declaration
stringlengths 27
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| file
stringlengths 52
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| context
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listlengths 1
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|
|---|---|---|---|
example {X Y : Semigrp} {Z : Type u} [Semigroup Z] (f : X ⟶ Y) (g : Y ⟶ of 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 : Semigrp\nZ : Type u\ninst✝ : Semigroup 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 : Semigrp} {X : Type u} [Semigroup X] (f : of 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": "Y Z : Semigrp\nX : Type u\ninst✝ : Semigroup 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 : Semigrp} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := 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\nx : ↑X\n⊢ (ConcreteCategory.hom (f ≫ g)) x = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) x)",
"after_state": "No Goals!"
}
] |
example {X Y : Semigrp} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean
|
{
"open": [
"CategoryTheory Semigrp"
],
"variables": []
}
|
[
{
"line": "simp",
"before_state": "X Y : Semigrp\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : Semigrp} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean
|
{
"open": [
"CategoryTheory Semigrp"
],
"variables": []
}
|
[
{
"line": "simp",
"before_state": "X Y : Semigrp\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : Semigrp) : ⇑(𝟙 X) = id := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean
|
{
"open": [
"CategoryTheory Semigrp"
],
"variables": []
}
|
[
{
"line": "simp",
"before_state": "X : Semigrp\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {X : Type*} [Semigroup X] : ⇑(MulHom.id X) = id := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean
|
{
"open": [
"CategoryTheory Semigrp"
],
"variables": []
}
|
[
{
"line": "simp",
"before_state": "X : Type u_1\ninst✝ : Semigroup X\n⊢ ⇑(MulHom.id X) = id",
"after_state": "No Goals!"
}
] |
example {M N : Semigrp} (f : M ⟶ N) (x y : M) : f (x * y) = f x * f y := by
simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/Semigrp.lean
|
{
"open": [
"CategoryTheory Semigrp"
],
"variables": []
}
|
[
{
"line": "simp",
"before_state": "M N : Semigrp\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 (X : Type u) [Ring X] [Algebra R X] : ⇑(𝟙 (of R X)) = id := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nX : Type u\ninst✝¹ : Ring X\ninst✝ : Algebra R X\n⊢ ⇑(ConcreteCategory.hom (𝟙 (of R X))) = id",
"after_state": "No Goals!"
}
] |
example {X Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] (f : X →ₐ[R] Y) :
⇑(ofHom f) = ⇑f := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : Ring X\ninst✝² : Algebra R X\ninst✝¹ : Ring Y\ninst✝ : Algebra R Y\nf : X →ₐ[R] Y\n⊢ ⇑(ConcreteCategory.hom (ofHom f)) = ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] (f : X →ₐ[R] Y)
(x : X) : (ofHom f) x = f x := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : Ring X\ninst✝² : Algebra R X\ninst✝¹ : Ring Y\ninst✝ : Algebra R Y\nf : X →ₐ[R] Y\nx : X\n⊢ (ConcreteCategory.hom (ofHom f)) x = f x",
"after_state": "No Goals!"
}
] |
example {X Y Z : AlgebraCat R} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(f ≫ g) = ⇑g ∘ ⇑f := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat 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} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] [Ring Z]
[Algebra R Z] (f : X →ₐ[R] Y) (g : Y →ₐ[R] Z) :
⇑(ofHom f ≫ ofHom g) = g ∘ f := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁶ : CommRing R\nX Y Z : Type v\ninst✝⁵ : Ring X\ninst✝⁴ : Algebra R X\ninst✝³ : Ring Y\ninst✝² : Algebra R Y\ninst✝¹ : Ring Z\ninst✝ : Algebra R Z\nf : X →ₐ[R] Y\ng : Y →ₐ[R] Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ ofHom g)) = ⇑g ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] {Z : AlgebraCat R}
(f : X →ₐ[R] Y) (g : of R Y ⟶ Z) :
⇑(ofHom f ≫ g) = g ∘ f := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝⁴ : CommRing R\nX Y : Type v\ninst✝³ : Ring X\ninst✝² : Algebra R X\ninst✝¹ : Ring Y\ninst✝ : Algebra R Y\nZ : AlgebraCat R\nf : X →ₐ[R] Y\ng : of R Y ⟶ Z\n⊢ ⇑(ConcreteCategory.hom (ofHom f ≫ g)) = ⇑(ConcreteCategory.hom g) ∘ ⇑f",
"after_state": "No Goals!"
}
] |
example {X Y : AlgebraCat 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/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nX Y : AlgebraCat 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 : AlgebraCat R} {X : Type v} [Ring X] [Algebra R X] (f : of R X ⟶ Y) (g : Y ⟶ Z) :
⇑(f ≫ g) = g ∘ f := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝² : CommRing R\nY Z : AlgebraCat R\nX : Type v\ninst✝¹ : Ring X\ninst✝ : Algebra 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 : AlgebraCat R} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat 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 : AlgebraCat R} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y : AlgebraCat R\ne : X ≅ Y\nx : ↑X\n⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x",
"after_state": "No Goals!"
}
] |
example {X Y : AlgebraCat R} (e : X ≅ Y) (y : Y) : e.hom (e.inv y) = y := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX Y : AlgebraCat R\ne : X ≅ Y\ny : ↑Y\n⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) y) = y",
"after_state": "No Goals!"
}
] |
example (X : AlgebraCat R) : ⇑(𝟙 X) = id := by simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nX : AlgebraCat R\n⊢ ⇑(ConcreteCategory.hom (𝟙 X)) = id",
"after_state": "No Goals!"
}
] |
example {M N : AlgebraCat.{v} R} (f : M ⟶ N) (x y : M) : f (x + y) = f x + f y := by
simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : AlgebraCat 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 : AlgebraCat.{v} R} (f : M ⟶ N) : f 0 = 0 := by
simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : AlgebraCat R\nf : M ⟶ N\n⊢ (ConcreteCategory.hom f) 0 = 0",
"after_state": "No Goals!"
