logicNatLift_unique_fun

formal source

  99    intro x
 100    rfl
 101
 102private theorem logicNatLift_unique_fun (B : PeanoObject)
 103    (f : PeanoObject.Hom logicNatPeano B) :
 104    f.toFun = (logicNatLift B).toFun := by
 105  funext n
 106  induction n with
 107  | identity =>
 108      exact f.map_zero
 109  | step n ih =>
 110      calc
 111        f.toFun (LogicNat.step n) = B.step (f.toFun n) := f.map_step n
 112        _ = B.step ((logicNatLift B).toFun n) := by rw [ih]
 113        _ = (logicNatLift B).toFun (LogicNat.step n) := rfl
 114
 115/-- `LogicNat` is initial among Peano objects. -/
 116def logicNat_initial : PeanoObject.IsInitial logicNatPeano where
 117  lift := logicNatLift
 118  uniq := by
 119    intro B f g
 120    rw [logicNatLift_unique_fun B f, logicNatLift_unique_fun B g]
 121
 122/-- The Peano object extracted from a realization's own orbit. -/
 123def realizationPeano (R : LogicRealization) : PeanoObject where
 124  carrier := R.Orbit
 125  zero := R.orbitZero
 126  step := R.orbitStep
 127
 128/-- Fold from a realization orbit into any Peano object, through the
 129realization's certified equivalence with `LogicNat`. -/
 130def realizationFold (R : LogicRealization) (B : PeanoObject) : R.Orbit → B.carrier :=
 131  fun n => (logicNatLift B).toFun (R.orbitEquivLogicNat n)
 132