ring_theory.polynomial.quotient ⟷ Mathlib.RingTheory.Polynomial.Quotient

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 -/
-import Data.Polynomial.Div
+import Algebra.Polynomial.Div
 import RingTheory.Polynomial.Basic
 import RingTheory.Ideal.QuotientOperations
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -122,8 +122,8 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
     intro f
     apply Polynomial.induction_on' f
     · intro p q hp hq
-      simp only [coe_eval₂_ring_hom] at hp 
-      simp only [coe_eval₂_ring_hom] at hq 
+      simp only [coe_eval₂_ring_hom] at hp
+      simp only [coe_eval₂_ring_hom] at hq
       simp only [coe_eval₂_ring_hom, hp, hq, RingHom.map_add]
     · rintro n ⟨x⟩
       simp only [← smul_X_eq_monomial, C_mul', Quotient.lift_mk, Submodule.Quotient.quot_mk_eq_mk,
@@ -183,10 +183,10 @@ theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk
     by
     refine' fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => _
     rw [mem_comap, ← quotient.eq_zero_iff_mem, ← RingHom.comp_apply]
-    rw [RingHom.mem_ker, coe_map_ring_hom] at hf 
+    rw [RingHom.mem_ker, coe_map_ring_hom] at hf
     replace hf := congr_arg (fun f : Polynomial _ => f.coeff n) hf
-    simp only [coeff_map, coeff_zero] at hf 
-    rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf 
+    simp only [coeff_map, coeff_zero] at hf
+    rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf
   obtain ⟨x, hx'⟩ := x
   obtain ⟨y, rfl⟩ := RingHom.mem_range.1 hx'
   refine' Subtype.eq _
@@ -230,9 +230,8 @@ theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
   refine' Finset.sum_eq_zero fun n hn => _
   simp only [eval₂_monomial, Function.comp_apply, RingHom.coe_comp]
   refine' mul_eq_zero_of_left _ _
-  suffices coeff n a ∈ I
-    by
-    rw [← @Ideal.mk_ker R _ I, RingHom.mem_ker] at this 
+  suffices coeff n a ∈ I by
+    rw [← @Ideal.mk_ker R _ I, RingHom.mem_ker] at this
     simp only [this, C_0]
   exact mem_map_C_iff.1 ha n
 #align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zero

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 -/
-import Mathbin.Data.Polynomial.Div
-import Mathbin.RingTheory.Polynomial.Basic
-import Mathbin.RingTheory.Ideal.QuotientOperations
+import Data.Polynomial.Div
+import RingTheory.Polynomial.Basic
+import RingTheory.Ideal.QuotientOperations
 
 #align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 4f840b8d28320b20c87db17b3a6eef3d325fca87
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Polynomial.Div
 import Mathbin.RingTheory.Polynomial.Basic
 import Mathbin.RingTheory.Ideal.QuotientOperations
 
+#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
+
 /-!
 # Quotients of polynomial rings
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -37,11 +37,13 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
 #align polynomial.quotient_span_X_sub_C_alg_equiv Polynomial.quotientSpanXSubCAlgEquiv
 -/
 
+#print Polynomial.quotientSpanXSubCAlgEquiv_mk /-
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :
     quotientSpanXSubCAlgEquiv x (Ideal.Quotient.mk _ p) = p.eval x :=
   rfl
 #align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mk
+-/
 
 #print Polynomial.quotientSpanXSubCAlgEquiv_symm_apply /-
 @[simp]
@@ -84,6 +86,7 @@ theorem quotient_map_C_eq_zero {I : Ideal R} :
 #align ideal.quotient_map_C_eq_zero Ideal.quotient_map_C_eq_zero
 -/
 
+#print Ideal.eval₂_C_mk_eq_zero /-
 theorem eval₂_C_mk_eq_zero {I : Ideal R} :
     ∀ f ∈ (map (C : R →+* R[X]) I : Ideal R[X]), eval₂RingHom (C.comp (Quotient.mk I)) X f = 0 :=
   by
@@ -100,6 +103,7 @@ theorem eval₂_C_mk_eq_zero {I : Ideal R} :
   · simpa [h] using quotient.eq_zero_iff_mem.2 ((mem_map_C_iff.1 ha) n)
   · simp [h]
 #align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zero
+-/
 
 #print Ideal.polynomialQuotientEquivQuotientPolynomial /-
 /-- If `I` is an ideal of `R`, then the ring polynomials over the quotient ring `I.quotient` is
@@ -168,6 +172,7 @@ theorem isDomain_map_C_quotient {P : Ideal R} (H : IsPrime P) :
 #align ideal.is_domain_map_C_quotient Ideal.isDomain_map_C_quotient
 -/
 
+#print Ideal.eq_zero_of_polynomial_mem_map_range /-
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
   If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
   In particular we can map `I` across this map, to get `I'` and a new map `R' → R'[x] → R'[x]/I`.
@@ -199,6 +204,7 @@ theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk
     rw [RingHom.mem_ker, RingHom.map_sub, hf.2, sub_eq_zero, coe_map_ring_hom, map_C]
   exact hx
 #align ideal.eq_zero_of_polynomial_mem_map_range Ideal.eq_zero_of_polynomial_mem_map_range
+-/
 
 end Ideal
 
@@ -206,6 +212,7 @@ namespace MvPolynomial
 
 variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 
+#print MvPolynomial.quotient_map_C_eq_zero /-
 theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
         i =
@@ -214,7 +221,9 @@ theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
   simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]
   exact Ideal.mem_map_of_mem _ hi
 #align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zero
+-/
 
+#print MvPolynomial.eval₂_C_mk_eq_zero /-
 theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :
     eval₂Hom (C.comp (Ideal.Quotient.mk I)) X a = 0 :=
@@ -230,6 +239,7 @@ theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     simp only [this, C_0]
   exact mem_map_C_iff.1 ha n
 #align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zero
+-/
 
 #print MvPolynomial.quotientEquivQuotientMvPolynomial /-
 /-- If `I` is an ideal of `R`, then the ring `mv_polynomial σ I.quotient` is isomorphic as an

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -51,6 +51,7 @@ theorem quotientSpanXSubCAlgEquiv_symm_apply (x : R) (y : R) :
 #align polynomial.quotient_span_X_sub_C_alg_equiv_symm_apply Polynomial.quotientSpanXSubCAlgEquiv_symm_apply
 -/
 