}
] |
example {M N : AlgebraCat.{v} R} (f : M ⟶ N) (r : R) (m : M) : f (r • m) = r • f m := by
simp
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/ConcreteCategory/AlgebraCat.lean
|
{
"open": [
"CategoryTheory AlgebraCat"
],
"variables": [
"(R : Type u) [CommRing R]"
]
}
|
[
{
"line": "simp",
"before_state": "R : Type u\ninst✝ : CommRing R\nM N : AlgebraCat R\nf : M ⟶ N\nr : R\nm : ↑M\n⊢ (ConcreteCategory.hom f) (r • m) = r • (ConcreteCategory.hom f) m",
"after_state": "No Goals!"
}
] |
example (f : U ⟶ V ⊗ (W ⊗ X)) (g : (V ⊗ W) ⊗ X ⟶ Y) :
f ⊗≫ g = f ≫ (α_ _ _ _).inv ≫ g := by
monoidal
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Monoidal/Basic.lean
|
{
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)"
]
}
|
[
{
"line": "monoidal",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\nf✝ : X ⟶ Y\ng✝ : Y ⟶ Z\nU V : 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!"
}
] |
example (f : Z ⟶ W) : (X ⊗ Y) ◁ f = (α_ _ _ _).hom ≫ X ◁ Y ◁ f ≫ (α_ _ _ _).inv := by
monoidal
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Monoidal/Basic.lean
|
{
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)"
]
}
|
[
{
"line": "monoidal",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nf : Z ⟶ W\n⊢ (X ⊗ Y) ◁ f = (α_ X Y Z).hom ≫ X ◁ Y ◁ f ≫ (α_ X Y W).inv",
"after_state": "No Goals!"
}
] |
example : f ≫ g = f ≫ g := by
monoidal
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Monoidal/Basic.lean
|
{
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)"
]
}
|
[
{
"line": "monoidal",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ f ≫ g = f ≫ g",
"after_state": "No Goals!"
}
] |
example : (f ⊗ g) ▷ X = (α_ _ _ _).hom ≫ (f ⊗ g ▷ X) ≫ (α_ _ _ _).inv := by
monoidal
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Monoidal/Basic.lean
|
{
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)"
]
}
|
[
{
"line": "monoidal",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ (f ⊗ g) ▷ X = (α_ X Y X).hom ≫ (f ⊗ g ▷ X) ≫ (α_ Y Z X).inv",
"after_state": "No Goals!"
}
] |
example {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) :
R V₁ V₂ ▷ V₃ ⊗≫ V₂ ◁ R V₁ V₃ =
R V₁ V₂ ▷ V₃ ≫ (α_ _ _ _).hom ⊗≫ 𝟙 _ ≫ V₂ ◁ R V₁ V₃ := by
monoidal
|
/root/DuelModelResearch/mathlib4/MathlibTest/CategoryTheory/Monoidal/Basic.lean
|
{
"open": [
"CategoryTheory Mathlib.Tactic BicategoryLike",
"MonoidalCategory"
],
"variables": [
"{C : Type u} [Category.{v} C] [MonoidalCategory C]",
"{X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)"
]
}
|
[
{
"line": "monoidal",
"before_state": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z W : C\nf : X ⟶ Y\ng : Y ⟶ Z\nV₁ V₂ V₃ : C\nR : (V₁ V₂ : C) → V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁\n⊢ R V₁ V₂ ▷ V₃ ⊗≫ V₂ ◁ R V₁ V₃ = R V₁ V₂ ▷ V₃ ≫ (α_ V₂ V₁ V₃).hom ⊗≫ 𝟙 (V₂ ⊗ V₁ ⊗ V₃) ≫ V₂ ◁ R V₁ V₃",
"after_state": "No Goals!"
}
] |
example {x : ℤ} (hx : x ≥ 12) : x * x ^ 2 ≥ 12 * x ^ 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x ≥ 12\n⊢ x * x ^ 2 ≥ 12 * x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
}
] |
example {x y : ℤ} (hx : x ≥ 12) : y + x * x ≥ y + 12 * x := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℤ\nhx : x ≥ 12\n⊢ y + x * x ≥ y + 12 * x",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x",
"after_state": "No Goals!"
}
] |
example {x y : ℤ} (hx : x ≥ 12) : y + x * x ≥ y + 12 * x := by rel [hx]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "rel [hx]",
"before_state": "x y : ℤ\nhx : x ≥ 12\n⊢ y + x * x ≥ y + 12 * x",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ x",
"after_state": "No Goals!"
}
] |
example {x : ℤ} (hx : x > 12) : x * x ^ 2 > 12 * x ^ 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x > 12\n⊢ x * x ^ 2 > 12 * x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℤ\nhx : x > 12\n⊢ 0 < x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℤ\nhx : x > 12\n⊢ 0 < x ^ 2",
"after_state": "No Goals!"
}
] |
example {x y : ℤ} (hx : x > 12) : y + x * x > y + 12 * x := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℤ\nhx : x > 12\n⊢ y + x * x > y + 12 * x",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\nx y : ℤ\nhx : x > 12\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\nx y : ℤ\nhx : x > 12\n⊢ 0 < x",
"after_state": "No Goals!"
}
] |
example {n m : ℤ} (hn : n ≥ 10) : n * n ^ 3 - m ≥ 10 * n ^ 3 - m := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "n m : ℤ\nhn : n ≥ 10\n⊢ n * n ^ 3 - m ≥ 10 * n ^ 3 - m",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.a0\nn m : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 3",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.a0\nn m : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 3",
"after_state": "No Goals!"
}
] |
example {k m n : ℤ} (hn : n ≥ 10) : m + 7 * n * n ^ 2 - k ≥ m + 7 * 10 * n ^ 2 - k := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "k m n : ℤ\nhn : n ≥ 10\n⊢ m + 7 * n * n ^ 2 - k ≥ m + 7 * 10 * n ^ 2 - k",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.bc.h.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ 7",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.bc.h.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ 7",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.bc.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.bc.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 2",
"after_state": "No Goals!"
}
] |
example {k m n : ℤ} (hn : n ≥ 10) : m + 7 * n * n ^ 2 - k ≥ m + 7 * 10 * n ^ 2 - k := by rel [hn]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "rel [hn]",
"before_state": "k m n : ℤ\nhn : n ≥ 10\n⊢ m + 7 * n * n ^ 2 - k ≥ m + 7 * 10 * n ^ 2 - k",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.bc.h.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ 7",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.bc.h.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ 7",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.bc.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.bc.a0\nk m n : ℤ\nhn : n ≥ 10\n⊢ 0 ≤ n ^ 2",
"after_state": "No Goals!"
