toNat_add

formal source

 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. -/
 274theorem toNat_mul (a b : LogicNat) :
 275    toNat (a * b) = toNat a * toNat b := by
 276  induction b with
 277  | identity =>
 278    show toNat (a * zero) = toNat a * toNat zero
 279    rw [mul_zero, toNat_zero, Nat.mul_zero]
 280  | step b ih =>
 281    show toNat (a * succ b) = toNat a * toNat (succ b)
 282    rw [mul_succ, toNat_succ, toNat_add, ih, Nat.mul_succ]
 283
 284/-- Left cancellation: `a + b = a + c ⇒ b = c`. Proved by transferring
 285to `Nat` via the recovery isomorphism. -/
 286theorem add_left_cancel {a b c : LogicNat} (h : a + b = a + c) : b = c := by
 287  have hcast := congrArg toNat h
 288  rw [toNat_add, toNat_add] at hcast
 289  have hnat : toNat b = toNat c := by omega
 290  have := congrArg fromNat hnat
 291  rw [fromNat_toNat, fromNat_toNat] at this