+#print Polynomial.quotientSpanCXSubCAlgEquiv /-
 /-- For a commutative ring $R$, evaluating a polynomial at an element $y \in R$ induces an
 isomorphism of $R$-algebras $R[X] / \langle x, X - y \rangle \cong R / \langle x \rangle$. -/
 noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
@@ -61,6 +62,7 @@ noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
         Ideal.quotientEquivAlgOfEq R <| by simp only [Ideal.map_span, Set.image_singleton]; congr 2;
           exact eval_C
 #align polynomial.quotient_span_C_X_sub_C_alg_equiv Polynomial.quotientSpanCXSubCAlgEquiv
+-/
 
 end Polynomial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -119,8 +119,8 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
     intro f
     apply Polynomial.induction_on' f
     · intro p q hp hq
-      simp only [coe_eval₂_ring_hom] at hp
-      simp only [coe_eval₂_ring_hom] at hq
+      simp only [coe_eval₂_ring_hom] at hp 
+      simp only [coe_eval₂_ring_hom] at hq 
       simp only [coe_eval₂_ring_hom, hp, hq, RingHom.map_add]
     · rintro n ⟨x⟩
       simp only [← smul_X_eq_monomial, C_mul', Quotient.lift_mk, Submodule.Quotient.quot_mk_eq_mk,
@@ -179,10 +179,10 @@ theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk
     by
     refine' fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => _
     rw [mem_comap, ← quotient.eq_zero_iff_mem, ← RingHom.comp_apply]
-    rw [RingHom.mem_ker, coe_map_ring_hom] at hf
+    rw [RingHom.mem_ker, coe_map_ring_hom] at hf 
     replace hf := congr_arg (fun f : Polynomial _ => f.coeff n) hf
-    simp only [coeff_map, coeff_zero] at hf
-    rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf
+    simp only [coeff_map, coeff_zero] at hf 
+    rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf 
   obtain ⟨x, hx'⟩ := x
   obtain ⟨y, rfl⟩ := RingHom.mem_range.1 hx'
   refine' Subtype.eq _
@@ -222,8 +222,9 @@ theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
   refine' Finset.sum_eq_zero fun n hn => _
   simp only [eval₂_monomial, Function.comp_apply, RingHom.coe_comp]
   refine' mul_eq_zero_of_left _ _
-  suffices coeff n a ∈ I by
-    rw [← @Ideal.mk_ker R _ I, RingHom.mem_ker] at this
+  suffices coeff n a ∈ I
+    by
+    rw [← @Ideal.mk_ker R _ I, RingHom.mem_ker] at this 
     simp only [this, C_0]
   exact mem_map_C_iff.1 ha n
 #align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zero

mathlib commit https://github.com/leanprover-community/mathlib/commit/88a563b158f59f2983cfad685664da95502e8cdd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 50251fd6309cca5ca2e747882ffecd2729f38c5d
+! leanprover-community/mathlib commit 4f840b8d28320b20c87db17b3a6eef3d325fca87
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -51,6 +51,17 @@ theorem quotientSpanXSubCAlgEquiv_symm_apply (x : R) (y : R) :
 #align polynomial.quotient_span_X_sub_C_alg_equiv_symm_apply Polynomial.quotientSpanXSubCAlgEquiv_symm_apply
 -/
 
+/-- For a commutative ring $R$, evaluating a polynomial at an element $y \in R$ induces an
+isomorphism of $R$-algebras $R[X] / \langle x, X - y \rangle \cong R / \langle x \rangle$. -/
+noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
+    (R[X] ⧸ (Ideal.span {C x, X - C y} : Ideal R[X])) ≃ₐ[R] R ⧸ (Ideal.span {x} : Ideal R) :=
+  (Ideal.quotientEquivAlgOfEq R <| by rw [Ideal.span_insert, sup_comm]).trans <|
+    (DoubleQuot.quotQuotEquivQuotSupₐ R _ _).symm.trans <|
+      (Ideal.quotientEquivAlg _ _ (quotientSpanXSubCAlgEquiv y) rfl).trans <|
+        Ideal.quotientEquivAlgOfEq R <| by simp only [Ideal.map_span, Set.image_singleton]; congr 2;
+          exact eval_C
+#align polynomial.quotient_span_C_X_sub_C_alg_equiv Polynomial.quotientSpanCXSubCAlgEquiv
+
 end Polynomial
 
 namespace Ideal

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -20,7 +20,7 @@ import Mathbin.RingTheory.Ideal.QuotientOperations
 -/
 
 
-open Polynomial
+open scoped Polynomial
 
 namespace Polynomial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -37,9 +37,6 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
 #align polynomial.quotient_span_X_sub_C_alg_equiv Polynomial.quotientSpanXSubCAlgEquiv
 -/
 
-/- warning: polynomial.quotient_span_X_sub_C_alg_equiv_mk -> Polynomial.quotientSpanXSubCAlgEquiv_mk is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mkₓ'. -/
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :
     quotientSpanXSubCAlgEquiv x (Ideal.Quotient.mk _ p) = p.eval x :=
@@ -74,9 +71,6 @@ theorem quotient_map_C_eq_zero {I : Ideal R} :
 #align ideal.quotient_map_C_eq_zero Ideal.quotient_map_C_eq_zero
 -/
 
-/- warning: ideal.eval₂_C_mk_eq_zero -> Ideal.eval₂_C_mk_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} :
     ∀ f ∈ (map (C : R →+* R[X]) I : Ideal R[X]), eval₂RingHom (C.comp (Quotient.mk I)) X f = 0 :=
   by
@@ -161,9 +155,6 @@ theorem isDomain_map_C_quotient {P : Ideal R} (H : IsPrime P) :
 #align ideal.is_domain_map_C_quotient Ideal.isDomain_map_C_quotient
 -/
 
-/- warning: ideal.eq_zero_of_polynomial_mem_map_range -> Ideal.eq_zero_of_polynomial_mem_map_range is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.eq_zero_of_polynomial_mem_map_range Ideal.eq_zero_of_polynomial_mem_map_rangeₓ'. -/
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
   If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
   In particular we can map `I` across this map, to get `I'` and a new map `R' → R'[x] → R'[x]/I`.
@@ -202,9 +193,6 @@ namespace MvPolynomial
 
 variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 
-/- warning: mv_polynomial.quotient_map_C_eq_zero -> MvPolynomial.quotient_map_C_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zeroₓ'. -/
 theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
         i =
@@ -214,9 +202,6 @@ theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
   exact Ideal.mem_map_of_mem _ hi
 #align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zero
 
-/- warning: mv_polynomial.eval₂_C_mk_eq_zero -> MvPolynomial.eval₂_C_mk_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :
     eval₂Hom (C.comp (Ideal.Quotient.mk I)) X a = 0 :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -38,10 +38,7 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
 -/
 
 /- warning: polynomial.quotient_span_X_sub_C_alg_equiv_mk -> Polynomial.quotientSpanXSubCAlgEquiv_mk is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (AlgEquiv.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) R (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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(Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R 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(Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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(CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ideal.Quotient.algebra.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} R _inst_1) (Polynomial.commRing.{u1} R _inst_1) (Polynomial.algebraOfAlgebra.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} 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+<too large>
 Case conversion may be inaccurate. Consider using '#align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mkₓ'. -/
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :
@@ -78,10 +75,7 @@ theorem quotient_map_C_eq_zero {I : Ideal R} :
 -/
 
 /- warning: ideal.eval₂_C_mk_eq_zero -> Ideal.eval₂_C_mk_eq_zero is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} :
     ∀ f ∈ (map (C : R →+* R[X]) I : Ideal R[X]), eval₂RingHom (C.comp (Quotient.mk I)) X f = 0 :=
@@ -168,10 +162,7 @@ theorem isDomain_map_C_quotient {P : Ideal R} (H : IsPrime P) :
 -/
 
