Theorem: Lean4Axiomatic.Rational.Impl.Fraction.Naive.add_preserves_positive_denominators (Lean4Axiomatic/Rational/Impl/Fraction/Naive.lean)

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
p q : Fraction ℤ
⊢ Positive p.denominator → Positive q.denominator → Positive (p + q).denominator
Tactic:
revert p
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
q : Fraction ℤ
⊢ ∀ {p : Fraction ℤ}, Positive p.denominator → Positive q.denominator → Positive (p + q).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
q : Fraction ℤ
⊢ ∀ {p : Fraction ℤ}, Positive p.denominator → Positive q.denominator → Positive (p + q).denominator
Tactic:
intro (pn//pd)
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
q : Fraction ℤ
pn pd : ℤ
⊢ Positive (pn//pd).denominator → Positive q.denominator → Positive (pn//pd + q).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
q : Fraction ℤ
pn pd : ℤ
⊢ Positive (pn//pd).denominator → Positive q.denominator → Positive (pn//pd + q).denominator
Tactic:
revert q
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd : ℤ
⊢ ∀ {q : Fraction ℤ}, Positive (pn//pd).denominator → Positive q.denominator → Positive (pn//pd + q).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd : ℤ
⊢ ∀ {q : Fraction ℤ}, Positive (pn//pd).denominator → Positive q.denominator → Positive (pn//pd + q).denominator
Tactic:
intro (qn//qd)
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
⊢ Positive (pn//pd).denominator → Positive (qn//qd).denominator → Positive (pn//pd + qn//qd).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
⊢ Positive (pn//pd).denominator → Positive (qn//qd).denominator → Positive (pn//pd + qn//qd).denominator
Tactic:
intro (_ : Positive pd) (_ : Positive qd)
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive (pn//pd + qn//qd).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive (pn//pd + qn//qd).denominator
Tactic:
show Positive ((pn * qd + pd * qn)//(pd * qd)).denominator
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive ((pn * qd + pd * qn)//(pd * qd)).denominator

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive ((pn * qd + pd * qn)//(pd * qd)).denominator
Tactic:
show Positive (pd * qd)
State after:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive (pd * qd)

State before:
ℕ : Type
inst✝¹ : Natural ℕ
ℤ : Type
inst✝ : Integer ℤ
pn pd qn qd : ℤ
x✝¹ : Positive pd
x✝ : Positive qd
⊢ Positive (pd * qd)
Tactic:
exact Integer.mul_preserves_positive ‹Positive pd› ‹Positive qd›
State after:
no goals