zero_divisors_in_add_monoid_algebras ⟷ Counterexamples.ZeroDivisorsInAddMonoidAlgebras

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

@@ -7,7 +7,7 @@ import Algebra.GeomSum
 import Algebra.Group.UniqueProds
 import Algebra.MonoidAlgebra.Basic
 import Data.Finsupp.Lex
-import Data.Zmod.Basic
+import Data.ZMod.Basic
 
 #align_import zero_divisors_in_add_monoid_algebras from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 
@@ -66,7 +66,7 @@ theorem zero_divisors_of_periodic {R A} [Nontrivial R] [Ring R] [AddMonoid A] {n
   · exact sub_ne_zero.mpr (by simpa [single_eq_single_iff])
   ·
     rw [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single,
-      sub_eq_zero, add_zero, ← succ_nsmul, Nat.sub_add_cancel (one_le_two.trans n2), na]
+      sub_eq_zero, add_zero, ← succ_nsmul', Nat.sub_add_cancel (one_le_two.trans n2), na]
 #align counterexample.zero_divisors_of_periodic Counterexample.zero_divisors_of_periodic
 
 theorem single_zero_one {R A} [Semiring R] [Zero A] :

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

@@ -3,11 +3,11 @@ Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathbin.Algebra.GeomSum
-import Mathbin.Algebra.Group.UniqueProds
-import Mathbin.Algebra.MonoidAlgebra.Basic
-import Mathbin.Data.Finsupp.Lex
-import Mathbin.Data.Zmod.Basic
+import Algebra.GeomSum
+import Algebra.Group.UniqueProds
+import Algebra.MonoidAlgebra.Basic
+import Data.Finsupp.Lex
+import Data.Zmod.Basic
 
 #align_import zero_divisors_in_add_monoid_algebras from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 
@@ -176,7 +176,7 @@ instance : Zero F :=
 instance : One F :=
   ⟨F.one⟩
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:337:4: warning: unsupported (TODO): `[tacs] -/
 /-- A tactic to prove trivial goals by enumeration. -/
 unsafe def boom : tactic Unit :=
   sorry

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
-
-! This file was ported from Lean 3 source module zero_divisors_in_add_monoid_algebras
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.GeomSum
 import Mathbin.Algebra.Group.UniqueProds
@@ -14,6 +9,8 @@ import Mathbin.Algebra.MonoidAlgebra.Basic
 import Mathbin.Data.Finsupp.Lex
 import Mathbin.Data.Zmod.Basic
 
+#align_import zero_divisors_in_add_monoid_algebras from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # Examples of zero-divisors in `add_monoid_algebra`s
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 
 ! This file was ported from Lean 3 source module zero_divisors_in_add_monoid_algebras
-! leanprover-community/mathlib commit 328375597f2c0dd00522d9c2e5a33b6a6128feeb
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.Data.Zmod.Basic
 /-!
 # Examples of zero-divisors in `add_monoid_algebra`s
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file contains an easy source of zero-divisors in an `add_monoid_algebra`.
 If `k` is a field and `G` is an additive group containing a non-zero torsion element, then
 `add_monoid_algebra k G` contains non-zero zero-divisors: this is lemma `zero_divisors_of_torsion`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -176,7 +176,7 @@ instance : Zero F :=
 instance : One F :=
   ⟨F.one⟩
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
 /-- A tactic to prove trivial goals by enumeration. -/
 unsafe def boom : tactic Unit :=
   sorry

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

Scatter the content of Data.Nat.Basic across:

• Data.Nat.Defs for the lemmas having no dependencies
• Algebra.Group.Nat for the monoid instances and the few miscellaneous lemmas needing them.
• Algebra.Ring.Nat for the semiring instance and the few miscellaneous lemmas following it.

Similarly, scatter

• Data.Int.Basic across Data.Int.Defs, Algebra.Group.Int, Algebra.Ring.Int
• Data.Nat.Order.Basic across Data.Nat.Defs, Algebra.Order.Group.Nat, Algebra.Order.Ring.Nat
• Data.Int.Order.Basic across Data.Int.Defs, Algebra.Order.Group.Int, Algebra.Order.Ring.Int

Also move a few lemmas from Data.Nat.Order.Lemmas to Data.Nat.Defs.

