6f4e711171
This PR adds a few unification hints that we will need after `backward.isDefEq.respectTransparency` is `true` by default. See #12338 It was part of #12179.
46 lines
1.4 KiB
Text
46 lines
1.4 KiB
Text
structure PreInt where
|
|
minuend : Nat
|
|
subtrahend : Nat
|
|
|
|
/-- Definition 4.1.1 -/
|
|
instance PreInt.instSetoid : Setoid PreInt where
|
|
r a b := a.minuend + b.subtrahend = b.minuend + a.subtrahend
|
|
iseqv := {
|
|
refl := by grind
|
|
symm := by grind
|
|
trans := by grind
|
|
}
|
|
|
|
abbrev MyInt := Quotient PreInt.instSetoid
|
|
|
|
abbrev MyInt.formalDiff (a b : Nat) : MyInt := Quotient.mk PreInt.instSetoid ⟨ a, b ⟩
|
|
|
|
theorem MyInt.eq (a b c d : Nat) : formalDiff a b = formalDiff c d ↔ a + d = c + b :=
|
|
⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩
|
|
|
|
instance MyInt.instOfNat {n : Nat} : OfNat MyInt n where
|
|
ofNat := formalDiff n 0
|
|
|
|
instance MyInt.instNatCast : NatCast MyInt where
|
|
natCast n := formalDiff n 0
|
|
|
|
theorem MyInt.natCast_eq (n : Nat) : (n : MyInt) = formalDiff n 0 := rfl
|
|
|
|
theorem MyInt.natCast_inj (n m : Nat) :
|
|
(n : MyInt) = (m : MyInt) ↔ n = m := by
|
|
rw [natCast_eq, natCast_eq, eq]
|
|
|
|
example (n m : Nat) : (n : MyInt) = (m : MyInt) ↔ n = m := by
|
|
grind [MyInt.natCast_eq, MyInt.eq]
|
|
|
|
@[grind]
|
|
theorem MyInt.eq_0_of_cast_eq_0 (n : Nat) (h : (n : MyInt) = 0) : n = 0 := by
|
|
rw [show (0 : MyInt) = ((0 : Nat) : MyInt) by rfl] at h
|
|
rwa [natCast_inj] at h
|
|
|
|
theorem MyInt.pos_iff_gt_0 {a : MyInt} : (∃ (n:Nat), n > 0 ∧ a = n) → a ≠ 0 := by
|
|
intro ⟨ w, hw ⟩ h
|
|
grind
|
|
|
|
example {a : MyInt} : (∃ (n:Nat), n > 0 ∧ a = n) → a ≠ 0 := by
|
|
grind
|