algebra.star.free
⟷
Mathlib.Algebra.Star.Free
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2020 Eric Weiser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Weiser
-/
-import Mathbin.Algebra.Star.Basic
-import Mathbin.Algebra.FreeAlgebra
+import Algebra.Star.Basic
+import Algebra.FreeAlgebra
#align_import algebra.star.free from "leanprover-community/mathlib"@"cb3ceec8485239a61ed51d944cb9a95b68c6bafc"
mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d
@@ -26,8 +26,7 @@ namespace FreeMonoid
variable {α : Type _}
-instance : StarSemigroup (FreeMonoid α)
- where
+instance : StarMul (FreeMonoid α) where
unit := List.reverse
star_involutive := List.reverse_reverse
star_hMul := List.reverse_append
mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504
@@ -30,7 +30,7 @@ instance : StarSemigroup (FreeMonoid α)
where
unit := List.reverse
star_involutive := List.reverse_reverse
- star_mul := List.reverse_append
+ star_hMul := List.reverse_append
#print FreeMonoid.star_of /-
@[simp]
@@ -65,7 +65,7 @@ instance : StarRing (FreeAlgebra R X)
· intros; simp only [lift_ι_apply, MulOpposite.unop_op]
· intros; simp only [*, map_mul, MulOpposite.unop_mul]
· intros; simp only [*, map_add, MulOpposite.unop_add]
- star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
+ star_hMul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
#print FreeAlgebra.star_ι /-
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2020 Eric Weiser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Weiser
-
-! This file was ported from Lean 3 source module algebra.star.free
-! leanprover-community/mathlib commit cb3ceec8485239a61ed51d944cb9a95b68c6bafc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Algebra.Star.Basic
import Mathbin.Algebra.FreeAlgebra
+#align_import algebra.star.free from "leanprover-community/mathlib"@"cb3ceec8485239a61ed51d944cb9a95b68c6bafc"
+
/-!
# A *-algebra structure on the free algebra.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -35,16 +35,20 @@ instance : StarSemigroup (FreeMonoid α)
star_involutive := List.reverse_reverse
star_mul := List.reverse_append
+#print FreeMonoid.star_of /-
@[simp]
theorem star_of (x : α) : star (of x) = of x :=
rfl
#align free_monoid.star_of FreeMonoid.star_of
+-/
+#print FreeMonoid.star_one /-
/-- Note that `star_one` is already a global simp lemma, but this one works with dsimp too -/
@[simp]
theorem star_one : star (1 : FreeMonoid α) = 1 :=
rfl
#align free_monoid.star_one FreeMonoid.star_one
+-/
end FreeMonoid
@@ -67,19 +71,25 @@ instance : StarRing (FreeAlgebra R X)
star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
+#print FreeAlgebra.star_ι /-
@[simp]
theorem star_ι (x : X) : star (ι R x) = ι R x := by simp [star, Star.star]
#align free_algebra.star_ι FreeAlgebra.star_ι
+-/
+#print FreeAlgebra.star_algebraMap /-
@[simp]
theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by
simp [star, Star.star]
#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMap
+-/
+#print FreeAlgebra.starHom /-
/-- `star` as an `alg_equiv` -/
def starHom : FreeAlgebra R X ≃ₐ[R] (FreeAlgebra R X)ᵐᵒᵖ :=
{ starRingEquiv with commutes' := fun r => by simp [star_algebra_map] }
#align free_algebra.star_hom FreeAlgebra.starHom
+-/
end FreeAlgebra
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -60,10 +60,10 @@ instance : StarRing (FreeAlgebra R X)
unfold Star.star
simp only [Function.comp_apply]
refine' FreeAlgebra.induction R X _ _ _ _ x
- · intros ; simp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]
- · intros ; simp only [lift_ι_apply, MulOpposite.unop_op]
- · intros ; simp only [*, map_mul, MulOpposite.unop_mul]
- · intros ; simp only [*, map_add, MulOpposite.unop_add]
+ · intros; simp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]
+ · intros; simp only [lift_ι_apply, MulOpposite.unop_op]
+ · intros; simp only [*, map_mul, MulOpposite.unop_mul]
+ · intros; simp only [*, map_add, MulOpposite.unop_add]
