l3lab/ntp-mathlib huggingface dataset:
theorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S)
    (g : S ⟶ T) [IsIso g] : P (f ≫ g) ↔ P f :=
  ⟨fun H => by
    

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
inst✝ : IsIso g
H : P (f ≫ g)
⊢ P f

Next tactic:
convert hP.1 (f ≫ g) (asIso g).symm.commRingCatIsoToRingEquiv H
theorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S)
    (g : S ⟶ T) [IsIso g] : P (f ≫ g) ↔ P f :=
  ⟨fun H => by
    convert hP.1 (f ≫ g) (asIso g).symm.commRingCatIsoToRingEquiv H
    

Proof state:
case h.e'_5
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
inst✝ : IsIso g
H : P (f ≫ g)
⊢ f = comp (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso g).symm)) (f ≫ g)

Next tactic:
change f = f ≫ g ≫ inv g
theorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S)
    (g : S ⟶ T) [IsIso g] : P (f ≫ g) ↔ P f :=
  ⟨fun H => by
    convert hP.1 (f ≫ g) (asIso g).symm.commRingCatIsoToRingEquiv H
    change f = f ≫ g ≫ inv g
    

Proof state:
case h.e'_5
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
inst✝ : IsIso g
H : P (f ≫ g)
⊢ f = f ≫ g ≫ inv g

Next tactic:
simp
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
⊢ P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r)

Next tactic:
let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
⊢ P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r)

Next tactic:
let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r)

Next tactic:
refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ P
      ((RingEquiv.toCommRingCatIso e₁).hom ≫
        CommRingCat.ofHom
          (IsLocalization.map (Localization.Away (f r)) f
            (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ↔
    P (IsLocalization.Away.map R' S' f r)

Next tactic:
refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ P
      (CommRingCat.ofHom
          ((RingEquiv.toCommRingCatIso e₁).hom ≫
            CommRingCat.ofHom
              (IsLocalization.map (Localization.Away (f r)) f
                (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ≫
        (RingEquiv.toCommRingCatIso e₂).hom) ↔
    P (IsLocalization.Away.map R' S' f r)

Next tactic:
rw [← eq_iff_iff]
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ P
      (CommRingCat.ofHom
          ((RingEquiv.toCommRingCatIso e₁).hom ≫
            CommRingCat.ofHom
              (IsLocalization.map (Localization.Away (f r)) f
                (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ≫
        (RingEquiv.toCommRingCatIso e₂).hom) =
    P (IsLocalization.Away.map R' S' f r)

Next tactic:
congr 1
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  

Proof state:
case e_x
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ CommRingCat.ofHom
        ((RingEquiv.toCommRingCatIso e₁).hom ≫
          CommRingCat.ofHom
            (IsLocalization.map (Localization.Away (f r)) f
              (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ≫
      (RingEquiv.toCommRingCatIso e₂).hom =
    IsLocalization.Away.map R' S' f r

Next tactic:
let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by rintro x ⟨n, rfl⟩; use n; simp : Submonoid.powers r ≤ Submonoid.comap f
                (Submonoid.powers (f r)))) : Localization.Away r →+* Localization.Away (f r)).comp
                (e₁: R' →+* Localization.Away r))
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by 

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
⊢ Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))

Next tactic:
rintro x ⟨n, rfl⟩
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by rintro x ⟨n, rfl⟩; 

Proof state:
case intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
n : ℕ
⊢ (fun x => r ^ x) n ∈ Submonoid.comap f (Submonoid.powers (f r))

Next tactic:
use n
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by rintro x ⟨n, rfl⟩; use n; 

Proof state:
case h
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
n : ℕ
⊢ (fun x => f r ^ x) n = f ((fun x => r ^ x) n)

Next tactic:
simp
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by rintro x ⟨n, rfl⟩; use n; simp : Submonoid.powers r ≤ Submonoid.comap f
                (Submonoid.powers (f r)))) : Localization.Away r →+* Localization.Away (f r)).comp
                (e₁: R' →+* Localization.Away r))
  

Proof state:
case e_x
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
e : R' →+* S' :=
  comp (↑e₂)
    (comp
      (IsLocalization.map (Localization.Away (f r)) f
        (_ : ∀ ⦃x : R⦄, x ∈ Submonoid.powers r → x ∈ Submonoid.comap f (Submonoid.powers (f r))))
      ↑e₁)
⊢ CommRingCat.ofHom
        ((RingEquiv.toCommRingCatIso e₁).hom ≫
          CommRingCat.ofHom
            (IsLocalization.map (Localization.Away (f r)) f
              (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ≫
      (RingEquiv.toCommRingCatIso e₂).hom =
    IsLocalization.Away.map R' S' f r

Next tactic:
suffices e = IsLocalization.Away.map R' S' f r by
    convert this
theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
    [Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
    P (Localization.awayMap f r) ↔ P (IsLocalization.Away.map R' S' f r) := by
  let e₁ : R' ≃+* Localization.Away r :=
    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
  let e₂ : Localization.Away (f r) ≃+* S' :=
    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
  refine' (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _
  refine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e₂.toCommRingCatIso.hom).symm.trans _
  rw [← eq_iff_iff]
  congr 1
  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't
  -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring
  -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the
  -- problem by converting the goal to a more straightforward form.
  let e := (e₂ : Localization.Away (f r) →+* S').comp
      (((IsLocalization.map (Localization.Away (f r)) f
            (by rintro x ⟨n, rfl⟩; use n; simp : Submonoid.powers r ≤ Submonoid.comap f
                (Submonoid.powers (f r)))) : Localization.Away r →+* Localization.Away (f r)).comp
                (e₁: R' →+* Localization.Away r))
  suffices e = IsLocalization.Away.map R' S' f r by
    

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
f : R →+* S
r : R
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
e₁ : R' ≃+* Localization.Away r :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))
e₂ : Localization.Away (f r) ≃+* S' :=
  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (f r)) (Localization.Away (f r)) S')
e : R' →+* S' :=
  comp (↑e₂)
    (comp
      (IsLocalization.map (Localization.Away (f r)) f
        (_ : ∀ ⦃x : R⦄, x ∈ Submonoid.powers r → x ∈ Submonoid.comap f (Submonoid.powers (f r))))
      ↑e₁)
this : e = IsLocalization.Away.map R' S' f r
⊢ CommRingCat.ofHom
        ((RingEquiv.toCommRingCatIso e₁).hom ≫
          CommRingCat.ofHom
            (IsLocalization.map (Localization.Away (f r)) f
              (_ : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))))) ≫
      (RingEquiv.toCommRingCatIso e₂).hom =
    IsLocalization.Away.map R' S' f r

Next tactic:
convert this