊢ x * (y * z) = x * y * z
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | have h' : ∀ x y : Int, x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x := by
intros x y h1 h2 h3
by_cases h4 : x > 0
. apply le_mul_of_one_le_right (Int.nonneg_of_pos h4)
have h5 : y < 0 ∨ 1 ≤ y := by
rw [← Int.zero_add 1, Int.add_one_le_iff]
exact Int.lt_or_gt_of_ne h2
rcases h5 with h5 | h5
. have := Int.mul_neg_of_pos_of_neg h4 h5
linarith
. exact h5
. have h5 : x < 0 := by
simp only [gt_iff_lt, not_lt] at h4
exact lt_of_le_of_ne h4 h1
linarith | x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ x * y ≥ x ∧ x * y ≥ y | x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x ∧ x * y ≥ y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ x * y ≥ x ∧ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | constructor | x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x ∧ x * y ≥ y | case left
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x ∧ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . exact h' x y h1 h2 h3 | case left
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . rw [mul_comm] at h3
rw [mul_comm]
exact h' y x h2 h1 h3 | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | intros x y h1 h2 h3 | x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ x * y ≥ x | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | by_cases h4 : x > 0 | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ x * y ≥ x | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ x * y ≥ x
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . apply le_mul_of_one_le_right (Int.nonneg_of_pos h4)
have h5 : y < 0 ∨ 1 ≤ y := by
rw [← Int.zero_add 1, Int.add_one_le_iff]
exact Int.lt_or_gt_of_ne h2
rcases h5 with h5 | h5
. have := Int.mul_neg_of_pos_of_neg h4 h5
linarith
. exact h5 | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ x * y ≥ x
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x | case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ x * y ≥ x
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . have h5 : x < 0 := by
simp only [gt_iff_lt, not_lt] at h4
exact lt_of_le_of_ne h4 h1
linarith | case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | apply le_mul_of_one_le_right (Int.nonneg_of_pos h4) | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ x * y ≥ x | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ 1 ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | have h5 : y < 0 ∨ 1 ≤ y := by
rw [← Int.zero_add 1, Int.add_one_le_iff]
exact Int.lt_or_gt_of_ne h2 | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ 1 ≤ y | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0 ∨ 1 ≤ y
⊢ 1 ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | rcases h5 with h5 | h5 | case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0 ∨ 1 ≤ y
⊢ 1 ≤ y | case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
⊢ 1 ≤ y
case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0 ∨ 1 ≤ y
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . have := Int.mul_neg_of_pos_of_neg h4 h5
linarith | case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
⊢ 1 ≤ y
case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y | case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
⊢ 1 ≤ y
case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | . exact h5 | case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | rw [← Int.zero_add 1, Int.add_one_le_iff] | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ y < 0 ∨ 1 ≤ y | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ y < 0 ∨ 0 < y | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ y < 0 ∨ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | exact Int.lt_or_gt_of_ne h2 | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ y < 0 ∨ 0 < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
⊢ y < 0 ∨ 0 < y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | have := Int.mul_neg_of_pos_of_neg h4 h5 | case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
⊢ 1 ≤ y | case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
this : x * y < 0
⊢ 1 ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | linarith | case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
this : x * y < 0
⊢ 1 ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inl
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : y < 0
this : x * y < 0
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | exact h5 | case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x > 0
h5 : 1 ≤ y
⊢ 1 ≤ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | have h5 : x < 0 := by
simp only [gt_iff_lt, not_lt] at h4
exact lt_of_le_of_ne h4 h1 | case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x | case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
h5 : x < 0
⊢ x * y ≥ x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | linarith | case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
h5 : x < 0
⊢ x * y ≥ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
h5 : x < 0
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | simp only [gt_iff_lt, not_lt] at h4 | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x < 0 | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x ≤ 0
⊢ x < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : ¬x > 0