star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -35,23 +35,11 @@ instance : StarSemigroup (FreeMonoid α)
star_involutive := List.reverse_reverse
star_mul := List.reverse_append
-/- warning: free_monoid.star_of -> FreeMonoid.star_of is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} (x : α), Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toHasStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toHasInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.cancelMonoid.{u1} α)))) (FreeMonoid.starSemigroup.{u1} α))) (FreeMonoid.of.{u1} α x)) (FreeMonoid.of.{u1} α x)
-but is expected to have type
- forall {α : Type.{u1}} (x : α), Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.instCancelMonoidFreeMonoid.{u1} α)))) (FreeMonoid.instStarSemigroupFreeMonoidToSemigroupToMonoidToRightCancelMonoidInstCancelMonoidFreeMonoid.{u1} α))) (FreeMonoid.of.{u1} α x)) (FreeMonoid.of.{u1} α x)
-Case conversion may be inaccurate. Consider using '#align free_monoid.star_of FreeMonoid.star_ofₓ'. -/
@[simp]
theorem star_of (x : α) : star (of x) = of x :=
rfl
#align free_monoid.star_of FreeMonoid.star_of
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/-- Note that `star_one` is already a global simp lemma, but this one works with dsimp too -/
@[simp]
theorem star_one : star (1 : FreeMonoid α) = 1 :=
@@ -79,30 +67,15 @@ instance : StarRing (FreeAlgebra R X)
star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
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@[simp]
theorem star_ι (x : X) : star (ι R x) = ι R x := by simp [star, Star.star]
#align free_algebra.star_ι FreeAlgebra.star_ι
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@[simp]
theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by
simp [star, Star.star]
#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMap
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/-- `star` as an `alg_equiv` -/
def starHom : FreeAlgebra R X ≃ₐ[R] (FreeAlgebra R X)ᵐᵒᵖ :=
{ starRingEquiv with commutes' := fun r => by simp [star_algebra_map] }
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -72,14 +72,10 @@ instance : StarRing (FreeAlgebra R X)
unfold Star.star
simp only [Function.comp_apply]
refine' FreeAlgebra.induction R X _ _ _ _ x
- · intros
- simp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]
- · intros
- simp only [lift_ι_apply, MulOpposite.unop_op]
- · intros
- simp only [*, map_mul, MulOpposite.unop_mul]
- · intros
- simp only [*, map_add, MulOpposite.unop_add]
+ · intros ; simp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]
+ · intros ; simp only [lift_ι_apply, MulOpposite.unop_op]
+ · intros ; simp only [*, map_mul, MulOpposite.unop_mul]
+ · intros ; simp only [*, map_add, MulOpposite.unop_add]
star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -94,10 +94,7 @@ theorem star_ι (x : X) : star (ι R x) = ι R x := by simp [star, Star.star]
#align free_algebra.star_ι FreeAlgebra.star_ι
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+<too large>
Case conversion may be inaccurate. Consider using '#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMapₓ'. -/
@[simp]
theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by
mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4
@@ -97,7 +97,7 @@ theorem star_ι (x : X) : star (ι R x) = ι R x := by simp [star, Star.star]
lean 3 declaration is
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u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))))) (algebraMap.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) r)