}
] |
example {x : ℤ} (hx : x ≥ 12) : x ≥ 12 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x ≥ 12\n⊢ x ≥ 12",
"after_state": "No Goals!"
}
] |
example {x y : ℤ} (hx : x ≥ 12) : y + 8 * x ≥ y + 8 * 12 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℤ\nhx : x ≥ 12\n⊢ y + 8 * x ≥ y + 8 * 12",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ 8",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\nx y : ℤ\nhx : x ≥ 12\n⊢ 0 ≤ 8",
"after_state": "No Goals!"
}
] |
example {a b x c d : ℝ} (h1 : a ≤ b) (h2 : c ≤ d) : x ^ 2 * a + c ≤ x ^ 2 * b + d := by
rel [h1, h2]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "rel [h1, h2]",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\n⊢ x ^ 2 * a + c ≤ x ^ 2 * b + d",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
}
] |
example {a b c x y : ℤ} (hb : b ≥ 4) (hxy : x ≤ y) :
c + (3 * |a| ^ 3 * b + b * 7 + 14) * x ≤ c + (3 * |a| ^ 3 * b + b * 7 + 14) * y := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b c x y : ℤ\nhb : b ≥ 4\nhxy : x ≤ y\n⊢ c + (3 * |a| ^ 3 * b + b * 7 + 14) * x ≤ c + (3 * |a| ^ 3 * b + b * 7 + 14) * y",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\na b c x y : ℤ\nhb : b ≥ 4\nhxy : x ≤ y\n⊢ 0 ≤ 3 * |a| ^ 3 * b + b * 7 + 14",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\na b c x y : ℤ\nhb : b ≥ 4\nhxy : x ≤ y\n⊢ 0 ≤ 3 * |a| ^ 3 * b + b * 7 + 14",
"after_state": "No Goals!"
}
] |
example {x y : ℤ} (hy : 3 ≤ y) (hx : x ≥ 9) : y + 2 * x ≥ 3 + 2 * 9 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℤ\nhy : 3 ≤ y\nhx : x ≥ 9\n⊢ y + 2 * x ≥ 3 + 2 * 9",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₂.a0\nx y : ℤ\nhy : 3 ≤ y\nhx : x ≥ 9\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₂.a0\nx y : ℤ\nhy : 3 ≤ y\nhx : x ≥ 9\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {b : ℤ} (h2 : b ≥ 3) : 2 * b + 5 ≥ 2 * 3 + 5 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "b : ℤ\nh2 : b ≥ 3\n⊢ 2 * b + 5 ≥ 2 * 3 + 5",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\nb : ℤ\nh2 : b ≥ 3\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\nb : ℤ\nh2 : b ≥ 3\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h1 : x ≤ 3) : 4 * x - 3 ≤ 4 * 3 - 3 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh1 : x ≤ 3\n⊢ 4 * x - 3 ≤ 4 * 3 - 3",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.a0\nx : ℝ\nh1 : x ≤ 3\n⊢ 0 ≤ 4",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.a0\nx : ℝ\nh1 : x ≤ 3\n⊢ 0 ≤ 4",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h : x < 1) : 3 * x ≤ 3 * 1 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh : x < 1\n⊢ 3 * x ≤ 3 * 1",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℝ\nh : x < 1\n⊢ 0 ≤ 3",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℝ\nh : x < 1\n⊢ 0 ≤ 3",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h1 : x < 3) : 4 * x - 3 < 4 * 3 - 3 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh1 : x < 3\n⊢ 4 * x - 3 < 4 * 3 - 3",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h.a0\nx : ℝ\nh1 : x < 3\n⊢ 0 < 4",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.a0\nx : ℝ\nh1 : x < 3\n⊢ 0 < 4",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h : x < 1) : 3 * x < 3 * 1 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh : x < 1\n⊢ 3 * x < 3 * 1",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℝ\nh : x < 1\n⊢ 0 < 3",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℝ\nh : x < 1\n⊢ 0 < 3",
"after_state": "No Goals!"
}
] |
example {x y : ℝ} (h1 : 1 ≤ y) (h2 : x < 2) : x * y ≤ 2 * y := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℝ\nh1 : 1 ≤ y\nh2 : x < 2\n⊢ x * y ≤ 2 * y",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y : ℝ\nh1 : 1 ≤ y\nh2 : x < 2\n⊢ 0 ≤ y",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y : ℝ\nh1 : 1 ≤ y\nh2 : x < 2\n⊢ 0 ≤ y",
"after_state": "No Goals!"
}
] |
example {a b c : ℚ} (h2 : 2 ≤ a + b) : 2 + c ≤ (a + b) + c := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b c : ℚ\nh2 : 2 ≤ a + b\n⊢ 2 + c ≤ a + b + c",
"after_state": "No Goals!"
}
] |
example {a b : ℚ} (h1 : 3 ≤ a) (h2 : 4 ≤ b) : (3 + 4) / 2 ≤ (a + b) / 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℚ\nh1 : 3 ≤ a\nh2 : 4 ≤ b\n⊢ (3 + 4) / 2 ≤ (a + b) / 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\na b : ℚ\nh1 : 3 ≤ a\nh2 : 4 ≤ b\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\na b : ℚ\nh1 : 3 ≤ a\nh2 : 4 ≤ b\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {a : ℚ} (h1 : 3 ≤ a) : 3 / 2 ≤ a / 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a : ℚ\nh1 : 3 ≤ a\n⊢ 3 / 2 ≤ a / 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\na : ℚ\nh1 : 3 ≤ a\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\na : ℚ\nh1 : 3 ≤ a\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {a : ℝ} (h1 : 3 ≤ a) : 3 / 2 ≤ a / 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a : ℝ\nh1 : 3 ≤ a\n⊢ 3 / 2 ≤ a / 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\na : ℝ\nh1 : 3 ≤ a\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\na : ℝ\nh1 : 3 ≤ a\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {x y : ℝ} (h : 3 ≤ x) (h' : 1 ≤ y) : (3 + 1) / 2 ≤ (x + y) / 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y : ℝ\nh : 3 ≤ x\nh' : 1 ≤ y\n⊢ (3 + 1) / 2 ≤ (x + y) / 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx y : ℝ\nh : 3 ≤ x\nh' : 1 ≤ y\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx y : ℝ\nh : 3 ≤ x\nh' : 1 ≤ y\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h : x ≤ 3) : 0.1 * x ≤ 0.1 * 3 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh : x ≤ 3\n⊢ 0.1 * x ≤ 0.1 * 3",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 0.1",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 0.1",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h : x ≤ 3) : x / 10 ≤ 3 / 10 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh : x ≤ 3\n⊢ x / 10 ≤ 3 / 10",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 10",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 10",
"after_state": "No Goals!"