⊢ x < 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | exact lt_of_le_of_ne h4 h1 | x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x ≤ 0
⊢ x < 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ : ℤ
h1✝ : x✝ ≠ 0
h2✝ : y✝ ≠ 0
h3✝ : 0 ≤ x✝ * y✝
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h4 : x ≤ 0
⊢ x < 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | exact h' x y h1 h2 h3 | case left
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | rw [mul_comm] at h3 | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ x * y
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | rw [mul_comm] | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ y * x ≥ y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ x * y ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternalsNonlinear.lean | verus_lemma_mul_ordering | [56, 1] | [76, 26] | exact h' y x h2 h1 h3 | case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ y * x ≥ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℤ
h1 : x ≠ 0
h2 : y ≠ 0
h3 : 0 ≤ y * x
h' : ∀ (x y : ℤ), x ≠ 0 → y ≠ 0 → 0 ≤ x * y → x * y ≥ x
⊢ y * x ≥ y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/DivInternals.lean | verus_lemma_div_basics | [71, 1] | [76, 8] | sorry | n : ℤ
h : n > 0
⊢ n / n = 1 ∧
(-(-n / n) == 1) = true ∧
(∀ (x : ℤ), 0 ≤ x ∧ x < n ↔ x / n = 0) ∧ (∀ (x : ℤ), (x + n) / n = x / n + 1) ∧ ∀ (x : ℤ), (x - n) / n = x / n - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
h : n > 0
⊢ n / n = 1 ∧
(-(-n / n) == 1) = true ∧
(∀ (x : ℤ), 0 ≤ x ∧ x < n ↔ x / n = 0) ∧ (∀ (x : ℤ), (x + n) / n = x / n + 1) ∧ ∀ (x : ℤ), (x - n) / n = x / n - 1
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternals.lean | verus_lemma_mul_induction | [63, 1] | [66, 8] | sorry | f : ℤ → Bool
h1 : f 0 = true
h2 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + 1) = true
h3 : ∀ (i : ℤ), i ≤ 0 ∧ f i = true → f (i - 1) = true
⊢ ∀ (i : ℤ), f i = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → Bool
h1 : f 0 = true
h2 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + 1) = true
h3 : ∀ (i : ℤ), i ≤ 0 ∧ f i = true → f (i - 1) = true
⊢ ∀ (i : ℤ), f i = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternals.lean | verus_lemma_mul_successor | [100, 1] | [102, 82] | exact ⟨add_one_mul, sub_one_mul⟩ | ⊢ (∀ (x y : ℤ), (x + 1) * y = x * y + y) ∧ ∀ (x y : ℤ), (x - 1) * y = x * y - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (∀ (x y : ℤ), (x + 1) * y = x * y + y) ∧ ∀ (x y : ℤ), (x - 1) * y = x * y - y
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/MulInternals.lean | verus_lemma_mul_distributes | [149, 1] | [151, 86] | exact ⟨Int.add_mul, Int.sub_mul⟩ | ⊢ (∀ (x y z : ℤ), (x + y) * z = x * z + y * z) ∧ ∀ (x y z : ℤ), (x - y) * z = x * z - y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (∀ (x y z : ℤ), (x + y) * z = x * z + y * z) ∧ ∀ (x y z : ℤ), (x - y) * z = x * z - y * z
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper_pos | [43, 1] | [47, 8] | sorry | n : ℤ
f : ℤ → Bool
x : ℤ
h1 : x ≥ 0
h2 : n > 0
h3 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h4 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h5 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : x ≥ 0
h2 : n > 0
h3 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h4 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h5 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper_neg | [74, 1] | [78, 8] | sorry | n : ℤ
f : ℤ → Bool
x : ℤ
h1 : x < 0
h2 : n > 0
h3 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h4 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h5 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : x < 0
h2 : n > 0
h3 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h4 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h5 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | by_cases h0 : x ≥ 0 | n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true | case pos
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x ≥ 0
⊢ f x = true
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | . exact verus_lemma_induction_helper_pos n f x h0 h1 h2 h3 h4 | case pos
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x ≥ 0
⊢ f x = true
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true | case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x ≥ 0
⊢ f x = true
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | . simp only [ge_iff_le, not_le] at h0
exact verus_lemma_induction_helper_neg n f x h0 h1 h2 h3 h4 | case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | exact verus_lemma_induction_helper_pos n f x h0 h1 h2 h3 h4 | case pos
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x ≥ 0
⊢ f x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x ≥ 0
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | simp only [ge_iff_le, not_le] at h0 | case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true | case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x < 0
⊢ f x = true | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : ¬x ≥ 0
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/GeneralInternals.lean | verus_lemma_induction_helper | [99, 1] | [106, 64] | exact verus_lemma_induction_helper_neg n f x h0 h1 h2 h3 h4 | case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x < 0
⊢ f x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℤ
