Documentation

Mathlib.Algebra.PUnitInstances

Instances on PUnit #

This file collects facts about algebraic structures on the one-element type, e.g. that it is a commutative ring.

theorem PUnit.addCommGroup.proof_2 :

∀ (a : PUnit.{u_1 + 1} ), 0 + a = 0 + a

instance PUnit.addCommGroup :

AddCommGroup PUnit.{u_1 + 1}

Equations

PUnit.addCommGroup = AddCommGroup.mk PUnit.addCommGroup.proof_11

theorem PUnit.addCommGroup.proof_3 :

∀ (a : PUnit.{u_1 + 1} ), a + 0 = a + 0

theorem PUnit.addCommGroup.proof_9 :

∀ (n : ℕ) (a : PUnit.{u_1 + 1} ), (fun x x => PUnit.unit) (Int.negSucc n) a = (fun x x => PUnit.unit) (Int.negSucc n) a

theorem PUnit.addCommGroup.proof_7 :

∀ (a : PUnit.{u_1 + 1} ), (fun x x => PUnit.unit) 0 a = (fun x x => PUnit.unit) 0 a

theorem PUnit.addCommGroup.proof_1 :

∀ (a b c : PUnit.{u_1 + 1} ), a + b + c = a + b + c

theorem PUnit.addCommGroup.proof_4 :

∀ (x : PUnit.{u_1 + 1} ), (fun x x => PUnit.unit) 0 x = (fun x x => PUnit.unit) 0 x

theorem PUnit.addCommGroup.proof_8 :

∀ (n : ℕ) (a : PUnit.{u_1 + 1} ), (fun x x => PUnit.unit) (Int.ofNat (Nat.succ n)) a = (fun x x => PUnit.unit) (Int.ofNat (Nat.succ n)) a

theorem PUnit.addCommGroup.proof_6 :

∀ (a b : PUnit.{u_1 + 1} ), a - b = a - b

theorem PUnit.addCommGroup.proof_5 :

∀ (n : ℕ) (x : PUnit.{u_1 + 1} ), (fun x x => PUnit.unit) (n + 1) x = (fun x x => PUnit.unit) (n + 1) x

theorem PUnit.addCommGroup.proof_10 :

∀ (a : PUnit.{u_1 + 1} ), -a + a = -a + a

theorem PUnit.addCommGroup.proof_11 :

∀ (a b : PUnit.{u_1 + 1} ), a + b = a + b

instance PUnit.commGroup :

CommGroup PUnit.{u_1 + 1}

Equations

PUnit.commGroup = CommGroup.mk PUnit.commGroup.proof_11

instance PUnit.instZeroPUnit :

Equations

PUnit.instZeroPUnit = { zero := () }

instance PUnit.instOnePUnit :

Equations

PUnit.instOnePUnit = { one := () }

instance PUnit.instAddPUnit :

Equations

PUnit.instAddPUnit = { add := fun x x => () }

instance PUnit.instMulPUnit :

Equations

PUnit.instMulPUnit = { mul := fun x x => () }

instance PUnit.instSubPUnit :

Equations

PUnit.instSubPUnit = { sub := fun x x => () }

instance PUnit.instDivPUnit :

Equations

PUnit.instDivPUnit = { div := fun x x => () }

instance PUnit.instNegPUnit :

Equations

PUnit.instNegPUnit = { neg := fun x => () }

instance PUnit.instInvPUnit :

Equations

PUnit.instInvPUnit = { inv := fun x => () }

@[simp]

theorem PUnit.zero_eq :

@[simp]

theorem PUnit.one_eq :

theorem PUnit.add_eq {x : PUnit.{1}} {y : PUnit.{1}} :

x + y = PUnit.unit

theorem PUnit.mul_eq {x : PUnit.{1}} {y : PUnit.{1}} :

x * y = PUnit.unit

@[simp]

theorem PUnit.sub_eq {x : PUnit.{1}} {y : PUnit.{1}} :

x - y = PUnit.unit

@[simp]

theorem PUnit.div_eq {x : PUnit.{1}} {y : PUnit.{1}} :

x / y = PUnit.unit

@[simp]

theorem PUnit.neg_eq {x : PUnit.{1}} :

