awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I J : Ideal S
refine : (↑I : Set S) * (↑J : Set S) ≤ (↑(I * J) : Set S)
⊢ LE.le.{u_1} (α := Ideal R) (span ((⇑(Algebra.intNorm R S) : (a : S) → R) '' ((↑I : Set S) * (↑J : Set S))))
    (map (F := S →* R) (Algebra.intNorm R S) (I * J))

Tactic:
refine Ideal.span_mono (Set.monotone_image ?_)

Premises used: Ideal.span_mono, Set.monotone_image
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I J : Ideal S
x : S
hxI : x ∈ (↑I : Set S)
y : S
hyJ : y ∈ (↑J : Set S)
⊢ (fun (x1 x2 : S) ↦ x1 * x2) x y ∈ (↑(I * J) : Set S)

Tactic:
Ideal.mul_mem_mul hxI hyJ

Premises used: Set.Mem, Ideal.mul_mem_mul
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I J : Ideal S
x : S
hxI : x ∈ (↑I : Set S)
y : S
hyJ : y ∈ (↑J : Set S)
⊢ (fun (x1 x2 : S) ↦ x1 * x2) x y ∈ (↑(I * J) : Set S)

Tactic:
exact Ideal.mul_mem_mul hxI hyJ

Premises used: Ideal.mul_mem_mul
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
le_antisymm ?_ (spanNorm_mul_spanNorm_le R _ _)

Premises used: le_antisymm, Ideal.spanNorm_mul_spanNorm_le
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
refine : spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
refine le_antisymm ?_ (spanNorm_mul_spanNorm_le R _ _)

Premises used: le_antisymm, Ideal.spanNorm_mul_spanNorm_le
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
inl :
  ∀ (hI : Eq (α := Ideal R) (spanNorm R I) ⊥),
    spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)
inr :
  ∀ (hI : Eq (α := Ideal R) (spanNorm R I) ⊤),
    spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)
⊢ spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
rcases eq_bot_or_top (spanNorm R I) with hI | hI

Premises used: Ideal.spanNorm
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊥
inl : ⊥ ≤ HMul.hMul (α := Ideal R) ⊥ (spanNorm R J)
⊢ spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
rw [hI, spanNorm_eq_bot_iff.mp hI, bot_mul, spanNorm_bot]

Premises used: Ideal.bot_mul, Iff.mp, Ideal.spanNorm_bot, Ideal.spanNorm_eq_bot_iff
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊥
⊢ ⊥ ≤ HMul.hMul (α := Ideal R) ⊥ (spanNorm R J)

Tactic:
bot_le

Premises used: Set.Mem, bot_le
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊥
⊢ ⊥ ≤ HMul.hMul (α := Ideal R) ⊥ (spanNorm R J)

Tactic:
exact bot_le

Premises used: bot_le
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
inr : LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)
⊢ spanNorm R (I * J) ≤ HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
rw [hI, Ideal.top_mul]

Premises used: Ideal.top_mul
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
inr : ∀ (hJ : Eq (α := Ideal R) (spanNorm R J) ⊥), LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)
inr₁ : ∀ (hJ : Eq (α := Ideal R) (spanNorm R J) ⊤), LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)
⊢ LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)

Tactic:
rcases eq_bot_or_top (spanNorm R J) with hJ | hJ

Premises used: Ideal.spanNorm
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
hJ : Eq (α := Ideal R) (spanNorm R J) ⊥
inr : LE.le (α := Ideal R) (spanNorm R (I * J)) ⊥
inr₁ : LE.le (α := Ideal R) (spanNorm R (I * ⊥)) ⊥
inr₂ : LE.le (α := Ideal R) (spanNorm.{u_1, u_3} R (S := S) ⊥) ⊥
inr₃ inr₄ : LE.le.{u_1} (α := Ideal R) ⊥ ⊥
⊢ LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)

Tactic:
rw [hJ, spanNorm_eq_bot_iff.mp hJ, mul_bot, spanNorm_bot]