f : ℤ → Bool
x : ℤ
h1 : n > 0
h2 : ∀ (i : ℤ), 0 ≤ i ∧ i < n → f i = true
h3 : ∀ (i : ℤ), i ≥ 0 ∧ f i = true → f (i + n) = true
h4 : ∀ (i : ℤ), i < n ∧ f i = true → f (i - n) = true
h0 : x < 0
⊢ f x = true
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/DivInternalsNonlinear.lean | verus_lemma_small_div | [32, 1] | [34, 41] | intro x d h | ⊢ ∀ (x d : ℤ), 0 ≤ x ∧ x < d ∧ d > 0 → x / d = 0 | x d : ℤ
h : 0 ≤ x ∧ x < d ∧ d > 0
⊢ x / d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (x d : ℤ), 0 ≤ x ∧ x < d ∧ d > 0 → x / d = 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/DivInternalsNonlinear.lean | verus_lemma_small_div | [32, 1] | [34, 41] | exact Int.ediv_eq_zero_of_lt h.1 h.2.1 | x d : ℤ
h : 0 ≤ x ∧ x < d ∧ d > 0
⊢ x / d = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x d : ℤ
h : 0 ≤ x ∧ x < d ∧ d > 0
⊢ x / d = 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_mod_of_zero_is_zero | [20, 1] | [20, 84] | rfl | m : ℤ
h : 0 < m
⊢ 0 % m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℤ
h : 0 < m
⊢ 0 % m = 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_0_mod_any | [42, 1] | [42, 79] | intros | ⊢ ∀ m > 0, 0 % m = 0 | m✝ : ℤ
a✝ : m✝ > 0
⊢ 0 % m✝ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ m > 0, 0 % m = 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_0_mod_any | [42, 1] | [42, 79] | rfl | m✝ : ℤ
a✝ : m✝ > 0
⊢ 0 % m✝ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m✝ : ℤ
a✝ : m✝ > 0
⊢ 0 % m✝ = 0
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | have h1 : 0 ≤ (x : Int) := by simp only [Nat.cast_nonneg] | x m : ℕ
h : x < m
h' : 0 < m
⊢ ↑x % ↑m = ↑x | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
⊢ ↑x % ↑m = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
⊢ ↑x % ↑m = ↑x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | have h2 : 0 ≤ (m : Int) := by simp only [Nat.cast_nonneg] | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
⊢ ↑x % ↑m = ↑x | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑x % ↑m = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
⊢ ↑x % ↑m = ↑x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | rw [← Int.mod_eq_emod h1 h2, ← Int.ofNat_mod x m] | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑x % ↑m = ↑x | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑(x % m) = ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑x % ↑m = ↑x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | exact ((fun a b => (Mathlib.Tactic.Zify.nat_cast_eq a b).mp) x (x % m) (Nat.mod_eq_of_lt h).symm).symm | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑(x % m) = ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
h2 : 0 ≤ ↑m
⊢ ↑(x % m) = ↑x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | simp only [Nat.cast_nonneg] | x m : ℕ
h : x < m
h' : 0 < m
⊢ 0 ≤ ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
⊢ 0 ≤ ↑x
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_small_mod | [56, 1] | [60, 105] | simp only [Nat.cast_nonneg] | x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
⊢ 0 ≤ ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℕ
h : x < m
h' : 0 < m
h1 : 0 ≤ ↑x
⊢ 0 ≤ ↑m
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_mod_range | [71, 1] | [73, 55] | have h' : m ≠ 0 := Int.ne_of_gt h | x m : ℤ
h : m > 0
⊢ 0 ≤ x % m ∧ x % m < m | x m : ℤ
h : m > 0
h' : m ≠ 0
⊢ 0 ≤ x % m ∧ x % m < m | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℤ
h : m > 0
⊢ 0 ≤ x % m ∧ x % m < m
TACTIC:
|
https://github.com/JOSHCLUNE/VerusLeanStd.git | 5d6e2b282c6edc077aa15430efe347a31616b377 | VerusLeanStd/ModInternalsNonlinear.lean | verus_lemma_mod_range | [71, 1] | [73, 55] | exact ⟨Int.emod_nonneg x h', Int.emod_lt_of_pos x h⟩ | x m : ℤ
h : m > 0
h' : m ≠ 0
⊢ 0 ≤ x % m ∧ x % m < m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x m : ℤ
h : m > 0
h' : m ≠ 0
⊢ 0 ≤ x % m ∧ x % m < m
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | eliminate h1 with h4 h5 | p q r al_ac betty_beg carl_cac : Prop
h1 : al_ac ∧ (betty_beg ∨ carl_cac)
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
⊢ False | case intro
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h5 : betty_beg ∨ carl_cac
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
p q r al_ac betty_beg carl_cac : Prop
h1 : al_ac ∧ (betty_beg ∨ carl_cac)
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | eliminate h5 with h6 h7 | case intro
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h5 : betty_beg ∨ carl_cac
⊢ False | case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
⊢ False
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h5 : betty_beg ∨ carl_cac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | { have h8 : ¬ al_ac := h2 h6 contradiction } | case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
⊢ False
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
⊢ False
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | { have h9 : betty_beg := h3 h7 have h10 : ¬ al_ac := h2 h9 contradiction } | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | have h8 : ¬ al_ac := h2 h6 | case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
⊢ False | case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
h8 : ¬al_ac
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | contradiction | case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