+ forall {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {X : Type.{u1}} (r : R), Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (Star.star.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (InvolutiveStar.toStar.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (StarAddMonoid.toInvolutiveStar.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (AddMonoidWithOne.toAddMonoid.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (AddCommMonoidWithOne.toAddMonoidWithOne.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (NonAssocSemiring.toAddCommMonoidWithOne.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (Semiring.toNonAssocSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))))) (StarRing.toStarAddMonoid.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (Semiring.toNonUnitalSemiring.{max u2 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) r) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) (FreeAlgebra.instStarRingFreeAlgebraToNonUnitalSemiringInstSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, max (succ u2) (succ u1)} (RingHom.{u2, max u1 u2} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) _x) (MulHomClass.toFunLike.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u1 u2} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u1 u2} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u1 u2} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u1 u2} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))))) (algebraMap.{u2, max u1 u2} R (FreeAlgebra.{u2, u1} R _inst_1 X) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) r)) (FunLike.coe.{max (succ u2) (succ (max u2 u1)), succ u2, succ (max u2 u1)} (RingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => FreeAlgebra.{u2, u1} R _inst_1 X) _x) (MulHomClass.toFunLike.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, max u2 u1} (RingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X))) R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) (RingHom.instRingHomClassRingHom.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{max u2 u1} (FreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X)))))) (algebraMap.{u2, max u2 u1} R (FreeAlgebra.{u2, u1} R _inst_1 X) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u2, u1} R _inst_1 X) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u2, u1} R _inst_1 X)) r)
Case conversion may be inaccurate. Consider using '#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMapₓ'. -/
@[simp]
theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by
mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Weiser
! This file was ported from Lean 3 source module algebra.star.free
-! leanprover-community/mathlib commit 07c3cf2d851866ff7198219ed3fedf42e901f25c
+! leanprover-community/mathlib commit cb3ceec8485239a61ed51d944cb9a95b68c6bafc
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.Algebra.FreeAlgebra
/-!
# A *-algebra structure on the free algebra.
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Reversing words gives a *-structure on the free monoid or on the free algebra on a type.
## Implementation note
mathlib commit https://github.com/leanprover-community/mathlib/commit/b19481deb571022990f1baa9cbf9172e6757a479
@@ -105,7 +105,7 @@ theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = alge
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {X : Type.{u2}}, AlgEquiv.{u1, max u1 u2, max u1 u2} R (FreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.{max u1 u2} (FreeAlgebra.{u1, u2} R _inst_1 X)) _inst_1 (FreeAlgebra.semiring.{u1, u2} R _inst_1 X) (MulOpposite.semiring.{max u1 u2} (FreeAlgebra.{u1, u2} R _inst_1 X) (FreeAlgebra.semiring.{u1, u2} R _inst_1 X)) (FreeAlgebra.algebra.{u1, u2} R _inst_1 X) (MulOpposite.algebra.{u1, max u1 u2} R (FreeAlgebra.{u1, u2} R _inst_1 X) _inst_1 (FreeAlgebra.semiring.{u1, u2} R _inst_1 X) (FreeAlgebra.algebra.{u1, u2} R _inst_1 X))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {X : Type.{u2}}, AlgEquiv.{u1, max u2 u1, max u2 u1} R (FreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.{max u2 u1} (FreeAlgebra.{u1, u2} R _inst_1 X)) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.instSemiringMulOpposite.{max u1 u2} (FreeAlgebra.{u1, u2} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X)) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.instAlgebraMulOppositeInstSemiringMulOpposite.{u1, max u1 u2} R (FreeAlgebra.{u1, u2} R _inst_1 X) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u1, u2} R _inst_1 X))
