set "baseuri" "cic:/matita/groups/".
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
-include "constructive_connectives.ma".
+include "excedence.ma".
-definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
-
-definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
-
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e.
+definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x.
+definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x.
+definition left_inverse ≝ λC:apartness.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x ≈ e.
+definition right_inverse ≝ λC:apartness.λop.λe:C.λ inv: C→ C. ∀x:C. op x (inv x) ≈ e.
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+(* ALLOW DEFINITION WITH SOME METAS *)
definition distributive_left ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f x (g y z) = g (f x y) (f x z).
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f x (g y z) ≈ g (f x y) (f x z).
definition distributive_right ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f (g x y) z = g (f x z) (f y z).
-
-record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
- { (* abelian additive semigroup properties *)
- plus_assoc_: associative ? plus;
- plus_comm_: symmetric ? plus;
- (* additive monoid properties *)
- zero_neutral_: left_neutral ? plus zero;
- (* additive group properties *)
- opp_inverse_: left_inverse ? plus zero opp
- }.
-
-record abelian_group : Type \def
- { carrier:> Type;
- plus: carrier → carrier → carrier;
- zero: carrier;
- opp: carrier → carrier;
- ag_abelian_group_properties: is_abelian_group ? plus zero opp
- }.
-
-notation "0" with precedence 89
-for @{ 'zero }.
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f (g x y) z ≈ g (f x z) (f y z).
+
+record abelian_group : Type ≝
+ { carr:> apartness;
+ plus: carr → carr → carr;
+ zero: carr;
+ opp: carr → carr;
+ plus_assoc_: associative ? plus (eq carr);
+ plus_comm_: commutative ? plus (eq carr);
+ zero_neutral_: left_neutral ? plus zero;
+ opp_inverse_: left_inverse ? plus zero opp;
+ plus_strong_ext: ∀z.strong_ext ? (plus z)
+}.
+
+notation "0" with precedence 89 for @{ 'zero }.
interpretation "Abelian group zero" 'zero =
(cic:/matita/groups/zero.con _).
interpretation "Abelian group minus" 'minus a b =
(cic:/matita/groups/minus.con _ a b).
-
-theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
- intro;
- apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
+
+lemma plus_assoc: ∀G:abelian_group.∀x,y,z:G.x+(y+z)≈x+y+z ≝ plus_assoc_.
+lemma plus_comm: ∀G:abelian_group.∀x,y:G.x+y≈y+x ≝ plus_comm_.
+lemma zero_neutral: ∀G:abelian_group.∀x:G.0+x≈x ≝ zero_neutral_.
+lemma opp_inverse: ∀G:abelian_group.∀x:G.-x+x≈0 ≝ opp_inverse_.
+
+definition ext ≝ λA:apartness.λop:A→A. ∀x,y. x ≈ y → op x ≈ op y.
+
+lemma strong_ext_to_ext: ∀A:apartness.∀op:A→A. strong_ext ? op → ext ? op.
+intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption;
+qed.
+
+lemma feq_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x));
+assumption;
+qed.
+
+coercion cic:/matita/groups/feq_plusl.con nocomposites.
+
+lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z).
+intros 5 (G z x y A); simplify in A;
+lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2;
+lapply (ap_rewl ???? E1 A) as A1; lapply (ap_rewr ???? E2 A1) as A2;
+apply (plus_strong_ext ???? A2);
+qed.
+
+lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → y+x ≈ z+x.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x));
+assumption;
+qed.
+
+coercion cic:/matita/groups/feq_plusr.con nocomposites.
+
+(* generation of coercions to make *_rew[lr] easier *)
+lemma feq_plusr_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → y+x ≈ z+x.
+compose feq_plusr with eq_sym (H); apply H; assumption;
+qed.
+coercion cic:/matita/groups/feq_plusr_sym_.con nocomposites.
+lemma feq_plusl_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → x+y ≈ x+z.
+compose feq_plusl with eq_sym (H); apply H; assumption;
+qed.
+coercion cic:/matita/groups/feq_plusl_sym_.con nocomposites.
+
+lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z → x+y # x+z.
+intros (G x y z Ayz); apply (plus_strong_ext ? (-x));
+apply (ap_rewl ??? ((-x + x) + y));
+[1: apply plus_assoc;
+|2: apply (ap_rewr ??? ((-x +x) +z));
+ [1: apply plus_assoc;
+ |2: apply (ap_rewl ??? (0 + y));
+ [1: apply (feq_plusr ???? (opp_inverse ??));
+ |2: apply (ap_rewl ???? (zero_neutral ? y));
+ apply (ap_rewr ??? (0 + z) (opp_inverse ??));
+ apply (ap_rewr ???? (zero_neutral ??)); assumption;]]]
qed.
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
+lemma fap_plusr: ∀G:abelian_group.∀x,y,z:G. y # z → y+x # z+x.