Before

After

@@ -134,7 +134,7 @@ def List.dropUntil {α} [DecidableEq α] : List α → List α → List α
 open Lean Elab in
 /-- `guard_decl na mod` makes sure that the declaration with name `na` is in the module `mod`.
 ```lean
-guard_decl Nat.nontrivial Mathlib.Data.Nat.Basic -- does nothing
+guard_decl Nat.nontrivial Mathlib.Algebra.Ring.Nat -- does nothing
 
 guard_decl Nat.nontrivial Not.In.Here
 -- the module Not.In.Here is not imported!

@@ -100,7 +100,7 @@ theorem zero_divisors_of_torsion {R A} [Nontrivial R] [Ring R] [AddMonoid A] (a
     · have a0 : a ≠ 0 :=
         ne_of_eq_of_ne (one_nsmul a).symm
           (nsmul_ne_zero_of_lt_addOrderOf' one_ne_zero (Nat.succ_le_iff.mp o2))
-      simp only [a0, single_eq_of_ne, Ne.def, not_false_iff]
+      simp only [a0, single_eq_of_ne, Ne, not_false_iff]
     · simpa only [single_eq_same] using zero_ne_one
   · convert Commute.geom_sum₂_mul (α := AddMonoidAlgebra R A) _ (addOrderOf a) using 3
     · rw [single_zero_one, one_pow, mul_one]
@@ -250,7 +250,7 @@ example {α} [Ring α] [Nontrivial α] : ∃ f g : AddMonoidAlgebra α F, f ≠
 
 example {α} [Zero α] :
     2 • (Finsupp.single 0 1 : α →₀ F) = Finsupp.single 0 1 ∧ (Finsupp.single 0 1 : α →₀ F) ≠ 0 :=
-  ⟨smul_single _ _ _, by simp [Ne.def, Finsupp.single_eq_zero, z01.ne]⟩
+  ⟨smul_single _ _ _, by simp [Ne, Finsupp.single_eq_zero, z01.ne]⟩
 
 end F
 

Reconstitute the file Algebra.Order.Monoid.WithZero from three files:

• Algebra.Order.Monoid.WithZero.Defs
• Algebra.Order.Monoid.WithZero.Basic
• Algebra.Order.WithZero

Avoid importing it in many files. Most uses were just to get le_zero_iff to work on Nat.

Before

After

@@ -92,7 +92,7 @@ theorem zero_divisors_of_torsion {R A} [Nontrivial R] [Ring R] [AddMonoid A] (a
     · intro b hb b0
       rw [single_pow, one_pow, single_eq_of_ne]
       exact nsmul_ne_zero_of_lt_addOrderOf' b0 (Finset.mem_range.mp hb)
-    · simp only [(zero_lt_two.trans_le o2).ne', Finset.mem_range, not_lt, le_zero_iff,
+    · simp only [(zero_lt_two.trans_le o2).ne', Finset.mem_range, not_lt, Nat.le_zero,
         false_imp_iff]
     · rw [single_pow, one_pow, zero_smul, single_eq_same]
   · apply_fun fun x : R[A] => x 0

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -64,7 +64,7 @@ theorem zero_divisors_of_periodic {R A} [Nontrivial R] [Ring R] [AddMonoid A] {n
   refine' ⟨single a 1, single ((n - 1) • a) 1 - single 0 1, by simp, _, _⟩
   · exact sub_ne_zero.mpr (by simpa [single, AddMonoidAlgebra, single_eq_single_iff])
   · rw [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single,
-      sub_eq_zero, add_zero, ← succ_nsmul, Nat.sub_add_cancel (one_le_two.trans n2), na]
+      sub_eq_zero, add_zero, ← succ_nsmul', Nat.sub_add_cancel (one_le_two.trans n2), na]
 #align counterexample.zero_divisors_of_periodic Counterexample.zero_divisors_of_periodic
 
 theorem single_zero_one {R A} [Semiring R] [Zero A] :

This PR removes the default values for nsmul and zsmul, forcing the user to populate them manually. The previous behavior can be obtained by writing nsmul := nsmulRec and zsmul := zsmulRec, which is now in the docstring for these fields.

The motivation here is to make it more obvious when module diamonds are being introduced, or at least where they might be hiding; you can now simply search for nsmulRec in the source code.

Arguably we should do the same thing for intCast, natCast, pow, and zpow too, but diamonds are less common in those fields, so I'll leave them to a subsequent PR.

Co-authored-by: Matthew Ballard <matt@mrb.email>

@@ -180,14 +180,17 @@ instance : LinearOrder F :=
 theorem z01 : (0 : F) < 1 := by decide
 #align counterexample.F.z01 Counterexample.F.z01
 
+instance : Add F where
+  add := max
+
 /-- `F` would be a `CommSemiring`, using `min` as multiplication.  Again, we do not need this. -/
 instance : AddCommMonoid F where
-  add := max
   add_assoc := by boom