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(Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))))))))) (AddSubmonoidClass.to_zeroMemClass.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoidWithOne.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocSemiring.toAddCommMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))))))) (Subring.setLike.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (SubsemiringClass.to_addSubmonoidClass.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R 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I) (fun (x : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) 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(Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.eq_zero_of_polynomial_mem_map_range Ideal.eq_zero_of_polynomial_mem_map_rangeₓ'. -/
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
   If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
@@ -212,10 +203,7 @@ namespace MvPolynomial
 variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 
 /- warning: mv_polynomial.quotient_map_C_eq_zero -> MvPolynomial.quotient_map_C_eq_zero is a dubious translation:
-lean 3 declaration is
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(CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) (fun (_x : RingHom.{u1, max u2 u1} R (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R 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(CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Ideal.Quotient.commRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) => R -> (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} 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(MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I))) 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(CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zeroₓ'. -/
 theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
@@ -227,10 +215,7 @@ theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
 #align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zero
 
 /- warning: mv_polynomial.eval₂_C_mk_eq_zero -> MvPolynomial.eval₂_C_mk_eq_zero is a dubious translation:
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(Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R 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R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R 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_inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} 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(CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (RingHom.instRingHomClassRingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))) (MvPolynomial.eval₂Hom.{u2, max u1 u2, u1} R (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) σ (CommRing.toCommSemiring.{u2} R _inst_1) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (RingHom.comp.{u2, u2, max u1 u2} R (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (MvPolynomial.C.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Ideal.Quotient.mk.{u2} R _inst_1 I)) (MvPolynomial.X.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommMonoidWithZero.toZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommSemiring.toCommMonoidWithZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommRing.toCommSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -41,7 +41,7 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (AlgEquiv.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) R (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) 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(Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) x p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) a) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) x p)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) a) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)))))))))))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) x p)
 Case conversion may be inaccurate. Consider using '#align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mkₓ'. -/
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -41,7 +41,7 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (AlgEquiv.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) R (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R 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(CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))))))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) x p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) a) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) => R) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} 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R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R 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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} 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(RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) a) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) a) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)))))))))))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instSingletonSet.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) x p)
 Case conversion may be inaccurate. Consider using '#align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mkₓ'. -/
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :
@@ -81,7 +81,7 @@ theorem quotient_map_C_eq_zero {I : Ideal R} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} (f : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (SetLike.hasMem.{u1, u1} (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.setLike.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Semiring.toModule.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) f (Ideal.map.{u1, u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.ringHomClass.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I)) -> (Eq.{succ u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) 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(CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) => (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) -> (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (RingHom.hasCoeToFun.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} (HasQuotient.Quotient.{u1, 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I)))) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) 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(HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (Polynomial.X.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) f) (OfNat.ofNat.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) 0 (Zero.zero.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.zero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.setLike.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Semiring.toModule.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) f (Ideal.map.{u1, u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) f) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.commSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (_x : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) 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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.setLike.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(Ideal.map.{u1, u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) f) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} 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(CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.commSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))))))) (Polynomial.eval₂RingHom.{u1, u1} R (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (RingHom.comp.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.semiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Polynomial.C.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (Polynomial.X.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) f) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) f) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) f) (Polynomial.zero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))))
 Case conversion may be inaccurate. Consider using '#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} :
     ∀ f ∈ (map (C : R →+* R[X]) I : Ideal R[X]), eval₂RingHom (C.comp (Quotient.mk I)) X f = 0 :=
@@ -171,7 +171,7 @@ theorem isDomain_map_C_quotient {P : Ideal R} (H : IsPrime P) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (x : coeSort.{succ u1, succ (succ u1)} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) 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(Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.setLike.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (AddZeroClass.toHasZero.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoidWithOne.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocSemiring.toAddCommMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} 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(Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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_inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} 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(AddMonoidWithOne.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} 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(CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)) (RingHom.comp.{u1, u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toNonAssocRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (x : Subtype.{succ u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (fun (x : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (SetLike.instMembership.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.instSetLikeSubring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) x (RingHom.range.{u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} R _inst_1) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, 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(CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)) (RingHom.comp.{u1, u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (x : Subtype.{succ u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (fun (x : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (SetLike.instMembership.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.instSetLikeSubring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) x (RingHom.range.{u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} R _inst_1) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, 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(CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)) (RingHom.comp.{u1, u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))))
 Case conversion may be inaccurate. Consider using '#align ideal.eq_zero_of_polynomial_mem_map_range Ideal.eq_zero_of_polynomial_mem_map_rangeₓ'. -/
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
   If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
@@ -215,7 +215,7 @@ variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 lean 3 declaration is
   forall {R : Type.{u1}} {σ : Type.{u2}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {i : R}, (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) i I) -> (Eq.{succ (max u2 u1)} (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (coeFn.{max (succ u1) (succ (max u2 u1)), max (succ u1) (succ (max u2 u1))} (RingHom.{u1, max u2 u1} R 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(CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) 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(CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R 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(CommRing.toRing.{max u2 u1} (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Ideal.Quotient.commRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) => R -> (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I))) (RingHom.hasCoeToFun.{u1, max u2 u1} R (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{max u2 u1} (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Ring.toNonAssocRing.{max u2 u1} (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (CommRing.toRing.{max u2 u1} (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Ideal.Quotient.commRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) (RingHom.comp.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Submodule.Quotient.HasQuotient.Quotient.hasZero.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (NonUnitalNonAssocRing.toAddCommGroup.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I))))))
 but is expected to have type
-  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {i : R}, (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) i I) -> (Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ 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(MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) i) (FunLike.coe.{max (succ u2) (succ u1), succ u2, max (succ u2) (succ u1)} (RingHom.{u2, max u2 u1} R (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ideal.map.{u2, max u2 u1, max u2 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(CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) i) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)))) 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R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ring.toAddCommGroup.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R 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+  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {i : R}, (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) i I) -> (Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ 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(MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) i) (OfNat.ofNat.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) i) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) i) (Submodule.Quotient.instZeroQuotientSubmoduleToSemiringToAddCommMonoidHasQuotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ring.toAddCommGroup.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)))))
 Case conversion may be inaccurate. Consider using '#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zeroₓ'. -/
 theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
@@ -230,7 +230,7 @@ theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
 lean 3 declaration is
   forall {R : Type.{u1}} {σ : Type.{u2}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {a : MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)}, (Membership.Mem.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (SetLike.hasMem.{max u2 u1, max u2 u1} (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.setLike.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))))) a (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) -> (Eq.{max (succ u2) (succ u1)} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (coeFn.{succ (max u2 u1), succ (max u2 u1)} (RingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) (fun (_x : RingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) => (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) -> (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (RingHom.hasCoeToFun.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) (MvPolynomial.eval₂Hom.{u1, max u2 u1, u2} R (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) σ (CommRing.toCommSemiring.{u1} R _inst_1) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (RingHom.comp.{u1, u1, max u2 u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (MvPolynomial.C.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (MvPolynomial.X.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) 0 (OfNat.mk.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) 0 (Zero.zero.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MulZeroClass.toHasZero.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonUnitalNonAssocSemiring.toMulZeroClass.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonAssocRing.toNonUnitalNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Ring.toNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commRing.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (Ideal.Quotient.commRing.{u1} R _inst_1 I))))))))))))
 but is expected to have type
-  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {a : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)}, (Membership.mem.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (SetLike.instMembership.{max u2 u1, max u2 u1} (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.setLike.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) a (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) -> (Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R 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(MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, max u2 u1, max u2 u1} (RingHom.{max u2 u1, max u1 u2} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (RingHom.instRingHomClassRingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))) (MvPolynomial.eval₂Hom.{u2, max u1 u2, u1} R (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) σ (CommRing.toCommSemiring.{u2} R _inst_1) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (RingHom.comp.{u2, u2, max u1 u2} R (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (MvPolynomial.C.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Ideal.Quotient.mk.{u2} R _inst_1 I)) (MvPolynomial.X.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommMonoidWithZero.toZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommSemiring.toCommMonoidWithZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommRing.toCommSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))
+  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {a : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)}, (Membership.mem.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (SetLike.instMembership.{max u2 u1, max u2 u1} (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.setLike.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) a (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) -> (Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (RingHom.{max u2 u1, max u1 u2} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) 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u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (NonUnitalNonAssocSemiring.toMul.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, max u2 u1, max u2 u1} (RingHom.{max u2 u1, max u1 u2} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, max u2 u1, max u2 u1} (RingHom.{max u2 u1, max u1 u2} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (RingHom.instRingHomClassRingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))) (MvPolynomial.eval₂Hom.{u2, max u1 u2, u1} R (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) σ (CommRing.toCommSemiring.{u2} R _inst_1) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (RingHom.comp.{u2, u2, max u1 u2} R (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (MvPolynomial.C.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Ideal.Quotient.mk.{u2} R _inst_1 I)) (MvPolynomial.X.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommMonoidWithZero.toZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommSemiring.toCommMonoidWithZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommRing.toCommSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))
 Case conversion may be inaccurate. Consider using '#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zeroₓ'. -/
 theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 61d8b8248633da198afea97ae7a90ee63bdf8c1c
+! leanprover-community/mathlib commit 50251fd6309cca5ca2e747882ffecd2729f38c5d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.RingTheory.Ideal.QuotientOperations
 