}
] |
example {x : ℝ} (h : x ≤ 3) : 1 / 10 * x ≤ 1 / 10 * 3 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℝ\nh : x ≤ 3\n⊢ 1 / 10 * x ≤ 1 / 10 * 3",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 1 / 10",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx : ℝ\nh : x ≤ 3\n⊢ 0 ≤ 1 / 10",
"after_state": "No Goals!"
}
] |
example (a b c d : ℕ) (h1 : a ≤ b) (h2 : c ≤ d) : a * c ≤ b * d := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b c d : ℕ\nh1 : a ≤ b\nh2 : c ≤ d\n⊢ a * c ≤ b * d",
"after_state": "No Goals!"
}
] |
example {a b : ℚ} (h : 0 ≤ a ^ 6) : 0 + b ≤ a ^ 6 + b := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℚ\nh : 0 ≤ a ^ 6\n⊢ 0 + b ≤ a ^ 6 + b",
"after_state": "No Goals!"
}
] |
example {a b : ℚ} (h₁ : a ≤ b) (c : ℝ) : (a + c : ℝ) ≤ b + c := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℚ\nh₁ : a ≤ b\nc : ℝ\n⊢ ↑a + c ≤ ↑b + c",
"after_state": "No Goals!"
}
] |
example {a b : ℚ} (h₁ : a < b) (c : ℝ) : (a + c : ℝ) < b + c := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℚ\nh₁ : a < b\nc : ℝ\n⊢ ↑a + c < ↑b + c",
"after_state": "No Goals!"
}
] |
example {k m n : ℤ} (H : m ^ 2 ≤ n ^ 2) : k + m ^ 2 ≤ k + n ^ 2 := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "k m n : ℤ\nH : m ^ 2 ≤ n ^ 2\n⊢ k + m ^ 2 ≤ k + n ^ 2",
"after_state": "No Goals!"
}
] |
example (n k : ℕ) (H : n % k + 1 ≤ k % n + 1) : n % k ≤ k % n := by
success_if_fail_with_msg
"gcongr did not make progress"
(gcongr)
linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "success_if_fail_with_msg \"gcongr did not make progress\"(gcongr)",
"before_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\n⊢ n % k ≤ k % n",
"after_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\n⊢ n % k ≤ k % n"
},
{
"line": "gcongr",
"before_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\n⊢ n % k ≤ k % n",
"after_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\n⊢ n % k ≤ k % n"
},
{
"line": "linarith",
"before_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\n⊢ n % k ≤ k % n",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n k : ℕ\nH : n % k + 1 ≤ k % n + 1\na✝ : n % k > k % n\n⊢ -1 + (↑n % ↑k + 1 - (↑k % ↑n + 1)) + (↑k % ↑n + 1 - ↑n % ↑k) = 0",
"after_state": "No Goals!"
}
] |
example {x : ℤ} (hx : x ≥ 12) (h : Even x) : Even x := by
success_if_fail_with_msg "rel failed, goal not a relation" (rel [hx])
exact h
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "success_if_fail_with_msg \"rel failed, goal not a relation\"(rel [hx])",
"before_state": "x : ℤ\nhx : x ≥ 12\nh : Even x\n⊢ Even x",
"after_state": "x : ℤ\nhx : x ≥ 12\nh : Even x\n⊢ Even x"
},
{
"line": "rel [hx]",
"before_state": "x : ℤ\nhx : x ≥ 12\nh : Even x\n⊢ Even x",
"after_state": "x : ℤ\nhx : x ≥ 12\nh : Even x\n⊢ Even x"
},
{
"line": "exact h",
"before_state": "x : ℤ\nhx : x ≥ 12\nh : Even x\n⊢ Even x",
"after_state": "No Goals!"
}
] |
example {a b x c d : ℝ} (h1 : a ≤ b) (h2 : c ≤ d) (h3 : 1 ≤ x + 1) : x * a + c ≤ x * b + d := by
success_if_fail_with_msg
"rel failed, cannot prove goal by 'substituting' the listed relationships. \
The steps which could not be automatically justified were:\n0 ≤ x\nc ≤ d"
(rel [h1])
have : 0 ≤ x := by linarith
rel [h1, h2]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "success_if_fail_with_msg\n \"rel failed, cannot prove goal by 'substituting' the listed relationships. \\\n The steps which could not be automatically justified were:\\n0 ≤ x\\nc ≤ d\"(rel [h1])",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d",
"after_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d"
},
{
"line": "rel [h1]",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d",
"after_state": "No Goals!"
},
{
"line": "have : 0 ≤ x := by linarith",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d",
"after_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : 0 ≤ x := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (linarith)",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d",
"after_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 0 ≤ x := ?body✝;\n ?_)",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ x * a + c ≤ x * b + d",
"after_state": "case body\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ 0 ≤ x\n---\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d"
},
{
"line": "case body✝ => with_annotate_state\"by\" (linarith)",
"before_state": "case body\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ 0 ≤ x\n---\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d",
"after_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d"
},
{
"line": "with_annotate_state\"by\" (linarith)",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ 0 ≤ x",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\n⊢ 0 ≤ x",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\na✝ : 0 > x\n⊢ 1 - (x + 1) + (x - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "rel [h1, h2]",
"before_state": "a b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ x * a + c ≤ x * b + d",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ 0 ≤ x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a ≤ b\nh2 : c ≤ d\nh3 : 1 ≤ x + 1\nthis : 0 ≤ x\n⊢ 0 ≤ x",
"after_state": "No Goals!"