   zero := 0
   zero_add := by boom
   add_zero := by boom
   add_comm := by boom
+  nsmul := nsmulRec
 
 /-- The `CovariantClass`es asserting monotonicity of addition hold for `F`. -/
 instance covariantClass_add_le : CovariantClass F F (· + ·) (· ≤ ·) :=

@@ -239,9 +239,8 @@ example : ¬CovariantClass (Lex (F →₀ F)) (Lex (F →₀ F)) (· + ·) (· 
   refine (not_lt (α := Lex (F →₀ F))).mpr (@h (Finsupp.single (0 : F) (1 : F))
     (Finsupp.single 1 1) (Finsupp.single 0 1) ?_) ⟨1, ?_⟩
   · exact Or.inr ⟨0, by simp [(by boom : ∀ j : F, j < 0 ↔ False)]⟩
-  · simp only [(by boom : ∀ j : F, j < 1 ↔ j = 0), Function.comp, ofLex_add, toDFinsupp_add,
-      toLex_add, ofLex_toLex, DFinsupp.coe_add, toDFinsupp_coe, Pi.toLex_apply, Pi.add_apply,
-      forall_eq, f010, f1, f110, add_zero, f011, f111, zero_add, and_self]
+  · simp only [(by boom : ∀ j : F, j < 1 ↔ j = 0), ofLex_add, coe_add, Pi.add_apply, forall_eq,
+      f010, f1, f110, add_zero, f011, f111, zero_add, and_self]
 
 example {α} [Ring α] [Nontrivial α] : ∃ f g : AddMonoidAlgebra α F, f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 :=
   zero_divisors_of_periodic (1 : F) le_rfl (by simp [two_smul]) z01.ne'

Introduce the notation R[A] for AddMonoidAlgebra R A. This is to align Mathlibs notation with the standard notation for group ring.

The notation is scoped in AddMonoidAlgebra and there is no analogous notation for MonoidAlgebra.

I only used the notation for single-character R and As and only in the range [a-zA-Z].

The extra lines are all in Mathlib/Algebra/MonoidAlgebra/Basic.lean. They are accounted for by extra text in the doc-module and the actual notation.

Affected files:

Counterexamples/ZeroDivisorsInAddMonoidAlgebras
Algebra/AlgebraicCard
Algebra/MonoidAlgebra/Basic
Algebra/MonoidAlgebra/Degree
Algebra/MonoidAlgebra/Division
Algebra/MonoidAlgebra/Grading
Algebra/MonoidAlgebra/NoZeroDivisors
Algebra/MonoidAlgebra/Support
Data/Polynomial/AlgebraMap
Data/Polynomial/Basic
Data/Polynomial/Eval
Data/Polynomial/Laurent
RingTheory/FiniteType

@@ -16,7 +16,7 @@ import Mathlib.Data.ZMod.Basic
 
 This file contains an easy source of zero-divisors in an `AddMonoidAlgebra`.
 If `k` is a field and `G` is an additive group containing a non-zero torsion element, then
-`AddMonoidAlgebra k G` contains non-zero zero-divisors: this is lemma `zero_divisors_of_torsion`.
+`k[G]` contains non-zero zero-divisors: this is lemma `zero_divisors_of_torsion`.
 
 There is also a version for periodic elements of an additive monoid: `zero_divisors_of_periodic`.
 
@@ -52,15 +52,15 @@ namespace Counterexample
 
 /-- This is a simple example showing that if `R` is a non-trivial ring and `A` is an additive
 monoid with an element `a` satisfying `n • a = a` and `(n - 1) • a ≠ a`, for some `2 ≤ n`,
-then `AddMonoidAlgebra R A` contains non-zero zero-divisors.  The elements are easy to write down:
-`[a]` and `[a] ^ (n - 1) - 1` are non-zero elements of `AddMonoidAlgebra R A` whose product
+then `R[A]` contains non-zero zero-divisors.  The elements are easy to write down:
+`[a]` and `[a] ^ (n - 1) - 1` are non-zero elements of `R[A]` whose product
 is zero.
 
 Observe that such an element `a` *cannot* be invertible.  In particular, this lemma never applies
 if `A` is a group. -/
 theorem zero_divisors_of_periodic {R A} [Nontrivial R] [Ring R] [AddMonoid A] {n : ℕ} (a : A)
     (n2 : 2 ≤ n) (na : n • a = a) (na1 : (n - 1) • a ≠ 0) :
-    ∃ f g : AddMonoidAlgebra R A, f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 := by
+    ∃ f g : R[A], f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 := by
   refine' ⟨single a 1, single ((n - 1) • a) 1 - single 0 1, by simp, _, _⟩
   · exact sub_ne_zero.mpr (by simpa [single, AddMonoidAlgebra, single_eq_single_iff])
   · rw [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single,
