`fromNat`

definition

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module: IndisputableMonolith.Foundation.ArithmeticFromLogic
domain: Foundation
line: 226 · github
papers citing: none yet

open explainer

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open lean source

IndisputableMonolith.Foundation.ArithmeticFromLogic on GitHub at line 226.

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explainer page

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open explainer

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formal source

 223  | .step n   => Nat.succ (toNat n)
 224
 225/-- The inverse map: build the orbit by iterating the step. -/
 226def fromNat : Nat → LogicNat
 227  | 0          => .identity
 228  | Nat.succ n => .step (fromNat n)
 229
 230@[simp] theorem toNat_zero : toNat zero = 0 := rfl
 231@[simp] theorem toNat_succ (n : LogicNat) : toNat (succ n) = Nat.succ (toNat n) := rfl
 232@[simp] theorem fromNat_zero : fromNat 0 = zero := rfl
 233@[simp] theorem fromNat_succ (n : Nat) : fromNat (Nat.succ n) = succ (fromNat n) := rfl
 234
 235theorem fromNat_toNat : ∀ n : LogicNat, fromNat (toNat n) = n := by
 236  intro n
 237  induction n with
 238  | identity => rfl
 239  | step n ih =>
 240    show fromNat (toNat (succ n)) = succ n
 241    rw [toNat_succ, fromNat_succ, ih]
 242
 243theorem toNat_fromNat : ∀ n : Nat, toNat (fromNat n) = n := by
 244  intro n
 245  induction n with
 246  | zero => rfl
 247  | succ n ih =>
 248    show toNat (fromNat (Nat.succ n)) = Nat.succ n
 249    rw [fromNat_succ, toNat_succ, ih]
 250
 251/-- **Recovery theorem (carrier)**: `LogicNat` and `Nat` have the same
 252underlying set, witnessed by the round-trip equalities. -/
 253def equivNat : LogicNat ≃ Nat where
 254  toFun := toNat
 255  invFun := fromNat
 256  left_inv := fromNat_toNat

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