-- Mathlib/Data/Rat/Cast/CharZero.lean
lemma cast_injective : Injective ((↑) : ℚ≥0 → α)

Proof state:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p = ↑q
⊢ p = q

Tactic:
rw [NNRat.cast_def, NNRat.cast_def, Commute.div_eq_div_iff] at hpq

Result:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ p = q

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑q.den ≠ 0

Proof state:
case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

Tactic:
exact Nat.cast_commute ..

Result:
no goals

Proof state:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
⊢ Injective NNRat.cast

Tactic:
rintro p q hpq

Result:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p = ↑q
⊢ p = q

Proof state:
case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ¬p.den = 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ¬q.den = 0

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬p.den = 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬q.den = 0

Tactic:
any_goals apply den_ne_zero

Result:
case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

Proof state:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ p = q

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑q.den ≠ 0

Tactic:
on_goal 1 => rw [← p.num_div_den, ← q.num_div_den, div_eq_div_iff]

Result:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.num * ↑q.den = ↑q.num * ↑p.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑q.den ≠ 0

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑q.den ≠ 0

Proof state:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ p = q

Tactic:
rw [← p.num_div_den, ← q.num_div_den, div_eq_div_iff]

Result:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.num * ↑q.den = ↑q.num * ↑p.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑q.den ≠ 0

Proof state:
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.num * ↑q.den = ↑q.num * ↑p.den

Tactic:
norm_cast at hpq ⊢

Result:
no goals

Proof state:
case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑q.den ≠ 0

Tactic:
norm_cast

Result:
case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬q.den = 0

Proof state:
case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ↑q.den ≠ 0

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑p.den ≠ 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ↑q.den ≠ 0

Tactic:
any_goals norm_cast

Result:
case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ¬p.den = 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num * ↑q.den = ↑q.num * ↑p.den
⊢ ¬q.den = 0

case hbd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ Commute ↑p.den ↑q.den

case hb
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬p.den = 0

case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬q.den = 0

Proof state:
case hd
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
p q : ℚ≥0
hpq : ↑p.num / ↑p.den = ↑q.num / ↑q.den
⊢ ¬q.den = 0

Tactic:
apply den_ne_zero

Result:
no goals