+intros (G x y z Ayz); apply (plus_strong_extr ? (-x));
+apply (ap_rewl ??? (y + (x + -x)));
+[1: apply (eq_sym ??? (plus_assoc ????));
+|2: apply (ap_rewr ??? (z + (x + -x)));
+ [1: apply (eq_sym ??? (plus_assoc ????));
+ |2: apply (ap_rewl ??? (y + (-x+x)) (plus_comm ? x (-x)));
+ apply (ap_rewl ??? (y + 0) (opp_inverse ??));
+ apply (ap_rewl ??? (0 + y) (plus_comm ???));
+ apply (ap_rewl ??? y (zero_neutral ??));
+ apply (ap_rewr ??? (z + (-x+x)) (plus_comm ? x (-x)));
+ apply (ap_rewr ??? (z + 0) (opp_inverse ??));
+ apply (ap_rewr ??? (0 + z) (plus_comm ???));
+ apply (ap_rewr ??? z (zero_neutral ??));
+ assumption]]
+qed.
+
+lemma plus_cancl: ∀G:abelian_group.∀y,z,x:G. x+y ≈ x+z → y ≈ z.
+intros 6 (G y z x E Ayz); apply E; apply fap_plusl; assumption;
+qed.
+
+lemma plus_cancr: ∀G:abelian_group.∀y,z,x:G. y+x ≈ z+x → y ≈ z.
+intros 6 (G y z x E Ayz); apply E; apply fap_plusr; assumption;
qed.
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
+theorem eq_opp_plus_plus_opp_opp:
+ ∀G:abelian_group.∀x,y:G. -(x+y) ≈ -x + -y.
+intros (G x y); apply (plus_cancr ??? (x+y));
+apply (eq_trans ?? 0 ? (opp_inverse ??));
+apply (eq_trans ?? (-x + -y + x + y)); [2: apply (eq_sym ??? (plus_assoc ????))]
+apply (eq_trans ?? (-y + -x + x + y)); [2: repeat apply feq_plusr; apply plus_comm]
+apply (eq_trans ?? (-y + (-x + x) + y)); [2: apply feq_plusr; apply plus_assoc;]
+apply (eq_trans ?? (-y + 0 + y));
+ [2: apply feq_plusr; apply feq_plusl; apply eq_sym; apply opp_inverse]
+apply (eq_trans ?? (-y + y));
+ [2: apply feq_plusr; apply eq_sym;
+ apply (eq_trans ?? (0+-y)); [apply plus_comm|apply zero_neutral]]
+apply eq_sym; apply opp_inverse.
qed.
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
+theorem eq_opp_opp_x_x: ∀G:abelian_group.∀x:G.--x ≈ x.
+intros (G x); apply (plus_cancl ??? (-x));
+apply (eq_trans ?? (--x + -x)); [apply plus_comm]
+apply (eq_trans ?? 0); [apply opp_inverse]
+apply eq_sym; apply opp_inverse;
qed.
-lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z.
-intros;
-generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
-intros; clear H;
-rewrite < plus_assoc in H1;
-rewrite < plus_assoc in H1;
-rewrite > opp_inverse in H1;
-rewrite > zero_neutral in H1;
-rewrite > zero_neutral in H1;
-assumption.
+theorem eq_zero_opp_zero: ∀G:abelian_group.0 ≈ -0. [assumption]
+intro G; apply (plus_cancr ??? 0);
+apply (eq_trans ?? 0); [apply zero_neutral;]
+apply eq_sym; apply opp_inverse;
+qed.
+
+lemma feq_oppr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x ≈ -y → x ≈ -z.
+intros (G x y z H1 H2); apply (plus_cancr ??? z);
+apply (eq_trans ?? 0 ?? (opp_inverse ?z));
+apply (eq_trans ?? (-y + z) ? H2);
+apply (eq_trans ?? (-y + y) ? H1);
+apply (eq_trans ?? 0 ? (opp_inverse ??));
+apply eq_reflexive;
+qed.
+
+lemma feq_oppl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → -y ≈ x → -z ≈ x.
+intros (G x y z H1 H2); apply eq_sym; apply (feq_oppr ??y);
+[2:apply eq_sym] assumption;
+qed.
+
+lemma feq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y.
+intros (G x y H); apply (feq_oppl ??y ? H); apply eq_reflexive;
+qed.
+
+coercion cic:/matita/groups/feq_opp.con nocomposites.
+
+lemma eq_opp_sym: ∀G:abelian_group.∀x,y:G. y ≈ x → -x ≈ -y.
+compose feq_opp with eq_sym (H); apply H; assumption;
+qed.
+
+coercion cic:/matita/groups/eq_opp_sym.con nocomposites.
+
+lemma eq_opp_plusr: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(x + z) ≈ -(y + z).
+compose feq_plusr with feq_opp(H); apply H; assumption;
+qed.
+
+coercion cic:/matita/groups/eq_opp_plusr.con nocomposites.
+
+lemma eq_opp_plusl: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(z + x) ≈ -(z + y).
+compose feq_plusl with feq_opp(H); apply H; assumption;
qed.
+coercion cic:/matita/groups/eq_opp_plusl.con nocomposites.
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