 /-!
 # Quotients of polynomial rings
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 -/
 
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/f8c79b0a623404854a2902b836eac32156fd7712

@@ -23,6 +23,7 @@ namespace Polynomial
 
 variable {R : Type _} [CommRing R]
 
+#print Polynomial.quotientSpanXSubCAlgEquiv /-
 /-- For a commutative ring $R$, evaluating a polynomial at an element $x \in R$ induces an
 isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. -/
 noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
@@ -31,18 +32,27 @@ noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
           (ker_eval_ring_hom x : RingHom.ker (aeval x).toRingHom = _)).symm.trans <|
     Ideal.quotientKerAlgEquivOfRightInverse fun _ => eval_C
 #align polynomial.quotient_span_X_sub_C_alg_equiv Polynomial.quotientSpanXSubCAlgEquiv
+-/
 
+/- warning: polynomial.quotient_span_X_sub_C_alg_equiv_mk -> Polynomial.quotientSpanXSubCAlgEquiv_mk is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (AlgEquiv.{u1, u1, u1} R (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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(Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) (Ideal.span.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSingleton.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R 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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) x p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Ideal.span.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x))))) p)) (Polynomial.eval.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) x p)
+Case conversion may be inaccurate. Consider using '#align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mkₓ'. -/
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_mk (x : R) (p : R[X]) :
     quotientSpanXSubCAlgEquiv x (Ideal.Quotient.mk _ p) = p.eval x :=
   rfl
 #align polynomial.quotient_span_X_sub_C_alg_equiv_mk Polynomial.quotientSpanXSubCAlgEquiv_mk
 
+#print Polynomial.quotientSpanXSubCAlgEquiv_symm_apply /-
 @[simp]
 theorem quotientSpanXSubCAlgEquiv_symm_apply (x : R) (y : R) :
     (quotientSpanXSubCAlgEquiv x).symm y = algebraMap R _ y :=
   rfl
 #align polynomial.quotient_span_X_sub_C_alg_equiv_symm_apply Polynomial.quotientSpanXSubCAlgEquiv_symm_apply
+-/
 
 end Polynomial
 
@@ -54,15 +64,23 @@ open Polynomial
 
 variable {R : Type _} [CommRing R]
 
-theorem quotient_map_c_eq_zero {I : Ideal R} :
+#print Ideal.quotient_map_C_eq_zero /-
+theorem quotient_map_C_eq_zero {I : Ideal R} :
     ∀ a ∈ I, ((Quotient.mk (map (C : R →+* R[X]) I : Ideal R[X])).comp C) a = 0 :=
   by
   intro a ha
   rw [RingHom.comp_apply, quotient.eq_zero_iff_mem]
   exact mem_map_of_mem _ ha
-#align ideal.quotient_map_C_eq_zero Ideal.quotient_map_c_eq_zero
+#align ideal.quotient_map_C_eq_zero Ideal.quotient_map_C_eq_zero
+-/
 
-theorem eval₂_c_mk_eq_zero {I : Ideal R} :
+/- warning: ideal.eval₂_C_mk_eq_zero -> Ideal.eval₂_C_mk_eq_zero is a dubious translation:
+lean 3 declaration is
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(CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.commSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) (Polynomial.eval₂RingHom.{u1, u1} R (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R 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I)))) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.semiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Polynomial.C.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (Polynomial.X.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) f) (OfNat.ofNat.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) 0 (Zero.zero.{u1} (Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Polynomial.zero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.setLike.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Semiring.toModule.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) f (Ideal.map.{u1, u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => Polynomial.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} 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+Case conversion may be inaccurate. Consider using '#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zeroₓ'. -/
+theorem eval₂_C_mk_eq_zero {I : Ideal R} :
     ∀ f ∈ (map (C : R →+* R[X]) I : Ideal R[X]), eval₂RingHom (C.comp (Quotient.mk I)) X f = 0 :=
   by
   intro a ha
@@ -77,8 +95,9 @@ theorem eval₂_c_mk_eq_zero {I : Ideal R} :
   by_cases h : m = 0
   · simpa [h] using quotient.eq_zero_iff_mem.2 ((mem_map_C_iff.1 ha) n)
   · simp [h]
-#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_c_mk_eq_zero
+#align ideal.eval₂_C_mk_eq_zero Ideal.eval₂_C_mk_eq_zero
 