}
] |
example {a b x c d : ℝ} (h1 : a + 1 ≤ b + 1) (h2 : c + 2 ≤ d + 2) :
x ^ 2 * a + c ≤ x ^ 2 * b + d := by
gcongr <;> linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "focus\n gcongr\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "a b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ x ^ 2 * a + c ≤ x ^ 2 * b + d",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "a b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ x ^ 2 * a + c ≤ x ^ 2 * b + d",
"after_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.a0\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d",
"after_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d"
},
{
"line": "skip",
"before_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d",
"after_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d"
},
{
"line": "all_goals linarith",
"before_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b\n---\ncase h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₁.h\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ a ≤ b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\na✝ : a > b\n⊢ a + 1 - (b + 1) + (b - a) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₂\na b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\n⊢ c ≤ d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b x c d : ℝ\nh1 : a + 1 ≤ b + 1\nh2 : c + 2 ≤ d + 2\na✝ : c > d\n⊢ c + 2 - (d + 2) + (d - c) = 0",
"after_state": "No Goals!"
}
] |
example {a b c d x : ℝ} (h : a + c + 1 ≤ b + d + 1) :
x ^ 2 * (a + c) + 5 ≤ x ^ 2 * (b + d) + 5 := by
gcongr x ^ 2 * ?_ + 5
linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr x ^ 2 * ?_ + 5",
"before_state": "a b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\n⊢ x ^ 2 * (a + c) + 5 ≤ x ^ 2 * (b + d) + 5",
"after_state": "case bc.h\na b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\n⊢ a + c ≤ b + d"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.a0\na b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.a0\na b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case bc.h\na b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\n⊢ a + c ≤ b + d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d x : ℝ\nh : a + c + 1 ≤ b + d + 1\na✝ : a + c > b + d\n⊢ a + c + 1 - (b + d + 1) + (b + d - (a + c)) = 0",
"after_state": "No Goals!"
}
] |
example {x y z : ℝ} (h : 2 ≤ z) : z * |x + y| ≤ z * (|x| + |y|) := by gcongr; apply abs_add
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x y z : ℝ\nh : 2 ≤ z\n⊢ z * |x + y| ≤ z * (|x| + |y|)",
"after_state": "case h\nx y z : ℝ\nh : 2 ≤ z\n⊢ |x + y| ≤ |x| + |y|"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y z : ℝ\nh : 2 ≤ z\n⊢ 0 ≤ z",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y z : ℝ\nh : 2 ≤ z\n⊢ 0 ≤ z",
"after_state": "No Goals!"
},
{
"line": "apply abs_add",
"before_state": "case h\nx y z : ℝ\nh : 2 ≤ z\n⊢ |x + y| ≤ |x| + |y|",
"after_state": "No Goals!"
}
] |
example (A B C : ℝ) : |A + B| + C ≤ |A| + |B| + C := by gcongr; apply abs_add
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "A B C : ℝ\n⊢ |A + B| + C ≤ |A| + |B| + C",
"after_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|"
},
{
"line": "apply abs_add",
"before_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|",
"after_state": "No Goals!"
}
] |
example (A B C : ℝ) : |A + B| + C ≤ |A| + |B| + C := by gcongr ?_ + _; apply abs_add
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr ?_ + _",
"before_state": "A B C : ℝ\n⊢ |A + B| + C ≤ |A| + |B| + C",
"after_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|"
},
{
"line": "apply abs_add",
"before_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|",
"after_state": "No Goals!"
}
] |
example (A B C : ℝ) : |A + B| + C ≤ |A| + |B| + C := by gcongr ?_ + (A : ℝ); apply abs_add
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr ?_ + (A : ℝ)",
"before_state": "A B C : ℝ\n⊢ |A + B| + C ≤ |A| + |B| + C",
"after_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|"
},
{
"line": "apply abs_add",
"before_state": "case bc\nA B C : ℝ\n⊢ |A + B| ≤ |A| + |B|",
"after_state": "No Goals!"
}
] |
example {n i : ℕ} (hi : i ∈ range n) : 2 ^ i ≤ 2 ^ n := by
gcongr
· norm_num
· apply le_of_lt
simpa using hi
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "n i : ℕ\nhi : i ∈ range n\n⊢ 2 ^ i ≤ 2 ^ n",
"after_state": "case ha\nn i : ℕ\nhi : i ∈ range n\n⊢ 1 ≤ 2\n---\ncase hmn\nn i : ℕ\nhi : i ∈ range n\n⊢ i ≤ n"
},
{
"line": "norm_num",
"before_state": "case ha\nn i : ℕ\nhi : i ∈ range n\n⊢ 1 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "apply le_of_lt",
"before_state": "case hmn\nn i : ℕ\nhi : i ∈ range n\n⊢ i ≤ n",
"after_state": "case hmn.hab\nn i : ℕ\nhi : i ∈ range n\n⊢ i < n"
},
{
"line": "simpa using hi",
"before_state": "case hmn.hab\nn i : ℕ\nhi : i ∈ range n\n⊢ i < n",
"after_state": "No Goals!"
}
] |
example {n' : ℕ} (hn' : 6 ≤ n') : 2 ^ ((n' + 1) * (n' + 1)) ≤ 2 ^ (n' * n' + 4 * n') := by
gcongr
· norm_num
· linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "n' : ℕ\nhn' : 6 ≤ n'\n⊢ 2 ^ ((n' + 1) * (n' + 1)) ≤ 2 ^ (n' * n' + 4 * n')",
"after_state": "case ha\nn' : ℕ\nhn' : 6 ≤ n'\n⊢ 1 ≤ 2\n---\ncase hmn\nn' : ℕ\nhn' : 6 ≤ n'\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'"
},
{
"line": "norm_num",
"before_state": "case ha\nn' : ℕ\nhn' : 6 ≤ n'\n⊢ 1 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case hmn\nn' : ℕ\nhn' : 6 ≤ n'\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n' : ℕ\nhn' : 6 ≤ n'\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 12 * -1 + 2 * (6 - ↑n') + (↑n' * ↑n' + 4 * ↑n' + 1 - (↑n' + 1) * (↑n' + 1)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "n' : ℕ\nhn' : 6 ≤ n'\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "n' : ℕ\nhn' : 6 ≤ n'\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 2 > 0",
"after_state": "No Goals!"
