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Added definition for cyclic' to AbGroups/Cyclic.v #2050

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merged 4 commits into from
Aug 6, 2024
Merged

Added definition for cyclic' to AbGroups/Cyclic.v #2050

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55adc5b
Added definition for cyclic' to AbGroups/Cyclic.v
ndcroos Aug 6, 2024
5d59e74
Comment, newline and grp_image change
ndcroos Aug 6, 2024
a6c4ec7
Removing universe variables
ndcroos Aug 6, 2024
db5f035
Removing universe variables, now builds
ndcroos Aug 6, 2024
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12 changes: 9 additions & 3 deletions theories/Algebra/AbGroups/Cyclic.v
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Expand Up @@ -13,7 +13,7 @@ Definition Z1 := FreeGroup Unit.
Definition Z1_gen : Z1 := freegroup_in tt. (* The generator *)

(** The recursion principle of [Z1] and its computation rule. *)
Definition Z1_rec {G : Group@{u}} (g : G) : Z1 $-> G
Definition Z1_rec {G : Group} (g : G) : Z1 $-> G
:= FreeGroup_rec Unit G (unit_name g).

Definition Z1_rec_beta {G : Group} (g : G) : Z1_rec g Z1_gen = g
Expand Down Expand Up @@ -76,5 +76,11 @@ Definition Z1_to_Z `{Funext} : ab_Z1 $-> abgroup_Z
(** * Finite cyclic groups *)

(** The [n]-th cyclic group is the cokernel of [Z1_mul_nat n]. *)
Definition cyclic@{u v | u < v} `{Funext} (n : nat) : AbGroup@{u}
:= ab_cokernel@{u v} (Z1_mul_nat n).
Definition cyclic `{Funext} (n : nat) : AbGroup
:= ab_cokernel (Z1_mul_nat n).

(** ** Alternative definition of cyclic group *)

(** The [n]-th cyclic group is the cokernel of [ab_mul n]. *)
Definition cyclic' (n : nat) : AbGroup
:= ab_cokernel (ab_mul (A:=abgroup_Z) n).
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