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {X : Type.{u2}}, AlgEquiv.{u1, max u2 u1, max u2 u1} R (FreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.{max u2 u1} (FreeAlgebra.{u1, u2} R _inst_1 X)) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.semiring.{max u1 u2} (FreeAlgebra.{u1, u2} R _inst_1 X) (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X)) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (MulOpposite.instAlgebraMulOppositeSemiring.{u1, max u1 u2} R (FreeAlgebra.{u1, u2} R _inst_1 X) _inst_1 (FreeAlgebra.instSemiringFreeAlgebra.{u1, u2} R _inst_1 X) (FreeAlgebra.instAlgebraFreeAlgebraInstSemiringFreeAlgebra.{u1, u2} R _inst_1 X))
Case conversion may be inaccurate. Consider using '#align free_algebra.star_hom FreeAlgebra.starHomₓ'. -/
/-- `star` as an `alg_equiv` -/
def starHom : FreeAlgebra R X ≃ₐ[R] (FreeAlgebra R X)ᵐᵒᵖ :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/b19481deb571022990f1baa9cbf9172e6757a479
@@ -32,11 +32,23 @@ instance : StarSemigroup (FreeMonoid α)
star_involutive := List.reverse_reverse
star_mul := List.reverse_append
+/- warning: free_monoid.star_of -> FreeMonoid.star_of is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}} (x : α), Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toHasStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toHasInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.cancelMonoid.{u1} α)))) (FreeMonoid.starSemigroup.{u1} α))) (FreeMonoid.of.{u1} α x)) (FreeMonoid.of.{u1} α x)
+but is expected to have type
+ forall {α : Type.{u1}} (x : α), Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.instCancelMonoidFreeMonoid.{u1} α)))) (FreeMonoid.instStarSemigroupFreeMonoidToSemigroupToMonoidToRightCancelMonoidInstCancelMonoidFreeMonoid.{u1} α))) (FreeMonoid.of.{u1} α x)) (FreeMonoid.of.{u1} α x)
+Case conversion may be inaccurate. Consider using '#align free_monoid.star_of FreeMonoid.star_ofₓ'. -/
@[simp]
theorem star_of (x : α) : star (of x) = of x :=
rfl
#align free_monoid.star_of FreeMonoid.star_of
+/- warning: free_monoid.star_one -> FreeMonoid.star_one is a dubious translation:
+lean 3 declaration is
+ forall {α : Type.{u1}}, Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toHasStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toHasInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.cancelMonoid.{u1} α)))) (FreeMonoid.starSemigroup.{u1} α))) (OfNat.ofNat.{u1} (FreeMonoid.{u1} α) 1 (OfNat.mk.{u1} (FreeMonoid.{u1} α) 1 (One.one.{u1} (FreeMonoid.{u1} α) (MulOneClass.toHasOne.{u1} (FreeMonoid.{u1} α) (Monoid.toMulOneClass.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.cancelMonoid.{u1} α))))))))) (OfNat.ofNat.{u1} (FreeMonoid.{u1} α) 1 (OfNat.mk.{u1} (FreeMonoid.{u1} α) 1 (One.one.{u1} (FreeMonoid.{u1} α) (MulOneClass.toHasOne.{u1} (FreeMonoid.{u1} α) (Monoid.toMulOneClass.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.cancelMonoid.{u1} α))))))))
+but is expected to have type
+ forall {α : Type.{u1}}, Eq.{succ u1} (FreeMonoid.{u1} α) (Star.star.{u1} (FreeMonoid.{u1} α) (InvolutiveStar.toStar.{u1} (FreeMonoid.{u1} α) (StarSemigroup.toInvolutiveStar.{u1} (FreeMonoid.{u1} α) (Monoid.toSemigroup.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toMonoid.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.instCancelMonoidFreeMonoid.{u1} α)))) (FreeMonoid.instStarSemigroupFreeMonoidToSemigroupToMonoidToRightCancelMonoidInstCancelMonoidFreeMonoid.{u1} α))) (OfNat.ofNat.{u1} (FreeMonoid.{u1} α) 1 (One.toOfNat1.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toOne.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.instCancelMonoidFreeMonoid.{u1} α)))))) (OfNat.ofNat.{u1} (FreeMonoid.{u1} α) 1 (One.toOfNat1.{u1} (FreeMonoid.{u1} α) (RightCancelMonoid.toOne.{u1} (FreeMonoid.{u1} α) (CancelMonoid.toRightCancelMonoid.{u1} (FreeMonoid.{u1} α) (FreeMonoid.instCancelMonoidFreeMonoid.{u1} α)))))