}
] |
example {F : ℕ → ℕ} (le_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N) {n' : ℕ} (hn' : 6 ≤ n') :
let A := F n';
A ! * (15 + 1) ^ n' ≤ A ! * (A + 1) ^ n' := by
intro A
gcongr
exact le_sum hn'
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "intro A",
"before_state": "F : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\n⊢ let A := F n';\n A ! * (15 + 1) ^ n' ≤ A ! * (A + 1) ^ n'",
"after_state": "F : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ A ! * (15 + 1) ^ n' ≤ A ! * (A + 1) ^ n'"
},
{
"line": "gcongr",
"before_state": "F : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ A ! * (15 + 1) ^ n' ≤ A ! * (A + 1) ^ n'",
"after_state": "case bc.hab.bc\nF : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ 15 ≤ A"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.ha\nF : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ 0 ≤ 15 + 1",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.ha\nF : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ 0 ≤ 15 + 1",
"after_state": "No Goals!"
},
{
"line": "exact le_sum hn'",
"before_state": "case bc.hab.bc\nF : ℕ → ℕ\nle_sum : ∀ {N : ℕ}, 6 ≤ N → 15 ≤ F N\nn' : ℕ\nhn' : 6 ≤ n'\nA : ℕ := F n'\n⊢ 15 ≤ A",
"after_state": "No Goals!"
}
] |
example {a : ℤ} {n : ℕ} (ha : ∀ i < n, 2 ^ i ≤ a) :
∏ i ∈ range n, (a - 2 ^ i) ≤ ∏ _i ∈ range n, a := by
gcongr with i
· intro i hi
simp only [mem_range] at hi
linarith [ha i hi]
· have : 0 ≤ 2 ^ i := by positivity
linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr with i",
"before_state": "a : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\n⊢ ∏ i ∈ range n, (a - 2 ^ i) ≤ ∏ _i ∈ range n, a",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\n⊢ ∀ i ∈ range n, 0 ≤ a - 2 ^ i\n---\ncase h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ a - 2 ^ i ≤ a"
},
{
"line": "intro i hi",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\n⊢ ∀ i ∈ range n, 0 ≤ a - 2 ^ i",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ a - 2 ^ i"
},
{
"line": "intro i;\n intro hi",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\n⊢ ∀ i ∈ range n, 0 ≤ a - 2 ^ i",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ a - 2 ^ i"
},
{
"line": "intro i",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\n⊢ ∀ i ∈ range n, 0 ≤ a - 2 ^ i",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\n⊢ i ∈ range n → 0 ≤ a - 2 ^ i"
},
{
"line": "intro hi",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\n⊢ i ∈ range n → 0 ≤ a - 2 ^ i",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ a - 2 ^ i"
},
{
"line": "simp only [mem_range] at hi",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ a - 2 ^ i",
"after_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i < n\n⊢ 0 ≤ a - 2 ^ i"
},
{
"line": "linarith [ha i hi]",
"before_state": "case h0\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i < n\n⊢ 0 ≤ a - 2 ^ i",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\nhi : i < n\na✝ : 0 > a - 2 ^ i\n⊢ -1 + (a - 2 ^ i + 1 - 0) + (2 ^ i - a) = 0",
"after_state": "No Goals!"
},
{
"line": "have : 0 ≤ 2 ^ i := by positivity",
"before_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ a - 2 ^ i ≤ a",
"after_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : 0 ≤ 2 ^ i := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ a - 2 ^ i ≤ a",
"after_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 0 ≤ 2 ^ i := ?body✝;\n ?_)",
"before_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ a - 2 ^ i ≤ a",
"after_state": "case body\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ 0 ≤ 2 ^ i\n---\ncase h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ 0 ≤ 2 ^ i\n---\ncase h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a",
"after_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "a : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ 0 ≤ 2 ^ i",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "a : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\n⊢ 0 ≤ 2 ^ i",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h1\na : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝ : i ∈ range n\nthis : 0 ≤ 2 ^ i\n⊢ a - 2 ^ i ≤ a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a : ℤ\nn : ℕ\nha : ∀ i < n, 2 ^ i ≤ a\ni : ℕ\na✝¹ : i ∈ range n\nthis : 0 ≤ 2 ^ i\na✝ : a - 2 ^ i > a\n⊢ -1 + (0 - 2 ^ i) + (a + 1 - (a - 2 ^ i)) = 0",
"after_state": "No Goals!"
}
] |
example {a b c d e : ℝ} (_h1 : 0 ≤ b) (_h2 : 0 ≤ c) (hac : a * b + 1 ≤ c * d + 1) (_hbd : b ≤ d) :
a * b + e ≤ c * d + e := by
gcongr ?_ + _
guard_target =ₛ a * b ≤ c * d
linarith
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr ?_ + _",
"before_state": "a b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\n⊢ a * b + e ≤ c * d + e",
"after_state": "case bc\na b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\n⊢ a * b ≤ c * d"
},
{
"line": "guard_target =ₛ a * b ≤ c * d",
"before_state": "case bc\na b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\n⊢ a * b ≤ c * d",
"after_state": "case bc\na b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\n⊢ a * b ≤ c * d"
},
{
"line": "linarith",
"before_state": "case bc\na b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\n⊢ a * b ≤ c * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d e : ℝ\n_h1 : 0 ≤ b\n_h2 : 0 ≤ c\nhac : a * b + 1 ≤ c * d + 1\n_hbd : b ≤ d\na✝ : a * b > c * d\n⊢ a * b + 1 - (c * d + 1) + (c * d - a * b) = 0",
"after_state": "No Goals!"