+#print Ideal.polynomialQuotientEquivQuotientPolynomial /-
 /-- If `I` is an ideal of `R`, then the ring polynomials over the quotient ring `I.quotient` is
 isomorphic to the quotient of `R[X]` by the ideal `map C I`,
 where `map C I` contains exactly the polynomials whose coefficients all lie in `I` -/
@@ -87,11 +106,11 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
     where
   toFun :=
     eval₂RingHom
-      (Quotient.lift I ((Quotient.mk (map C I : Ideal R[X])).comp C) quotient_map_c_eq_zero)
+      (Quotient.lift I ((Quotient.mk (map C I : Ideal R[X])).comp C) quotient_map_C_eq_zero)
       (Quotient.mk (map C I : Ideal R[X]) X)
   invFun :=
     Quotient.lift (map C I : Ideal R[X]) (eval₂RingHom (C.comp (Quotient.mk I)) X)
-      eval₂_c_mk_eq_zero
+      eval₂_C_mk_eq_zero
   map_mul' f g := by simp only [coe_eval₂_ring_hom, eval₂_mul]
   map_add' f g := by simp only [eval₂_add, coe_eval₂_ring_hom]
   left_inv := by
@@ -115,7 +134,9 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
         Submodule.Quotient.quot_mk_eq_mk, quotient.mk_eq_mk, eval₂_X_pow, eval₂_smul,
         coe_eval₂_ring_hom, RingHom.map_pow, eval₂_C, RingHom.coe_comp, RingHom.map_mul, eval₂_X]
 #align ideal.polynomial_quotient_equiv_quotient_polynomial Ideal.polynomialQuotientEquivQuotientPolynomial
+-/
 
+#print Ideal.polynomialQuotientEquivQuotientPolynomial_symm_mk /-
 @[simp]
 theorem polynomialQuotientEquivQuotientPolynomial_symm_mk (I : Ideal R) (f : R[X]) :
     I.polynomialQuotientEquivQuotientPolynomial.symm (Quotient.mk _ f) = f.map (Quotient.mk I) := by
@@ -123,7 +144,9 @@ theorem polynomialQuotientEquivQuotientPolynomial_symm_mk (I : Ideal R) (f : R[X
     Ideal.Quotient.lift_mk, coe_eval₂_ring_hom, eval₂_eq_eval_map, ← Polynomial.map_map, ←
     eval₂_eq_eval_map, Polynomial.eval₂_C_X]
 #align ideal.polynomial_quotient_equiv_quotient_polynomial_symm_mk Ideal.polynomialQuotientEquivQuotientPolynomial_symm_mk
+-/
 
+#print Ideal.polynomialQuotientEquivQuotientPolynomial_map_mk /-
 @[simp]
 theorem polynomialQuotientEquivQuotientPolynomial_map_mk (I : Ideal R) (f : R[X]) :
     I.polynomialQuotientEquivQuotientPolynomial (f.map I.Quotient.mk) = Quotient.mk _ f :=
@@ -131,13 +154,22 @@ theorem polynomialQuotientEquivQuotientPolynomial_map_mk (I : Ideal R) (f : R[X]
   apply (polynomial_quotient_equiv_quotient_polynomial I).symm.Injective
   rw [RingEquiv.symm_apply_apply, polynomial_quotient_equiv_quotient_polynomial_symm_mk]
 #align ideal.polynomial_quotient_equiv_quotient_polynomial_map_mk Ideal.polynomialQuotientEquivQuotientPolynomial_map_mk
+-/
 
+#print Ideal.isDomain_map_C_quotient /-
 /-- If `P` is a prime ideal of `R`, then `R[x]/(P)` is an integral domain. -/
-theorem isDomain_map_c_quotient {P : Ideal R} (H : IsPrime P) :
+theorem isDomain_map_C_quotient {P : Ideal R} (H : IsPrime P) :
     IsDomain (R[X] ⧸ (map (C : R →+* R[X]) P : Ideal R[X])) :=
   RingEquiv.isDomain (Polynomial (R ⧸ P)) (polynomialQuotientEquivQuotientPolynomial P).symm
-#align ideal.is_domain_map_C_quotient Ideal.isDomain_map_c_quotient
+#align ideal.is_domain_map_C_quotient Ideal.isDomain_map_C_quotient
+-/
 
+/- warning: ideal.eq_zero_of_polynomial_mem_map_range -> Ideal.eq_zero_of_polynomial_mem_map_range is a dubious translation:
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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (AddZeroClass.toHasZero.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoidWithOne.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocSemiring.toAddCommMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))))))))) (AddSubmonoidClass.to_zeroMemClass.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddMonoidWithOne.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocSemiring.toAddCommMonoidWithOne.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))))))) (Subring.setLike.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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(Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toNonAssocRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ring.toSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (x : Subtype.{succ u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (fun (x : HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (SetLike.instMembership.{u1, u1} (Subring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I))) (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Subring.instSetLikeSubring.{u1} (HasQuotient.Quotient.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R 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(Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) I) (Ideal.Quotient.commRing.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I)))) (Ideal.Quotient.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1) I) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))))
+Case conversion may be inaccurate. Consider using '#align ideal.eq_zero_of_polynomial_mem_map_range Ideal.eq_zero_of_polynomial_mem_map_rangeₓ'. -/
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
   If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
   In particular we can map `I` across this map, to get `I'` and a new map `R' → R'[x] → R'[x]/I`.
@@ -176,16 +208,28 @@ namespace MvPolynomial
 
 variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 
-theorem quotient_map_c_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
+/- warning: mv_polynomial.quotient_map_C_eq_zero -> MvPolynomial.quotient_map_C_eq_zero is a dubious translation:
+lean 3 declaration is
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(CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) (fun (_x : RingHom.{u1, max u2 u1} R (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R 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(CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (Ideal.hasQuotient.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)) (Ideal.Quotient.commRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1) (Ideal.map.{u1, max u2 u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (RingHom.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I)))))) => R -> (HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} 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(MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1))) (RingHom.ringHomClass.{u1, max u2 u1} R (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)) I))) 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+but is expected to have type
+  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {i : R}, (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) i I) -> (Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => HasQuotient.Quotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ 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_inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)) i) (Submodule.Quotient.instZeroQuotientSubmoduleToSemiringToAddCommMonoidHasQuotient.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1)) (Ring.toAddCommGroup.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} R σ _inst_1))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (MvPolynomial.C.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)) I)))))
+Case conversion may be inaccurate. Consider using '#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zeroₓ'. -/
+theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
         i =
       0 :=
   by
   simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]
   exact Ideal.mem_map_of_mem _ hi
-#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_c_eq_zero
+#align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_C_eq_zero
 