h8 : ¬al_ac
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h6 : betty_beg
h8 : ¬al_ac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | have h9 : betty_beg := h3 h7 | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | have h10 : ¬ al_ac := h2 h9 | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
⊢ False | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
h10 : ¬al_ac
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Demos/Lecture04.lean | Lecture04.these_are_contradictory | [185, 1] | [202, 20] | contradiction | case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
h10 : ¬al_ac
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
p q r al_ac betty_beg carl_cac : Prop
h2 : betty_beg → ¬al_ac
h3 : carl_cac → betty_beg
h4 : al_ac
h7 : carl_cac
h9 : betty_beg
h10 : ¬al_ac
⊢ False
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.refl | [12, 11] | [12, 74] | ring | n a : ℤ
⊢ a - a = n * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a : ℤ
⊢ a - a = n * 0
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.add | [14, 11] | [18, 47] | obtain ⟨x, hx⟩ := h1 | n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a + c ≡ b + d [ZMOD n] | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a + c ≡ b + d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a + c ≡ b + d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.add | [14, 11] | [18, 47] | obtain ⟨y, hy⟩ := h2 | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a + c ≡ b + d [ZMOD n] | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a + c ≡ b + d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a + c ≡ b + d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.add | [14, 11] | [18, 47] | exact ⟨x + y, by linear_combination hx + hy⟩ | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a + c ≡ b + d [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a + c ≡ b + d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.add | [14, 11] | [18, 47] | linear_combination hx + hy | n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a + c - (b + d) = n * (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a + c - (b + d) = n * (x + y)
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.sub | [26, 11] | [30, 47] | obtain ⟨x, hx⟩ := h1 | n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a - c ≡ b - d [ZMOD n] | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a - c ≡ b - d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a - c ≡ b - d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.sub | [26, 11] | [30, 47] | obtain ⟨y, hy⟩ := h2 | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a - c ≡ b - d [ZMOD n] | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a - c ≡ b - d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a - c ≡ b - d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.sub | [26, 11] | [30, 47] | exact ⟨x - y, by linear_combination hx - hy⟩ | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a - c ≡ b - d [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a - c ≡ b - d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.sub | [26, 11] | [30, 47] | linear_combination hx - hy | n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a - c - (b - d) = n * (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a - c - (b - d) = n * (x - y)
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.neg | [38, 11] | [40, 40] | obtain ⟨x, hx⟩ := h1 | n a b : ℤ
h1 : a ≡ b [ZMOD n]
⊢ -a ≡ -b [ZMOD n] | case intro
n a b x : ℤ
hx : a - b = n * x
⊢ -a ≡ -b [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h1 : a ≡ b [ZMOD n]
⊢ -a ≡ -b [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.neg | [38, 11] | [40, 40] | exact ⟨-x, by linear_combination -hx⟩ | case intro
n a b x : ℤ
hx : a - b = n * x
⊢ -a ≡ -b [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b x : ℤ
hx : a - b = n * x
⊢ -a ≡ -b [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.neg | [38, 11] | [40, 40] | linear_combination -hx | n a b x : ℤ
hx : a - b = n * x
⊢ -a - -b = n * -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b x : ℤ
hx : a - b = n * x
⊢ -a - -b = n * -x
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.mul | [42, 11] | [46, 63] | obtain ⟨x, hx⟩ := h1 | n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a * c ≡ b * d [ZMOD n] | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a * c ≡ b * d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d : ℤ
h1 : a ≡ b [ZMOD n]
h2 : c ≡ d [ZMOD n]
⊢ a * c ≡ b * d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.mul | [42, 11] | [46, 63] | obtain ⟨y, hy⟩ := h2 | case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a * c ≡ b * d [ZMOD n] | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a * c ≡ b * d [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b c d : ℤ
h2 : c ≡ d [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a * c ≡ b * d [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.mul | [42, 11] | [46, 63] | exact ⟨x * c + b * y, by linear_combination c * hx + b * hy⟩ | case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a * c ≡ b * d [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a * c ≡ b * d [ZMOD n]
TACTIC:
|
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