}
] |
example (f g : ℕ → ℕ) (s : Finset ℕ) (h : ∀ i ∈ s, f i ≤ g i) :
∑ i ∈ s, (3 + f i ^ 2) ≤ ∑ i ∈ s, (3 + g i ^ 2) := by
gcongr with i hi
exact h i hi
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr with i hi",
"before_state": "f g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ≤ g i\n⊢ ∑ i ∈ s, (3 + f i ^ 2) ≤ ∑ i ∈ s, (3 + g i ^ 2)",
"after_state": "case h.bc.hab\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ≤ g i\ni : ℕ\nhi : i ∈ s\n⊢ f i ≤ g i"
},
{
"line": "gcongr_discharger",
"before_state": "case h.bc.ha\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ≤ g i\ni : ℕ\nhi : i ∈ s\n⊢ 0 ≤ f i",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h.bc.ha\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ≤ g i\ni : ℕ\nhi : i ∈ s\n⊢ 0 ≤ f i",
"after_state": "No Goals!"
},
{
"line": "exact h i hi",
"before_state": "case h.bc.hab\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ≤ g i\ni : ℕ\nhi : i ∈ s\n⊢ f i ≤ g i",
"after_state": "No Goals!"
}
] |
example (f g : ℕ → ℕ) (s : Finset ℕ) (h : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1) :
∑ i ∈ s, f i ^ 2 ≤ ∑ i ∈ s, g i ^ 2 := by
gcongr ∑ _i ∈ s, ?_ with i hi
linarith [h i hi]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr∑ _i ∈ s, ?_ with i hi",
"before_state": "f g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\n⊢ ∑ i ∈ s, f i ^ 2 ≤ ∑ i ∈ s, g i ^ 2",
"after_state": "case h\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\n⊢ f i ^ 2 ≤ g i ^ 2"
},
{
"line": "linarith [h i hi]",
"before_state": "case h\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\n⊢ f i ^ 2 ≤ g i ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\na✝ : f i ^ 2 > g i ^ 2\n⊢ -1 + (↑(g i) ^ 2 + 1 - ↑(f i) ^ 2) + (↑(f i) ^ 2 + 1 - (↑(g i) ^ 2 + 1)) = 0",
"after_state": "No Goals!"
}
] |
example (f g : ℕ → ℕ) (s : Finset ℕ) (h : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1) :
∑ i ∈ s, (3 + f i ^ 2) ≤ ∑ i ∈ s, (3 + g i ^ 2) := by
gcongr ∑ _i ∈ s, (3 + ?_) with i hi
linarith [h i hi]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr∑ _i ∈ s, (3 + ?_) with i hi",
"before_state": "f g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\n⊢ ∑ i ∈ s, (3 + f i ^ 2) ≤ ∑ i ∈ s, (3 + g i ^ 2)",
"after_state": "case h.bc\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\n⊢ f i ^ 2 ≤ g i ^ 2"
},
{
"line": "linarith [h i hi]",
"before_state": "case h.bc\nf g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\n⊢ f i ^ 2 ≤ g i ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f g : ℕ → ℕ\ns : Finset ℕ\nh : ∀ i ∈ s, f i ^ 2 + 1 ≤ g i ^ 2 + 1\ni : ℕ\nhi : i ∈ s\na✝ : f i ^ 2 > g i ^ 2\n⊢ -1 + (↑(g i) ^ 2 + 1 - ↑(f i) ^ 2) + (↑(f i) ^ 2 + 1 - (↑(g i) ^ 2 + 1)) = 0",
"after_state": "No Goals!"
}
] |
example {α β : Type*} [SemilatticeSup α] (f : β → α)
{s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) (h₁ : s₁.Nonempty) :
s₁.sup' h₁ f ≤ s₂.sup' (h₁.mono h) f := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "α : Type u_1\nβ : Type u_2\ninst✝ : SemilatticeSup α\nf : β → α\ns₁ s₂ : Finset β\nh : s₁ ⊆ s₂\nh₁ : s₁.Nonempty\n⊢ s₁.sup' h₁ f ≤ s₂.sup' ⋯ f",
"after_state": "No Goals!"
}
] |
example {α β : Type*} [SemilatticeSup α] (f : β → α)
{s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty) :
s₁.sup' h₁ f ≤ s₂.sup' h₂ f := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/inequalities.lean
|
{
"open": [
"Nat Finset"
],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "α : Type u_1\nβ : Type u_2\ninst✝ : SemilatticeSup α\nf : β → α\ns₁ s₂ : Finset β\nh : s₁ ⊆ s₂\nh₁ : s₁.Nonempty\nh₂ : s₂.Nonempty\n⊢ s₁.sup' h₁ f ≤ s₂.sup' h₂ f",
"after_state": "No Goals!"
}
] |
example (ha : a ≡ 2 [ZMOD 4]) : a * b ^ 2 + a ^ 2 * b + 3 ≡ 2 * b ^ 2 + 2 ^ 2 * b + 3 [ZMOD 4] := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℤ\nha : a ≡ 2 [ZMOD 4]\n⊢ a * b ^ 2 + a ^ 2 * b + 3 ≡ 2 * b ^ 2 + 2 ^ 2 * b + 3 [ZMOD 4]",
"after_state": "No Goals!"
}
] |
example (ha : a ≡ 4 [ZMOD 5]) (hb : b ≡ 3 [ZMOD 5]) :
a * b + b ^ 3 + 3 ≡ 4 * 3 + 3 ^ 3 + 3 [ZMOD 5] := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "a b : ℤ\nha : a ≡ 4 [ZMOD 5]\nhb : b ≡ 3 [ZMOD 5]\n⊢ a * b + b ^ 3 + 3 ≡ 4 * 3 + 3 ^ 3 + 3 [ZMOD 5]",
"after_state": "No Goals!"
}
] |
example (hb : 3 ≡ b [ZMOD 5]) : b ^ 2 ≡ 3 ^ 2 [ZMOD 5] := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "b : ℤ\nhb : 3 ≡ b [ZMOD 5]\n⊢ b ^ 2 ≡ 3 ^ 2 [ZMOD 5]",
"after_state": "No Goals!"
}
] |
example (hb : 3 ≡ b [ZMOD 5]) : b ^ 2 ≡ 3 ^ 2 [ZMOD 5] := by rel [hb]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rel [hb]",
"before_state": "b : ℤ\nhb : 3 ≡ b [ZMOD 5]\n⊢ b ^ 2 ≡ 3 ^ 2 [ZMOD 5]",
"after_state": "No Goals!"
