Documentation

Mathlib.Data.LawfulXor

The LawfulXor typeclass #

This file generalizes basic lemmas about the ^^^ operator across numeric types.

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class LawfulXor (α : Type u_1) [XorOp α] [Zero α] :
Prop

A typeclass indicating that the xor operation, ^^^, is lawful.

  • xor_assoc (a b c : α) : a ^^^ b ^^^ c = a ^^^ (b ^^^ c)
  • xor_self (a : α) : a ^^^ a = 0
  • xor_zero (a : α) : a ^^^ 0 = a
  • xor_comm (a b : α) : a ^^^ b = b ^^^ a
Instances
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    @[simp]
    theorem zero_xor {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a : α) :
    0 ^^^ a = a
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    instance instCommutativeXor_mathlib {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] :
    Std.Commutative XorOp.xor
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    instance instAssociativeXor_mathlib {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] :
    Std.Associative XorOp.xor
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    instance instLawfulCommIdentityXorOfNat_mathlib {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] :
    Std.LawfulCommIdentity XorOp.xor 0
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    @[simp]
    theorem xor_cancel_right {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a b : α) :
    a ^^^ b ^^^ b = a
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    @[simp]
    theorem xor_cancel_left {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a b : α) :
    a ^^^ (a ^^^ b) = b
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    instance instLawfulXorNat :
    LawfulXor Nat
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    instance instLawfulXorBitVec {w : Nat} :
    LawfulXor (BitVec w)
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    instance instLawfulXorUInt8 :
    LawfulXor UInt8
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    instance instLawfulXorUInt16 :
    LawfulXor UInt16
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    instance instLawfulXorUInt32 :
    LawfulXor UInt32
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    instance instLawfulXorUInt64 :
    LawfulXor UInt64
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    instance instLawfulXorUSize :
    LawfulXor USize
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    instance instLawfulXorInt8 :
    LawfulXor Int8
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    instance instLawfulXorInt16 :
    LawfulXor Int16
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    instance instLawfulXorInt32 :
    LawfulXor Int32
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    instance instLawfulXorInt64 :
    LawfulXor Int64
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    instance instLawfulXorISize :
    LawfulXor ISize
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    theorem xor_right_involutive {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a : α) :
    Function.Involutive fun (x : α) => a ^^^ x
    source
    theorem xor_left_involutive {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a : α) :
    Function.Involutive fun (x : α) => x ^^^ a
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    theorem xor_eq_iff_left_eq {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a b c : α) :
    a ^^^ b = c ↔ a = c ^^^ b
    source
    theorem xor_eq_iff_right_eq {α : Type u_1} [XorOp α] [Zero α] [LawfulXor α] (a b c : α) :
    a ^^^ b = c ↔ b = a ^^^ c