`toNat_fromNat`

proved

show as: view math explainer →

module: IndisputableMonolith.Foundation.ArithmeticFromLogic
domain: Foundation
line: 243 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.Foundation.ArithmeticFromLogic on GitHub at line 243.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

used by

formal source

 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
 266    rw [add_zero, toNat_zero, Nat.add_zero]
 267  | step b ih =>
 268    show toNat (a + succ b) = toNat a + toNat (succ b)
 269    rw [add_succ, toNat_succ, toNat_succ, ih, Nat.add_succ]
 270
 271/-- **Recovery theorem (multiplication)**: the multiplication
 272`LogicNat` carries agrees with `Nat` multiplication under the
 273equivalence. -/

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.