}
] |
example (hx : x ≡ 0 [ZMOD 3]) : x ^ 3 ≡ 0 ^ 3 [ZMOD 3] := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x ≡ 0 [ZMOD 3]\n⊢ x ^ 3 ≡ 0 ^ 3 [ZMOD 3]",
"after_state": "No Goals!"
}
] |
example (hx : x ≡ 2 [ZMOD 3]) : x ^ 3 ≡ 2 ^ 3 [ZMOD 3] := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x ≡ 2 [ZMOD 3]\n⊢ x ^ 3 ≡ 2 ^ 3 [ZMOD 3]",
"after_state": "No Goals!"
}
] |
example (hn : n ≡ 1 [ZMOD 3]) : n ^ 3 + 7 * n ≡ 1 ^ 3 + 7 * 1 [ZMOD 3] := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "n : ℤ\nhn : n ≡ 1 [ZMOD 3]\n⊢ n ^ 3 + 7 * n ≡ 1 ^ 3 + 7 * 1 [ZMOD 3]",
"after_state": "No Goals!"
}
] |
example (hn : n ≡ 1 [ZMOD 3]) : n ^ 3 + 7 * n ≡ 1 ^ 3 + 7 * 1 [ZMOD 3] := by rel [hn]
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rel [hn]",
"before_state": "n : ℤ\nhn : n ≡ 1 [ZMOD 3]\n⊢ n ^ 3 + 7 * n ≡ 1 ^ 3 + 7 * 1 [ZMOD 3]",
"after_state": "No Goals!"
}
] |
example (hn : n ≡ 1 [ZMOD 2]) : 5 * n ^ 2 + 3 * n + 7 ≡ 5 * 1 ^ 2 + 3 * 1 + 7 [ZMOD 2] := by
gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "n : ℤ\nhn : n ≡ 1 [ZMOD 2]\n⊢ 5 * n ^ 2 + 3 * n + 7 ≡ 5 * 1 ^ 2 + 3 * 1 + 7 [ZMOD 2]",
"after_state": "No Goals!"
}
] |
example (hx : x ≡ 3 [ZMOD 5]) : x ^ 5 ≡ 3 ^ 5 [ZMOD 5] := by gcongr
|
/root/DuelModelResearch/mathlib4/MathlibTest/GCongr/mod.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "gcongr",
"before_state": "x : ℤ\nhx : x ≡ 3 [ZMOD 5]\n⊢ x ^ 5 ≡ 3 ^ 5 [ZMOD 5]",
"after_state": "No Goals!"
}
] |
example (x : Nat) : x ≠ x.succ := ne_of_lt (by apply?)
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "x : ℕ\n⊢ x < x.succ",
"after_state": "No Goals!"
},
{
"line": "exact Nat.lt_add_one x",
"before_state": "x : ℕ\n⊢ x < x.succ",
"after_state": "No Goals!"
}
] |
example : 0 ≠ 1 + 1 := ne_of_lt (by apply?)
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "⊢ 0 < 1 + 1",
"after_state": "No Goals!"
},
{
"line": "exact Nat.zero_lt_succ 1",
"before_state": "⊢ 0 < 1 + 1",
"after_state": "No Goals!"
}
] |
example (x y : Nat) : x + y = y + x := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "x y : ℕ\n⊢ x + y = y + x",
"after_state": "No Goals!"
},
{
"line": "exact Nat.add_comm x y",
"before_state": "x y : ℕ\n⊢ x + y = y + x",
"after_state": "No Goals!"
}
] |
example (n m k : Nat) : n ≤ m → n + k ≤ m + k := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "n m k : ℕ\n⊢ n ≤ m → n + k ≤ m + k",
"after_state": "No Goals!"
},
{
"line": "exact fun a => Nat.add_le_add_right a k",
"before_state": "n m k : ℕ\n⊢ n ≤ m → n + k ≤ m + k",
"after_state": "No Goals!"
}
] |
example (_ha : a > 0) (w : b ∣ c) : a * b ∣ a * c := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "a b c : ℕ\n_ha : a > 0\nw : b ∣ c\n⊢ a * b ∣ a * c",
"after_state": "No Goals!"
},
{
"line": "exact Nat.mul_dvd_mul_left a w",
"before_state": "a b c : ℕ\n_ha : a > 0\nw : b ∣ c\n⊢ a * b ∣ a * c",
"after_state": "No Goals!"
}
] |
example : Int := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "exact Aesop.defaultSafePenalty",
"before_state": "⊢ ℤ",
"after_state": "No Goals!"
}
] |
example (P : Prop) (p : P) : P := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "P : Prop\np : P\n⊢ P",
"after_state": "No Goals!"
},
{
"line": "exact p",
"before_state": "P : Prop\np : P\n⊢ P",
"after_state": "No Goals!"
}
] |
example (P : Prop) (p : P) (np : ¬P) : false := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "P : Prop\np : P\nnp : ¬P\n⊢ false = true",
"after_state": "No Goals!"
},
{
"line": "exact False.elim (np p)",
"before_state": "P : Prop\np : P\nnp : ¬P\n⊢ false = true",
"after_state": "No Goals!"
}
] |
example (X : Type) (P : Prop) (x : X) (h : ∀ x : X, x = x → P) : P := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "X : Type\nP : Prop\nx : X\nh : ∀ (x : X), x = x → P\n⊢ P",
"after_state": "No Goals!"
},
{
"line": "exact h x rfl",
"before_state": "X : Type\nP : Prop\nx : X\nh : ∀ (x : X), x = x → P\n⊢ P",
"after_state": "No Goals!"
}
] |
example (α : Prop) : α → α := by apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "α : Prop\n⊢ α → α",
"after_state": "No Goals!"
},
{
"line": "exact fun a => a",
"before_state": "α : Prop\n⊢ α → α",
"after_state": "No Goals!"
}
] |
example (a b : ℕ) : a + b = b + a := by
apply?
|
/root/DuelModelResearch/mathlib4/MathlibTest/LibrarySearch/basic.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "apply?",
"before_state": "a b : ℕ\n⊢ a + b = b + a",
"after_state": "No Goals!"
},
{
"line": "exact Nat.add_comm a b",
"before_state": "a b : ℕ\n⊢ a + b = b + a",
"after_state": "No Goals!"
}
] |
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