-theorem eval₂_c_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
+/- warning: mv_polynomial.eval₂_C_mk_eq_zero -> MvPolynomial.eval₂_C_mk_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {σ : Type.{u2}} [_inst_1 : CommRing.{u1} R] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {a : MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)}, (Membership.Mem.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, u2} R σ _inst_1)))) (SetLike.hasMem.{max u2 u1, max u2 u1} (Ideal.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) 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R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) => (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) -> (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (RingHom.hasCoeToFun.{max u2 u1, max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, u2} R σ (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))) (MvPolynomial.eval₂Hom.{u1, max u2 u1, u2} R (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) σ (CommRing.toCommSemiring.{u1} R _inst_1) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (RingHom.comp.{u1, u1, max u2 u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commSemiring.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (MvPolynomial.C.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Ideal.Quotient.mk.{u1} R _inst_1 I)) (MvPolynomial.X.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) 0 (OfNat.mk.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) 0 (Zero.zero.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MulZeroClass.toHasZero.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonUnitalNonAssocSemiring.toMulZeroClass.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (NonAssocRing.toNonUnitalNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (Ring.toNonAssocRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (CommRing.toRing.{max u2 u1} (MvPolynomial.{u2, u1} σ (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (MvPolynomial.commRing.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) σ (Ideal.Quotient.commRing.{u1} R _inst_1 I))))))))))))
+but is expected to have type
+  forall {R : Type.{u2}} {σ : Type.{u1}} [_inst_1 : CommRing.{u2} R] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {a : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)}, (Membership.mem.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (SetLike.instMembership.{max u2 u1, max u2 u1} (Ideal.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.setLike.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Semiring.toModule.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))))) a (Ideal.map.{u2, max u2 u1, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.{u2, max u2 u1} R (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1))))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} 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_inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))) (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} 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(CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (RingHom.instRingHomClassRingHom.{max u2 u1, max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, u1} R σ (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))) (MvPolynomial.eval₂Hom.{u2, max u1 u2, u1} R (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) σ (CommRing.toCommSemiring.{u2} R _inst_1) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (RingHom.comp.{u2, u2, max u1 u2} R (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (MvPolynomial.commSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))))) (MvPolynomial.C.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) (Ideal.Quotient.mk.{u2} R _inst_1 I)) (MvPolynomial.X.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))) a) (OfNat.ofNat.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) 0 (Zero.toOfNat0.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommMonoidWithZero.toZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommSemiring.toCommMonoidWithZero.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (CommRing.toCommSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : MvPolynomial.{u1, u2} σ R (CommRing.toCommSemiring.{u2} R _inst_1)) => MvPolynomial.{u1, u2} σ (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I))) a) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u2, u1} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) σ (Ideal.Quotient.commRing.{u2} R _inst_1 I))))))))
+Case conversion may be inaccurate. Consider using '#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zeroₓ'. -/
+theorem eval₂_C_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :
     eval₂Hom (C.comp (Ideal.Quotient.mk I)) X a = 0 :=
   by
@@ -198,8 +242,9 @@ theorem eval₂_c_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
     rw [← @Ideal.mk_ker R _ I, RingHom.mem_ker] at this
     simp only [this, C_0]
   exact mem_map_C_iff.1 ha n
-#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_c_mk_eq_zero
+#align mv_polynomial.eval₂_C_mk_eq_zero MvPolynomial.eval₂_C_mk_eq_zero
 
+#print MvPolynomial.quotientEquivQuotientMvPolynomial /-
 /-- If `I` is an ideal of `R`, then the ring `mv_polynomial σ I.quotient` is isomorphic as an
 `R`-algebra to the quotient of `mv_polynomial σ R` by the ideal generated by `I`. -/
 def quotientEquivQuotientMvPolynomial (I : Ideal R) :
@@ -208,11 +253,11 @@ def quotientEquivQuotientMvPolynomial (I : Ideal R) :
   toFun :=
     eval₂Hom
       (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I : Ideal (MvPolynomial σ R))).comp C)
-        fun i hi => quotient_map_c_eq_zero hi)
+        fun i hi => quotient_map_C_eq_zero hi)
       fun i => Ideal.Quotient.mk (Ideal.map C I : Ideal (MvPolynomial σ R)) (X i)
   invFun :=
     Ideal.Quotient.lift (Ideal.map C I : Ideal (MvPolynomial σ R))
-      (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) fun a ha => eval₂_c_mk_eq_zero ha
+      (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) fun a ha => eval₂_C_mk_eq_zero ha
   map_mul' := RingHom.map_mul _
   map_add' := RingHom.map_add _
   left_inv := by
@@ -241,6 +286,7 @@ def quotientEquivQuotientMvPolynomial (I : Ideal R) :
       simp only [hp]
   commutes' r := eval₂Hom_C _ _ (Ideal.Quotient.mk I r)
 #align mv_polynomial.quotient_equiv_quotient_mv_polynomial MvPolynomial.quotientEquivQuotientMvPolynomial
+-/
 
 end MvPolynomial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 5120cf49cb659e2499edd7e4d336a04efd598f2f
+! leanprover-community/mathlib commit 61d8b8248633da198afea97ae7a90ee63bdf8c1c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -27,9 +27,8 @@ variable {R : Type _} [CommRing R]
 isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. -/
 noncomputable def quotientSpanXSubCAlgEquiv (x : R) :
     (R[X] ⧸ Ideal.span ({X - C x} : Set R[X])) ≃ₐ[R] R :=
-  (AlgEquiv.restrictScalars R <|
-          Ideal.quotientEquivAlgOfEq R
-            (ker_eval_ring_hom x : RingHom.ker (aeval x).toRingHom = _)).symm.trans <|
+  (Ideal.quotientEquivAlgOfEq R
+          (ker_eval_ring_hom x : RingHom.ker (aeval x).toRingHom = _)).symm.trans <|
     Ideal.quotientKerAlgEquivOfRightInverse fun _ => eval_C
 #align polynomial.quotient_span_X_sub_C_alg_equiv Polynomial.quotientSpanXSubCAlgEquiv
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/290a7ba01fbcab1b64757bdaa270d28f4dcede35

@@ -120,7 +120,7 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
 @[simp]
 theorem polynomialQuotientEquivQuotientPolynomial_symm_mk (I : Ideal R) (f : R[X]) :
     I.polynomialQuotientEquivQuotientPolynomial.symm (Quotient.mk _ f) = f.map (Quotient.mk I) := by
-  rw [polynomial_quotient_equiv_quotient_polynomial, RingEquiv.symm_mk, [anonymous],
+  rw [polynomial_quotient_equiv_quotient_polynomial, RingEquiv.symm_mk, RingEquiv.coe_mk,
     Ideal.Quotient.lift_mk, coe_eval₂_ring_hom, eval₂_eq_eval_map, ← Polynomial.map_map, ←
     eval₂_eq_eval_map, Polynomial.eval₂_C_X]
 #align ideal.polynomial_quotient_equiv_quotient_polynomial_symm_mk Ideal.polynomialQuotientEquivQuotientPolynomial_symm_mk

mathlib commit https://github.com/leanprover-community/mathlib/commit/da3fc4a33ff6bc75f077f691dc94c217b8d41559

