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_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 =
interpretation "Abelian group minus" 'minus a b =
(cic:/matita/groups/minus.con _ a b).
-
-lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
-intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
-cases (Exy (ap_symmetric ??? a));
-qed.
-
-lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
-intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
-apply ap_symmetric; assumption;
-qed.
+
+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.
intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption;
qed.
-lemma f_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z.
+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;
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_rewl ??? (0 + y));
[1: apply (feq_plusr ???? (opp_inverse ??));
- |2: apply (ap_rewl ???? (zero_neutral ? y)); apply (ap_rewr ??? (0 + z));
- [1: apply (feq_plusr ???? (opp_inverse ??));
- |2: apply (ap_rewr ???? (zero_neutral ? z)); assumption;]]]]
+ |2: apply (ap_rewl ???? (zero_neutral ? y));
+ apply (ap_rewr ??? (0 + z) (opp_inverse ??));
+ apply (ap_rewr ???? (zero_neutral ??)); assumption;]]]
qed.
-lemma plus_canc: ∀G:abelian_group.∀x,y,z:G. x+y ≈ x+z → y ≈ z.
-intros 6 (G x y z E Ayz); apply E; apply fap_plusl; assumption;
-qed.
+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 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 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.
+
+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.
-(*
-
-theorem eq_opp_plus_plus_opp_opp: ∀G:abelian_group.∀x,y:G. -(x+y) = -x + -y.
- intros;
- apply (cancellationlaw ? (x+y));
- rewrite < plus_comm;
- rewrite > opp_inverse;
- rewrite > plus_assoc;
- rewrite > plus_comm in ⊢ (? ? ? (? ? ? (? ? ? %)));
- rewrite < plus_assoc in ⊢ (? ? ? (? ? ? %));
- rewrite > plus_comm;
- rewrite > plus_comm in ⊢ (? ? ? (? ? (? ? % ?) ?));
- rewrite > opp_inverse;
- rewrite > zero_neutral;
- rewrite > opp_inverse;
- reflexivity.
-qed.
-
-theorem eq_opp_opp_x_x: ∀G:abelian_group.∀x:G.--x=x.
- intros;
- apply (cancellationlaw ? (-x));
- rewrite > opp_inverse;
- rewrite > plus_comm;
- rewrite > opp_inverse;
- reflexivity.
-qed.
-
-theorem eq_zero_opp_zero: ∀G:abelian_group.0=-0.
- [ assumption
- | intros;
- apply (cancellationlaw ? 0);
- rewrite < plus_comm in ⊢ (? ? ? %);
- rewrite > opp_inverse;
- rewrite > zero_neutral;
- reflexivity
- ].
-qed.
-
-*)
\ No newline at end of file
+coercion cic:/matita/groups/eq_opp_plusl.con nocomposites.
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