-x = PUnit.unit

@[simp]

theorem PUnit.inv_eq {x : PUnit.{1}} :

x⁻¹ = PUnit.unit

instance PUnit.commRing :

CommRing PUnit.{u_1 + 1}

Equations

PUnit.commRing = let src := PUnit.commGroup; let src_1 := PUnit.addCommGroup; CommRing.mk (_ : ∀ (a b : PUnit.{u_1 + 1} ), a * b = b * a)

instance PUnit.cancelCommMonoidWithZero :

CancelCommMonoidWithZero PUnit.{1}

Equations

PUnit.cancelCommMonoidWithZero = let src := PUnit.commRing; CancelCommMonoidWithZero.mk

instance PUnit.normalizedGCDMonoid :

NormalizedGCDMonoid PUnit.{1}

Equations

PUnit.normalizedGCDMonoid = NormalizedGCDMonoid.mk PUnit.normalizedGCDMonoid.proof_10 PUnit.normalizedGCDMonoid.proof_11

theorem PUnit.gcd_eq {x : PUnit.{1}} {y : PUnit.{1}} :

gcd x y = PUnit.unit

theorem PUnit.lcm_eq {x : PUnit.{1}} {y : PUnit.{1}} :

lcm x y = PUnit.unit

@[simp]

theorem PUnit.norm_unit_eq {x : PUnit.{1}} :

instance PUnit.canonicallyOrderedAddMonoid :

CanonicallyOrderedAddMonoid PUnit.{u_1 + 1}

Equations

One or more equations did not get rendered due to their size.

instance PUnit.linearOrderedCancelAddCommMonoid :

LinearOrderedCancelAddCommMonoid PUnit.{u_1 + 1}

Equations

One or more equations did not get rendered due to their size.

instance PUnit.instLinearOrderedAddCommMonoidWithTopPUnit :

LinearOrderedAddCommMonoidWithTop PUnit.{u_1 + 1}

Equations

One or more equations did not get rendered due to their size.

instance PUnit.vadd {R : Type u_1} :

VAdd R PUnit.{u_2 + 1}

Equations

PUnit.vadd = { vadd := fun x x => PUnit.unit }

instance PUnit.smul {R : Type u_1} :

SMul R PUnit.{u_2 + 1}

Equations

PUnit.smul = { smul := fun x x => PUnit.unit }

@[simp]

theorem PUnit.vadd_eq {R : Type u_1} (y : PUnit.{u_2 + 1} ) (r : R) :

r +ᵥ y = PUnit.unit

@[simp]

theorem PUnit.smul_eq {R : Type u_1} (y : PUnit.{u_2 + 1} ) (r : R) :

r • y = PUnit.unit

theorem PUnit.instIsCentralVAddPUnitVaddAddOpposite.proof_1 {R : Type u_1} :

IsCentralVAdd R PUnit.{u_2 + 1}

instance PUnit.instIsCentralVAddPUnitVaddAddOpposite {R : Type u_1} :

IsCentralVAdd R PUnit.{u_2 + 1}

Equations

@PUnit.instIsCentralVAddPUnitVaddAddOpposite = @PUnit.instIsCentralVAddPUnitVaddAddOpposite.proof_1

instance PUnit.instIsCentralScalarPUnitSmulMulOpposite {R : Type u_1} :

IsCentralScalar R PUnit.{u_2 + 1}

Equations

@PUnit.instIsCentralScalarPUnitSmulMulOpposite = @PUnit.instIsCentralScalarPUnitSmulMulOpposite.proof_1

instance PUnit.instVAddCommClassPUnitVaddVadd {R : Type u_1} {S : Type u_2} :

VAddCommClass R S PUnit.{u_3 + 1}

Equations

@PUnit.instVAddCommClassPUnitVaddVadd = @PUnit.instVAddCommClassPUnitVaddVadd.proof_1

theorem PUnit.instVAddCommClassPUnitVaddVadd.proof_1 {R : Type u_1} {S : Type u_2} :