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit da420a8c6dd5bdfb85c4ced85c34388f633bc6ff
+! leanprover-community/mathlib commit 5120cf49cb659e2499edd7e4d336a04efd598f2f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -221,26 +221,24 @@ def quotientEquivQuotientMvPolynomial (I : Ideal R) :
     apply induction_on f
     · rintro ⟨r⟩
       rw [coe_eval₂_hom, eval₂_C]
-      simp only [eval₂_hom_eq_bind₂, Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.lift_mk,
-        Ideal.Quotient.mk_eq_mk, bind₂_C_right, RingHom.coe_comp]
+      simp only [Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.lift_mk, MvPolynomial.eval₂Hom_C,
+        Function.comp_apply, Ideal.Quotient.mk_eq_mk, MvPolynomial.C_inj, RingHom.coe_comp]
     · simp_intro p q hp hq only [RingHom.map_add, MvPolynomial.coe_eval₂Hom, coe_eval₂_hom,
-        MvPolynomial.eval₂_add, MvPolynomial.eval₂Hom_eq_bind₂, eval₂_hom_eq_bind₂]
+        MvPolynomial.eval₂_add]
       rw [hp, hq]
-    · simp_intro p i hp only [eval₂_hom_eq_bind₂, coe_eval₂_hom]
-      simp only [hp, eval₂_hom_eq_bind₂, coe_eval₂_hom, Ideal.Quotient.lift_mk, bind₂_X_right,
-        eval₂_mul, RingHom.map_mul, eval₂_X]
+    · simp_intro p i hp only [coe_eval₂_hom]
+      simp only [hp, coe_eval₂_hom, Ideal.Quotient.lift_mk, eval₂_mul, RingHom.map_mul, eval₂_X]
   right_inv := by
     rintro ⟨f⟩
     apply induction_on f
     · intro r
       simp only [Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk,
         RingHom.coe_comp, eval₂_hom_C]
-    · simp_intro p q hp hq only [eval₂_hom_eq_bind₂, Submodule.Quotient.quot_mk_eq_mk, eval₂_add,
-        RingHom.map_add, coe_eval₂_hom, Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk]
+    · simp_intro p q hp hq only [Submodule.Quotient.quot_mk_eq_mk, eval₂_add, RingHom.map_add,
+        coe_eval₂_hom, Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk]
       rw [hp, hq]
-    · simp_intro p i hp only [eval₂_hom_eq_bind₂, Submodule.Quotient.quot_mk_eq_mk, coe_eval₂_hom,
-        Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk, bind₂_X_right, eval₂_mul, RingHom.map_mul,
-        eval₂_X]
+    · simp_intro p i hp only [Submodule.Quotient.quot_mk_eq_mk, coe_eval₂_hom,
+        Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk, eval₂_mul, RingHom.map_mul, eval₂_X]
       simp only [hp]
   commutes' r := eval₂Hom_C _ _ (Ideal.Quotient.mk I r)
 #align mv_polynomial.quotient_equiv_quotient_mv_polynomial MvPolynomial.quotientEquivQuotientMvPolynomial

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

@@ -178,7 +178,7 @@ namespace MvPolynomial
 variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
 
 theorem quotient_map_c_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
-    (Ideal.Quotient.mk (Ideal.map (c : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp c
+    (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))).comp C
         i =
       0 :=
   by
@@ -187,8 +187,8 @@ theorem quotient_map_c_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
 #align mv_polynomial.quotient_map_C_eq_zero MvPolynomial.quotient_map_c_eq_zero
 
 theorem eval₂_c_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
-    (ha : a ∈ (Ideal.map (c : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :
-    eval₂Hom (c.comp (Ideal.Quotient.mk I)) x a = 0 :=
+    (ha : a ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R))) :
+    eval₂Hom (C.comp (Ideal.Quotient.mk I)) X a = 0 :=
   by
   rw [as_sum a]
   rw [coe_eval₂_hom, eval₂_sum]
@@ -204,16 +204,16 @@ theorem eval₂_c_mk_eq_zero {I : Ideal R} {a : MvPolynomial σ R}
 /-- If `I` is an ideal of `R`, then the ring `mv_polynomial σ I.quotient` is isomorphic as an
 `R`-algebra to the quotient of `mv_polynomial σ R` by the ideal generated by `I`. -/
 def quotientEquivQuotientMvPolynomial (I : Ideal R) :
-    MvPolynomial σ (R ⧸ I) ≃ₐ[R] MvPolynomial σ R ⧸ (Ideal.map c I : Ideal (MvPolynomial σ R))
+    MvPolynomial σ (R ⧸ I) ≃ₐ[R] MvPolynomial σ R ⧸ (Ideal.map C I : Ideal (MvPolynomial σ R))
     where
   toFun :=
     eval₂Hom
-      (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map c I : Ideal (MvPolynomial σ R))).comp c)
+      (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I : Ideal (MvPolynomial σ R))).comp C)
         fun i hi => quotient_map_c_eq_zero hi)
-      fun i => Ideal.Quotient.mk (Ideal.map c I : Ideal (MvPolynomial σ R)) (x i)
+      fun i => Ideal.Quotient.mk (Ideal.map C I : Ideal (MvPolynomial σ R)) (X i)
   invFun :=
-    Ideal.Quotient.lift (Ideal.map c I : Ideal (MvPolynomial σ R))
-      (eval₂Hom (c.comp (Ideal.Quotient.mk I)) x) fun a ha => eval₂_c_mk_eq_zero ha
+    Ideal.Quotient.lift (Ideal.map C I : Ideal (MvPolynomial σ R))
+      (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) fun a ha => eval₂_c_mk_eq_zero ha
   map_mul' := RingHom.map_mul _
   map_add' := RingHom.map_add _
   left_inv := by
@@ -242,7 +242,7 @@ def quotientEquivQuotientMvPolynomial (I : Ideal R) :
         Ideal.Quotient.lift_mk, Ideal.Quotient.mk_eq_mk, bind₂_X_right, eval₂_mul, RingHom.map_mul,
         eval₂_X]
       simp only [hp]
-  commutes' r := eval₂Hom_c _ _ (Ideal.Quotient.mk I r)
+  commutes' r := eval₂Hom_C _ _ (Ideal.Quotient.mk I r)
 #align mv_polynomial.quotient_equiv_quotient_mv_polynomial MvPolynomial.quotientEquivQuotientMvPolynomial
 
 end MvPolynomial

mathlib commit https://github.com/leanprover-community/mathlib/commit/f24cc2891c0e328f0ee8c57387103aa462c44b5e

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -3,7 +3,7 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 -/
-import Mathlib.Data.Polynomial.Div
+import Mathlib.Algebra.Polynomial.Div
 import Mathlib.RingTheory.Polynomial.Basic
 import Mathlib.RingTheory.Ideal.QuotientOperations
 

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -32,7 +32,7 @@ noncomputable def quotientSpanXSubCAlgEquivAux1 (x : R) :
     (R[X] ⧸ Ideal.span {X - C x}) ≃ₐ[R] (R[X] ⧸ (RingHom.ker (aeval x).toRingHom : Ideal R[X])) :=
   @Ideal.quotientEquivAlgOfEq R R[X] _ _ _ _ _ (ker_evalRingHom x).symm
 
--- porting note: need to split this definition into two sub-definitions to prevent time out
+-- Porting note: need to split this definition into two sub-definitions to prevent time out
 /-- For a commutative ring $R$, evaluating a polynomial at an element $x \in R$ induces an
 isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. -/
 noncomputable def quotientSpanXSubCAlgEquiv (x : R) :

We clean up heart beat bumps after #6474.

