lean2/tests/lean/run/nat_bug3.lean
2014-07-25 15:03:57 -07:00

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import logic num
using num eq_proofs
inductive nat : Type :=
| zero : nat
| succ : nat → nat
abbreviation plus (x y : nat) : nat
:= nat_rec x (λn r, succ r) y
definition to_nat [coercion] [inline] (n : num) : nat
:= num_rec zero (λn, pos_num_rec (succ zero) (λn r, plus r (plus r (succ zero))) (λn r, plus r r) n) n
definition add (x y : nat) : nat
:= plus x y
variable le : nat → nat → Prop
infixl `+`:65 := add
infix `≤`:50 := le
axiom add_one (n:nat) : n + (succ zero) = succ n
axiom add_le_right_inv {n m k : nat} (H : n + k ≤ m + k) : n ≤ m
theorem succ_le_cancel {n m : nat} (H : succ n ≤ succ m) : n ≤ m
:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)