File size: 2,527 Bytes
9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import Mathlib.Algebra.Field.ZMod
import Mathlib.Logic.Equiv.Defs
namespace OAI
universe uDepth1 uDepth2 uDepth3
namespace DepthThreeLowerBound
namespace BinaryAlgebra
abbrev F2 := ZMod 2
def boolEquivF2 : Bool ≃ F2 := finTwoEquiv.symm
def bitValue (b : Bool) : F2 := boolEquivF2 b
def fieldBit (a : F2) : Bool := boolEquivF2.symm a
@[simp] theorem bitValue_false : bitValue false = 0 := rfl
@[simp] theorem bitValue_true : bitValue true = 1 := rfl
@[simp] theorem fieldBit_zero : fieldBit 0 = false := rfl
@[simp] theorem fieldBit_one : fieldBit 1 = true := rfl
@[simp] theorem fieldBit_bitValue (b : Bool) : fieldBit (bitValue b) = b :=
boolEquivF2.symm_apply_apply b
@[simp] theorem bitValue_fieldBit (a : F2) : bitValue (fieldBit a) = a :=
boolEquivF2.apply_symm_apply a
theorem bitValue_injective : Function.Injective bitValue :=
boolEquivF2.injective
theorem bitValue_surjective : Function.Surjective bitValue :=
boolEquivF2.surjective
theorem bitValue_xor (a b : Bool) :
bitValue (Bool.xor a b) = bitValue a + bitValue b := by
cases a <;> cases b <;> decide
theorem bitValue_and (a b : Bool) :
bitValue (a && b) = bitValue a * bitValue b := by
cases a <;> cases b <;> decide
theorem bitValue_not (a : Bool) : bitValue (!a) = 1 + bitValue a := by
cases a <;> decide
@[simp] theorem bitValue_eq_zero_iff (b : Bool) : bitValue b = 0 ↔ b = false := by
cases b <;> decide
@[simp] theorem bitValue_eq_one_iff (b : Bool) : bitValue b = 1 ↔ b = true := by
cases b <;> decide
theorem eq_zero_or_one (a : F2) : a = 0 ∨ a = 1 := by
obtain ⟨b, rfl⟩ := boolEquivF2.surjective a
cases b <;> decide
@[simp] theorem fieldBit_eq_false_iff (a : F2) : fieldBit a = false ↔ a = 0 := by
obtain ⟨b, rfl⟩ := boolEquivF2.surjective a
cases b <;> decide
@[simp] theorem fieldBit_eq_true_iff (a : F2) : fieldBit a = true ↔ a = 1 := by
obtain ⟨b, rfl⟩ := boolEquivF2.surjective a
cases b <;> decide
def bitsEquivF2 (V : Type uDepth1) : (V → Bool) ≃ (V → F2) where
toFun x i := bitValue (x i)
invFun x i := fieldBit (x i)
left_inv x := funext fun i => fieldBit_bitValue (x i)
right_inv x := funext fun i => bitValue_fieldBit (x i)
@[simp] theorem bitsEquivF2_apply (V : Type uDepth2) (x : V → Bool) (i : V) :
bitsEquivF2 V x i = bitValue (x i) := rfl
@[simp] theorem bitsEquivF2_symm_apply (V : Type uDepth3) (x : V → F2) (i : V) :
(bitsEquivF2 V).symm x i = fieldBit (x i) := rfl
end BinaryAlgebra
end DepthThreeLowerBound
end OAI
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