VAddCommClass R S PUnit.{u_3 + 1}

instance PUnit.instSMulCommClassPUnitSmulSmul {R : Type u_1} {S : Type u_2} :

SMulCommClass R S PUnit.{u_3 + 1}

Equations

@PUnit.instSMulCommClassPUnitSmulSmul = @PUnit.instSMulCommClassPUnitSmulSmul.proof_1

instance PUnit.instIsScalarTowerPUnitVaddVadd {R : Type u_1} {S : Type u_2} [VAdd R S] :

VAddAssocClass R S PUnit.{u_3 + 1}

Equations

@PUnit.instIsScalarTowerPUnitVaddVadd = @PUnit.instIsScalarTowerPUnitVaddVadd.proof_1

theorem PUnit.instIsScalarTowerPUnitVaddVadd.proof_1 {R : Type u_1} {S : Type u_2} [VAdd R S] :

VAddAssocClass R S PUnit.{u_3 + 1}

instance PUnit.instIsScalarTowerPUnitSmulSmul {R : Type u_1} {S : Type u_2} [SMul R S] :

IsScalarTower R S PUnit.{u_3 + 1}

Equations

@PUnit.instIsScalarTowerPUnitSmulSmul = @PUnit.instIsScalarTowerPUnitSmulSmul.proof_1

instance PUnit.smulWithZero {R : Type u_1} [Zero R] :

SMulWithZero R PUnit.{1}

Equations

PUnit.smulWithZero = let src := PUnit.smul; SMulWithZero.mk (_ : ∀ (m : PUnit.{1}), 0 • m = 0)

instance PUnit.mulAction {R : Type u_1} [Monoid R] :

MulAction R PUnit.{u_2 + 1}

Equations

PUnit.mulAction = let src := PUnit.smul; MulAction.mk (_ : ∀ (b : PUnit.{u_2 + 1} ), 1 • b = b) (_ : ∀ (x y : R) (b : PUnit.{u_2 + 1} ), (x * y) • b = x • y • b)

instance PUnit.distribMulAction {R : Type u_1} [Monoid R] :

DistribMulAction R PUnit.{u_2 + 1}

Equations

PUnit.distribMulAction = let src := PUnit.mulAction; DistribMulAction.mk (_ : ∀ (a : R), a • 0 = 0) (_ : ∀ (a : R) (x y : PUnit.{u_2 + 1} ), a • (x + y) = a • x + a • y)

instance PUnit.mulDistribMulAction {R : Type u_1} [Monoid R] :

MulDistribMulAction R PUnit.{u_2 + 1}

Equations

PUnit.mulDistribMulAction = let src := PUnit.mulAction; MulDistribMulAction.mk (_ : ∀ (r : R) (x y : PUnit.{u_2 + 1} ), r • (x * y) = r • x * r • y) (_ : ∀ (r : R), r • 1 = 1)

instance PUnit.mulSemiringAction {R : Type u_1} [Semiring R] :

MulSemiringAction R PUnit.{u_2 + 1}

Equations

One or more equations did not get rendered due to their size.

instance PUnit.mulActionWithZero {R : Type u_1} [MonoidWithZero R] :

MulActionWithZero R PUnit.{1}

Equations

PUnit.mulActionWithZero = let src := PUnit.mulAction; let src_1 := PUnit.smulWithZero; MulActionWithZero.mk (_ : ∀ (a : R), a • 0 = 0) (_ : ∀ (m : PUnit.{1}), 0 • m = 0)

instance PUnit.module {R : Type u_1} [Semiring R] :

Module R PUnit.{u_2 + 1}

Equations

PUnit.module = let src := PUnit.distribMulAction; Module.mk (_ : ∀ (r s : R) (x : PUnit.{u_2 + 1} ), (r + s) • x = r • x + s • x) (_ : ∀ (x : PUnit.{u_2 + 1} ), 0 • x = 0)