`fromNat_toNat`

proved

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module: IndisputableMonolith.Foundation.ArithmeticFromLogic
domain: Foundation
line: 235 · github
papers citing: none yet

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IndisputableMonolith.Foundation.ArithmeticFromLogic on GitHub at line 235.

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formal source

 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
 257  right_inv := toNat_fromNat
 258
 259/-- **Recovery theorem (addition)**: the addition `LogicNat` carries
 260agrees with `Nat` addition under the equivalence. -/
 261theorem toNat_add (a b : LogicNat) :
 262    toNat (a + b) = toNat a + toNat b := by
 263  induction b with
 264  | identity =>
 265    show toNat (a + zero) = toNat a + toNat zero

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