@@ -68,24 +68,24 @@ theorem zero_divisors_of_periodic {R A} [Nontrivial R] [Ring R] [AddMonoid A] {n
 #align counterexample.zero_divisors_of_periodic Counterexample.zero_divisors_of_periodic
 
 theorem single_zero_one {R A} [Semiring R] [Zero A] :
-    single (0 : A) (1 : R) = (1 : AddMonoidAlgebra R A) :=
+    single (0 : A) (1 : R) = (1 : R[A]) :=
   rfl
 #align counterexample.single_zero_one Counterexample.single_zero_one
 
 /-- This is a simple example showing that if `R` is a non-trivial ring and `A` is an additive
-monoid with a non-zero element `a` of finite order `oa`, then `AddMonoidAlgebra R A` contains
+monoid with a non-zero element `a` of finite order `oa`, then `R[A]` contains
 non-zero zero-divisors.  The elements are easy to write down:
-`∑ i in Finset.range oa, [a] ^ i` and `[a] - 1` are non-zero elements of `AddMonoidAlgebra R A`
+`∑ i in Finset.range oa, [a] ^ i` and `[a] - 1` are non-zero elements of `R[A]`
 whose product is zero.
 
 In particular, this applies whenever the additive monoid `A` is an additive group with a non-zero
 torsion element. -/
 theorem zero_divisors_of_torsion {R A} [Nontrivial R] [Ring R] [AddMonoid A] (a : A)
-    (o2 : 2 ≤ addOrderOf a) : ∃ f g : AddMonoidAlgebra R A, f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 := by
+    (o2 : 2 ≤ addOrderOf a) : ∃ f g : R[A], f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 := by
   refine'
     ⟨(Finset.range (addOrderOf a)).sum fun i : ℕ => single a 1 ^ i, single a 1 - single 0 1, _, _,
       _⟩
-  · apply_fun fun x : AddMonoidAlgebra R A => x 0
+  · apply_fun fun x : R[A] => x 0
     refine' ne_of_eq_of_ne (_ : (_ : R) = 1) one_ne_zero
     dsimp only; rw [Finset.sum_apply']
     refine (Finset.sum_eq_single 0 ?_ ?_).trans ?_
@@ -95,7 +95,7 @@ theorem zero_divisors_of_torsion {R A} [Nontrivial R] [Ring R] [AddMonoid A] (a
     · simp only [(zero_lt_two.trans_le o2).ne', Finset.mem_range, not_lt, le_zero_iff,
         false_imp_iff]
     · rw [single_pow, one_pow, zero_smul, single_eq_same]
-  · apply_fun fun x : AddMonoidAlgebra R A => x 0
+  · apply_fun fun x : R[A] => x 0
     refine' sub_ne_zero.mpr (ne_of_eq_of_ne (_ : (_ : R) = 0) _)
     · have a0 : a ≠ 0 :=
         ne_of_eq_of_ne (one_nsmul a).symm

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
-
-! This file was ported from Lean 3 source module zero_divisors_in_add_monoid_algebras
-! leanprover-community/mathlib commit 328375597f2c0dd00522d9c2e5a33b6a6128feeb
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.GeomSum
 import Mathlib.Algebra.Group.UniqueProds
@@ -14,6 +9,8 @@ import Mathlib.Algebra.MonoidAlgebra.Basic
 import Mathlib.Data.Finsupp.Lex
 import Mathlib.Data.ZMod.Basic
 
+#align_import zero_divisors_in_add_monoid_algebras from "leanprover-community/mathlib"@"328375597f2c0dd00522d9c2e5a33b6a6128feeb"
+
 /-!
 # Examples of zero-divisors in `AddMonoidAlgebra`s
 

See #4354

@@ -242,8 +242,8 @@ example : ¬CovariantClass (Lex (F →₀ F)) (Lex (F →₀ F)) (· + ·) (· 
   refine (not_lt (α := Lex (F →₀ F))).mpr (@h (Finsupp.single (0 : F) (1 : F))
     (Finsupp.single 1 1) (Finsupp.single 0 1) ?_) ⟨1, ?_⟩
   · exact Or.inr ⟨0, by simp [(by boom : ∀ j : F, j < 0 ↔ False)]⟩
-  · simp only [(by boom : ∀ j : F, j < 1 ↔ j = 0), Function.comp, ofLex_add, toDfinsupp_add,
-      toLex_add, ofLex_toLex, Dfinsupp.coe_add, toDfinsupp_coe, Pi.toLex_apply, Pi.add_apply,
+  · simp only [(by boom : ∀ j : F, j < 1 ↔ j = 0), Function.comp, ofLex_add, toDFinsupp_add,
+      toLex_add, ofLex_toLex, DFinsupp.coe_add, toDFinsupp_coe, Pi.toLex_apply, Pi.add_apply,
       forall_eq, f010, f1, f110, add_zero, f011, f111, zero_add, and_self]
 
 example {α} [Ring α] [Nontrivial α] : ∃ f g : AddMonoidAlgebra α F, f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 :=

Also fix bug in elementwise that wasn't caught while tests weren't running.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com>