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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 set "baseuri" "cic:/matita/algebra/semigroups".
17 include "higher_order_defs/functions.ma".
19 definition isSemiGroup ≝
20 λC:Type. λop: C → C → C.associative C op.
22 record SemiGroup : Type ≝
24 op: carrier → carrier → carrier;
25 semigroup_properties: isSemiGroup carrier op
28 coercion cic:/matita/algebra/semigroups/carrier.con.
30 notation "hvbox(a break \middot \sub S b)"
31 left associative with precedence 55
32 for @{ 'ptimes $S $a $b }.
34 notation "hvbox(a break \middot b)"
35 left associative with precedence 55
36 for @{ 'ptimesi $a $b }.
38 interpretation "Semigroup operation" 'ptimesi a b =
39 (cic:/matita/algebra/semigroups/op.con _ a b).
42 interpretation "Semigroup operation" 'ptimes S a b =
43 (cic:/matita/algebra/semigroups/op.con S a b). *)
45 definition is_left_unit ≝
46 λS:SemiGroup. λe:S. ∀x:S. e·x = x.
48 definition is_right_unit ≝
49 λS:SemiGroup. λe:S. ∀x:S. x·e = x.
51 theorem is_left_unit_to_is_right_unit_to_eq:
52 ∀S:SemiGroup. ∀e,e':S.
53 is_left_unit ? e → is_right_unit ? e' → e=e'.
56 rewrite < (H1 e) in \vdash (? ? % ?);