instance : Algebra R (v.adicCompletionIntegers K) where
  smul r x :=
    ⟨r • (x : v.adicCompletion K), by
      have h :
        (algebraMap R (adicCompletion K v)) r = (algebraMap R K r : adicCompletion K v) := rfl
      rw [Algebra.smul_def]
      refine' ValuationSubring.mul_mem _ _ _ _ x.2
      --Porting note: added instance
      letI : Valued K ℤₘ₀ := adicValued v
      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
      erw [mem_adicCompletionIntegers, h, Valued.valuedCompletion_apply]
      exact v.valuation_le_one _⟩
  toFun r :=
    ⟨(algebraMap R K r : adicCompletion K v), by
      -- Porting note: added instance
      letI : Valued K ℤₘ₀ := adicValued v
      --Porting note: rest of proof was `simpa only
      --   [mem_adicCompletionIntegers, Valued.valuedCompletion_apply] using
      --   v.valuation_le_one _
      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
      erw [mem_adicCompletionIntegers, Valued.valuedCompletion_apply]
      exact v.valuation_le_one _⟩
  map_one' := by simp only [map_one]; rfl
  map_mul' x y := by
    ext
    --Porting note: added instance
    letI : Valued K ℤₘ₀ := adicValued v
    simp_rw [RingHom.map_mul, Subring.coe_mul, UniformSpace.Completion.coe_mul]
  map_zero' := by simp only [map_zero]; rfl
  map_add' x y := by
    ext
    --Porting note: added instance
    letI : Valued K ℤₘ₀ := adicValued v
    

Proof state:
case a
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : R
this : Valued K ℤₘ₀ := adicValued v
⊢ ↑(OneHom.toFun
        (↑{
            toOneHom :=
              {
                toFun := fun r =>
                  { val := ↑K ((algebraMap R K) r),
                    property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                map_one' :=
                  (_ :
                    (fun r =>
                          { val := ↑K ((algebraMap R K) r),
                            property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                        1 =
                      1) },
            map_mul' :=
              (_ :
                ∀ (x y : R),
                  OneHom.toFun
                      {
                        toFun := fun r =>
                          { val := ↑K ((algebraMap R K) r),
                            property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                        map_one' :=
                          (_ :
                            (fun r =>
                                  { val := ↑K ((algebraMap R K) r),
                                    property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                1 =
                              1) }
                      (x * y) =
                    OneHom.toFun
                        {
                          toFun := fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                          map_one' :=
                            (_ :
                              (fun r =>
                                    { val := ↑K ((algebraMap R K) r),
                                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                  1 =
                                1) }
                        x *
                      OneHom.toFun
                        {
                          toFun := fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                          map_one' :=
                            (_ :
                              (fun r =>
                                    { val := ↑K ((algebraMap R K) r),
                                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                  1 =
                                1) }
                        y) })
        (x + y)) =
    ↑(OneHom.toFun
          (↑{
              toOneHom :=
                {
                  toFun := fun r =>
                    { val := ↑K ((algebraMap R K) r),
                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                  map_one' :=
                    (_ :
                      (fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                          1 =
                        1) },
              map_mul' :=
                (_ :
                  ∀ (x y : R),
                    OneHom.toFun
                        {
                          toFun := fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                          map_one' :=
                            (_ :
                              (fun r =>
                                    { val := ↑K ((algebraMap R K) r),
                                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                  1 =
                                1) }
                        (x * y) =
                      OneHom.toFun
                          {
                            toFun := fun r =>
                              { val := ↑K ((algebraMap R K) r),
                                property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                            map_one' :=
                              (_ :
                                (fun r =>
                                      { val := ↑K ((algebraMap R K) r),
                                        property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                    1 =
                                  1) }
                          x *
                        OneHom.toFun
                          {
                            toFun := fun r =>
                              { val := ↑K ((algebraMap R K) r),
                                property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                            map_one' :=
                              (_ :
                                (fun r =>
                                      { val := ↑K ((algebraMap R K) r),
                                        property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                    1 =
                                  1) }
                          y) })
          x +
        OneHom.toFun
          (↑{
              toOneHom :=
                {
                  toFun := fun r =>
                    { val := ↑K ((algebraMap R K) r),
                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                  map_one' :=
                    (_ :
                      (fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                          1 =
                        1) },
              map_mul' :=
                (_ :
                  ∀ (x y : R),
                    OneHom.toFun
                        {
                          toFun := fun r =>
                            { val := ↑K ((algebraMap R K) r),
                              property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                          map_one' :=
                            (_ :
                              (fun r =>
                                    { val := ↑K ((algebraMap R K) r),
                                      property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                  1 =
                                1) }
                        (x * y) =
                      OneHom.toFun
                          {
                            toFun := fun r =>
                              { val := ↑K ((algebraMap R K) r),
                                property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                            map_one' :=
                              (_ :
                                (fun r =>
                                      { val := ↑K ((algebraMap R K) r),
                                        property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                    1 =
                                  1) }
                          x *
                        OneHom.toFun
                          {
                            toFun := fun r =>
                              { val := ↑K ((algebraMap R K) r),
                                property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) },
                            map_one' :=
                              (_ :
                                (fun r =>
                                      { val := ↑K ((algebraMap R K) r),
                                        property := (_ : ↑K ((algebraMap R K) r) ∈ adicCompletionIntegers K v) })
                                    1 =
                                  1) }
                          y) })
          y)

Next tactic:
simp_rw [RingHom.map_add, Subring.coe_add, UniformSpace.Completion.coe_add]