@@ -63,7 +63,6 @@ noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
           simp only [Ideal.map_span, Set.image_singleton]; congr 2; exact eval_C
 #align polynomial.quotient_span_C_X_sub_C_alg_equiv Polynomial.quotientSpanCXSubCAlgEquiv
 
-set_option maxHeartbeats 250000 in
 /-- For a commutative ring $R$, evaluating a polynomial at elements $y(X) \in R[X]$ and $x \in R$
 induces an isomorphism of $R$-algebras $R[X, Y] / \langle X - x, Y - y(X) \rangle \cong R$. -/
 noncomputable def quotientSpanCXSubCXSubCAlgEquiv {x : R} {y : R[X]} :

Generalizes RingEquiv.noZeroDivisors and RingEquiv.isDomain to MulEquiv

Adds Function.Injective.isLeft/RightCancelMulZero, MulEquiv.toZeroHomClass, and MulEquiv.isField (the last one is useful for #6309)

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -162,7 +162,7 @@ theorem polynomialQuotientEquivQuotientPolynomial_map_mk (I : Ideal R) (f : R[X]
 /-- If `P` is a prime ideal of `R`, then `R[x]/(P)` is an integral domain. -/
 theorem isDomain_map_C_quotient {P : Ideal R} (_ : IsPrime P) :
     IsDomain (R[X] ⧸ (map (C : R →+* R[X]) P : Ideal R[X])) :=
-  RingEquiv.isDomain (Polynomial (R ⧸ P)) (polynomialQuotientEquivQuotientPolynomial P).symm
+  MulEquiv.isDomain (Polynomial (R ⧸ P)) (polynomialQuotientEquivQuotientPolynomial P).symm
 #align ideal.is_domain_map_C_quotient Ideal.isDomain_map_C_quotient
 
 /-- Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -20,7 +20,7 @@ open Polynomial
 
 namespace Polynomial
 
-variable {R : Type _} [CommRing R]
+variable {R : Type*} [CommRing R]
 
 noncomputable def quotientSpanXSubCAlgEquivAux2 (x : R) :
     (R[X] ⧸ (RingHom.ker (aeval x).toRingHom : Ideal R[X])) ≃ₐ[R] R :=
@@ -79,7 +79,7 @@ noncomputable section
 
 open Polynomial
 
-variable {R : Type _} [CommRing R]
+variable {R : Type*} [CommRing R]
 
 theorem quotient_map_C_eq_zero {I : Ideal R} :
     ∀ a ∈ I, ((Quotient.mk (map (C : R →+* R[X]) I : Ideal R[X])).comp C) a = 0 := by
@@ -197,7 +197,7 @@ end Ideal
 
 namespace MvPolynomial
 
-variable {R : Type _} {σ : Type _} [CommRing R] {r : R}
+variable {R : Type*} {σ : Type*} [CommRing R] {r : R}
 
 theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
     (Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I :

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 4f840b8d28320b20c87db17b3a6eef3d325fca87
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Polynomial.Div
 import Mathlib.RingTheory.Polynomial.Basic
 import Mathlib.RingTheory.Ideal.QuotientOperations
 
+#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
+
 /-!
 # Quotients of polynomial rings
 -/

Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com> Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -66,6 +66,14 @@ noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
           simp only [Ideal.map_span, Set.image_singleton]; congr 2; exact eval_C
 #align polynomial.quotient_span_C_X_sub_C_alg_equiv Polynomial.quotientSpanCXSubCAlgEquiv
 
+set_option maxHeartbeats 250000 in
+/-- For a commutative ring $R$, evaluating a polynomial at elements $y(X) \in R[X]$ and $x \in R$
+induces an isomorphism of $R$-algebras $R[X, Y] / \langle X - x, Y - y(X) \rangle \cong R$. -/
+noncomputable def quotientSpanCXSubCXSubCAlgEquiv {x : R} {y : R[X]} :
+    @AlgEquiv R (R[X][X] ⧸ (Ideal.span {C (X - C x), X - C y} : Ideal <| R[X][X])) R _ _ _
+      (Ideal.Quotient.algebra R) _ :=
+((quotientSpanCXSubCAlgEquiv (X - C x) y).restrictScalars R).trans <| quotientSpanXSubCAlgEquiv x
+
 end Polynomial
 
 namespace Ideal

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -129,7 +129,7 @@ def polynomialQuotientEquivQuotientPolynomial (I : Ideal R) :
     · -- Porting note: was `simp_intro p q hp hq`
       intros p q hp hq
       simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, map_add, Quotient.lift_mk,
-        coe_eval₂RingHom] at hp hq⊢
+        coe_eval₂RingHom] at hp hq ⊢
       rw [hp, hq]
     · intro n a
       simp only [← smul_X_eq_monomial, ← C_mul' a (X ^ n), Quotient.lift_mk,

ring_theory.polynomial.quotient@61d8b8248633da198afea97ae7a90ee63bdf8c1c..4f840b8d28320b20c87db17b3a6eef3d325fca87

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.quotient
-! leanprover-community/mathlib commit 61d8b8248633da198afea97ae7a90ee63bdf8c1c
+! leanprover-community/mathlib commit 4f840b8d28320b20c87db17b3a6eef3d325fca87
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -55,6 +55,17 @@ theorem quotientSpanXSubCAlgEquiv_symm_apply (x : R) (y : R) :
   rfl
 #align polynomial.quotient_span_X_sub_C_alg_equiv_symm_apply Polynomial.quotientSpanXSubCAlgEquiv_symm_apply
 
+/-- For a commutative ring $R$, evaluating a polynomial at an element $y \in R$ induces an
+isomorphism of $R$-algebras $R[X] / \langle x, X - y \rangle \cong R / \langle x \rangle$. -/
+noncomputable def quotientSpanCXSubCAlgEquiv (x y : R) :
+    (R[X] ⧸ (Ideal.span {C x, X - C y} : Ideal R[X])) ≃ₐ[R] R ⧸ (Ideal.span {x} : Ideal R) :=
+  (Ideal.quotientEquivAlgOfEq R <| by rw [Ideal.span_insert, sup_comm]).trans <|
+    (DoubleQuot.quotQuotEquivQuotSupₐ R _ _).symm.trans <|
+      (Ideal.quotientEquivAlg _ _ (quotientSpanXSubCAlgEquiv y) rfl).trans <|
+        Ideal.quotientEquivAlgOfEq R <| by
+          simp only [Ideal.map_span, Set.image_singleton]; congr 2; exact eval_C
+#align polynomial.quotient_span_C_X_sub_C_alg_equiv Polynomial.quotientSpanCXSubCAlgEquiv
+
 end Polynomial
 
 namespace Ideal

Coming from #3292

• depends on: #3414

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kevin Buzzard <k.buzzard@imperial.ac.uk> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>