Premises used: Ideal.mul_bot, Iff.mp, Ideal.spanNorm_bot, Ideal.spanNorm_eq_bot_iff
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
hJ : Eq (α := Ideal R) (spanNorm R J) ⊤
inr : LE.le (α := Ideal R) (spanNorm R (I * J)) ⊤
⊢ LE.le (α := Ideal R) (spanNorm R (I * J)) (spanNorm R J)

Tactic:
rw [hJ]
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
hJ : Eq (α := Ideal R) (spanNorm R J) ⊤
⊢ LE.le (α := Ideal R) (spanNorm R (I * J)) ⊤

Tactic:
le_top

Premises used: Set.Mem, le_top
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : Eq (α := Ideal R) (spanNorm R I) ⊤
hJ : Eq (α := Ideal R) (spanNorm R J) ⊤
⊢ LE.le (α := Ideal R) (spanNorm R (I * J)) ⊤

Tactic:
exact le_top

Premises used: le_top
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x x_1 : R
h : Membership.mem (γ := Set R) ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S)) x_1
mem :
  ∀ (x : S) (hx : x ∈ (↑I : Set S)),
    Membership.mem (α := R) (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I)
      ((Algebra.intNorm R S : (a : S) → R) x : R)
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x_1

Tactic:
obtain ⟨x, hx, rfl⟩ := h

Premises used: rfl
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x_1 : R
x : S
hx : x ∈ (↑I : Set S)
⊢ Membership.mem (α := R) (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I)
    ((Algebra.intNorm R S : (a : S) → R) x : R)

Tactic:
mem_comap.mpr <| mem_of_dvd _ (Algebra.dvd_algebraMap_intNorm_self _ _ x) hx

Premises used: Iff.mpr, Algebra.dvd_algebraMap_intNorm_self, Set.Mem, Ideal.mem_of_dvd, Ideal.mem_comap
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x_1 : R
x : S
hx : x ∈ (↑I : Set S)
⊢ Membership.mem (α := R) (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I)
    ((Algebra.intNorm R S : (a : S) → R) x : R)

Tactic:
exact mem_comap.mpr <| mem_of_dvd _ (Algebra.dvd_algebraMap_intNorm_self _ _ x) hx

Premises used: Iff.mpr, Algebra.dvd_algebraMap_intNorm_self, Ideal.mem_of_dvd, Ideal.mem_comap
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x : R
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) 0

Tactic:
simp
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x x_1 y : R
hx_1 :
  Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x_1
hy_1 : Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) y
hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x_1
hy : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) y
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (x_1 + y)

Tactic:
Submodule.add_mem _ hx hy

Premises used: Submodule.add_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x x_1 y : R
hx_1 :
  Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x_1
hy_1 : Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) y
hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x_1
hy : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) y
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (x_1 + y)

Tactic:
exact Submodule.add_mem _ hx hy

Premises used: Submodule.add_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x a x_1 : R
hx_1 :
  Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x_1
hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x_1
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (a • x_1)

Tactic:
Submodule.smul_mem _ _ hx

Premises used: Submodule.smul_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x a x_1 : R
hx_1 :
  Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x_1
hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x_1
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (a • x_1)

Tactic:
exact Submodule.smul_mem _ _ hx

Premises used: Submodule.smul_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
I : Ideal S
x : R
hx : Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x
mem :
  ∀ (x : R) (h : Membership.mem (γ := Set R) ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S)) x),
    Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x
zero : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) 0
add :
  ∀ (x y : R),
    Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x →
      Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S)))
          y →
        ∀ (hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x)
          (hy : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) y),
          Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (x + y)
smul :
  ∀ (a x : R),
    Membership.mem (γ := Submodule R R) (Submodule.span R ((⇑(Algebra.intNorm R S) : (a : S) → R) '' (↑I : Set S))) x →
      ∀ (hx : Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x),
        Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) (a • x)
⊢ Membership.mem (γ := Submodule R R) (comap (F := R →+* S) (algebraMap R S) I) x

Tactic:
induction hx using Submodule.span_induction with
  | mem _ h =>
      obtain ⟨x, hx, rfl⟩ := h
      exact mem_comap.mpr <| mem_of_dvd _ (Algebra.dvd_algebraMap_intNorm_self _ _ x) hx
  | zero => simp
  | add _ _ _ _ hx hy => exact Submodule.add_mem _ hx hy
  | smul _ _ _ hx => exact Submodule.smul_mem _ _ hx