+Case conversion may be inaccurate. Consider using '#align free_monoid.star_one FreeMonoid.star_oneₓ'. -/
/-- Note that `star_one` is already a global simp lemma, but this one works with dsimp too -/
@[simp]
theorem star_one : star (1 : FreeMonoid α) = 1 :=
@@ -68,15 +80,33 @@ instance : StarRing (FreeAlgebra R X)
star_mul a b := by simp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]
star_add a b := by simp only [Function.comp_apply, map_add, MulOpposite.unop_add]
+/- warning: free_algebra.star_ι -> FreeAlgebra.star_ι is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align free_algebra.star_ι FreeAlgebra.star_ιₓ'. -/
@[simp]
theorem star_ι (x : X) : star (ι R x) = ι R x := by simp [star, Star.star]
#align free_algebra.star_ι FreeAlgebra.star_ι
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+Case conversion may be inaccurate. Consider using '#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMapₓ'. -/
@[simp]
theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by
simp [star, Star.star]
#align free_algebra.star_algebra_map FreeAlgebra.star_algebraMap
+/- warning: free_algebra.star_hom -> FreeAlgebra.starHom is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align free_algebra.star_hom FreeAlgebra.starHomₓ'. -/
/-- `star` as an `alg_equiv` -/
def starHom : FreeAlgebra R X ≃ₐ[R] (FreeAlgebra R X)ᵐᵒᵖ :=
{ starRingEquiv with commutes' := fun r => by simp [star_algebra_map] }
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -34,7 +34,7 @@ theorem star_of (x : α) : star (of x) = of x :=
#align free_monoid.star_of FreeMonoid.star_of
/-- Note that `star_one` is already a global simp lemma, but this one works with dsimp too -/
-@[simp, nolint simpNF] -- Porting note: dsimp cannot prove this
+@[simp, nolint simpNF] -- Porting note (#10675): dsimp cannot prove this
theorem star_one : star (1 : FreeMonoid α) = 1 :=
rfl
#align free_monoid.star_one FreeMonoid.star_one
Typically a * operation on a mathematical structure R
equipped with a multiplication is an involutive anti-automorphism i.e.
∀ r s : R, star (r * s) = star s * star r
Currently mathlib defines a class StarSemigroup
to be a semigroup satisfying this property. However, the requirement for the multiplication to be associative is unnecessarily restrictive. There are important classes of star-algebra which are not associative (e.g. JB*-algebras).
This PR removes the requirement for a StarSemigroup
to be a semigroup, merely requiring it to have a multiplication.
I've changed the name from StarSemigroup
to StarMul
since it's no longer a semigroup.
Previously opened as a mathlib PR https://github.com/leanprover-community/mathlib/pull/17949
Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@@ -23,7 +23,7 @@ namespace FreeMonoid
variable {α : Type*}
-instance : StarSemigroup (FreeMonoid α) where
+instance : StarMul (FreeMonoid α) where
star := List.reverse
star_involutive := List.reverse_reverse
star_mul := List.reverse_append
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -21,7 +21,7 @@ to avoid importing `Algebra.Star.Basic` into the entire hierarchy.
namespace FreeMonoid
-variable {α : Type _}
+variable {α : Type*}
instance : StarSemigroup (FreeMonoid α) where
star := List.reverse
@@ -43,7 +43,7 @@ end FreeMonoid
namespace FreeAlgebra
-variable {R : Type _} [CommSemiring R] {X : Type _}
+variable {R : Type*} [CommSemiring R] {X : Type*}
/-- The star ring formed by reversing the elements of products -/
instance : StarRing (FreeAlgebra R X) where
@@ -2,15 +2,12 @@
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-
-! This file was ported from Lean 3 source module algebra.star.free
-! leanprover-community/mathlib commit 07c3cf2d851866ff7198219ed3fedf42e901f25c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.FreeAlgebra
+#align_import algebra.star.free from "leanprover-community/mathlib"@"07c3cf2d851866ff7198219ed3fedf42e901f25c"
+
/-!
# A *-algebra structure on the free algebra.
The unported dependencies are