Premises used: Iff.mpr, Algebra.dvd_algebraMap_intNorm_self, Ideal.mem_of_dvd, Submodule.smul, Submodule.zero, Submodule.add, Ideal.mem_comap, rfl, Submodule.smul_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
⊢ Nontrivial R

Tactic:
nontriviality R
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
inl : Subsingleton S → spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)
inr : Nontrivial S → spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
cases subsingleton_or_nontrivial S

Premises used: subsingleton_or_nontrivial
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Subsingleton S
⊢ ∀ (I : Ideal S), I = ⊤

Tactic:
have : ∀ I : Ideal S, I = ⊤ := fun I ↦ Subsingleton.elim I ⊤
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Subsingleton S
this : ∀ (I : Ideal S), I = ⊤
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
simp [this I, this J]
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Nontrivial S
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
eq_of_localization_maximal (fun P hP ↦ ?_)

Premises used: Ideal.eq_of_localization_maximal
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Nontrivial S
inr :
  ∀ (P : Ideal R) (hP : P.IsMaximal),
    Eq (α := Ideal (Localization.AtPrime P))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P))
        (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))
⊢ spanNorm R (I * J) = HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)

Tactic:
refine eq_of_localization_maximal (fun P hP ↦ ?_)

Premises used: Ideal.eq_of_localization_maximal
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Nontrivial S
P : Ideal R
hP : P.IsMaximal
pos :
  ∀ (hP0 : P = ⊥),
    Eq (α := Ideal (Localization.AtPrime P))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P))
        (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))
neg :
  ∀ (hP0 : ¬P = ⊥),
    Eq (α := Ideal (Localization.AtPrime P))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)))
      (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P))
        (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))
⊢ Eq (α := Ideal (Localization.AtPrime P))
    (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)))
    (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P))
      (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))

Tactic:
by_cases hP0 : P = ⊥
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Nontrivial S
P : Ideal R
hP : P.IsMaximal
hP0 : P = ⊥
pos :
  ∀ (hP : IsMaximal.{u_1} (α := R) ⊥),
    Eq (α := Ideal (Localization.AtPrime (R := R) ⊥))
      (map (F := R →+* Localization.AtPrime (R := R) ⊥) (algebraMap R (Localization.AtPrime (R := R) ⊥))
        (spanNorm R (I * J)))
      (map (F := R →+* Localization.AtPrime (R := R) ⊥) (algebraMap R (Localization.AtPrime (R := R) ⊥))
        (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))
⊢ Eq (α := Ideal (Localization.AtPrime P))
    (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)))
    (map (F := R →+* Localization.AtPrime P) (algebraMap R (Localization.AtPrime P))
      (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))

Tactic:
subst hP0
-- Module: Mathlib.RingTheory.Ideal.Norm.RelNorm
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
S : Type u_3
inst_2 : CommRing S
inst_3 : IsDomain S
inst_4 : IsIntegrallyClosed R
inst_5 : IsIntegrallyClosed S
inst_6 : Algebra R S
inst_7 : Module.Finite R S
inst_8 : IsTorsionFree R S
inst_9 : IsDedekindDomain R
inst_10 : IsDedekindDomain S
I J : Ideal S
inst_11 : Nontrivial R
h : Nontrivial S
hP : IsMaximal.{u_1} (α := R) ⊥
pos : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
⊢ Eq (α := Ideal (Localization.AtPrime (R := R) ⊥))
    (map (F := R →+* Localization.AtPrime (R := R) ⊥) (algebraMap R (Localization.AtPrime (R := R) ⊥))
      (spanNorm R (I * J)))
    (map (F := R →+* Localization.AtPrime (R := R) ⊥) (algebraMap R (Localization.AtPrime (R := R) ⊥))
      (HMul.hMul (α := Ideal R) (spanNorm R I) (spanNorm R J)))

Tactic:
rw [spanNorm_mul_of_bot_or_top]

Premises used: Ideal.spanNorm_mul_of_bot_or_top