snapshot

[helm.git] / matita / dama / groups.ma

diff --git a/matita/dama/groups.ma b/matita/dama/groups.ma
index da24dadc56ea7d0361f05a23af7dc848969148b5..f2b4bd01cfc47198e66f7a52130c38410778b7c9 100644 (file)
--- a/matita/dama/groups.ma
+++ b/matita/dama/groups.ma
@@ -36,13 +36,13 @@ record abelian_group : Type ≝
    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 =
@@ -59,16 +59,11 @@ definition minus ≝
 
 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.
 
@@ -80,6 +75,8 @@ 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;
@@ -93,6 +90,18 @@ 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));
@@ -101,23 +110,23 @@ 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 ??)); assumption;]]]]
+        |2: apply (ap_rewl ???? (zero_neutral ? y)); 
+            apply (ap_rewr ??? (0 + z) (opp_inverse ??)); 
+            apply (ap_rewr ???? (zero_neutral ??)); 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_symmetric ??? (plus_assoc ????)); 
+[1: apply (eq_sym ??? (plus_assoc ????)); 
 |2: apply (ap_rewr ??? (z + (x + -x)));
-    [1: apply (eq_symmetric ??? (plus_assoc ????)); 
-    |2: apply (ap_rewl ??? (y + (-x+x)) (feq_plusl ???? (plus_comm ???)));
-        apply (ap_rewl ??? (y + 0) (feq_plusl ???? (opp_inverse ??)));
+    [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)) (feq_plusl ???? (plus_comm ???)));
-        apply (ap_rewr ??? (z + 0) (feq_plusl ???? (opp_inverse ??)));
+        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]]
@@ -134,27 +143,65 @@ 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_transitive ?? 0); [apply (opp_inverse ??)]
-apply (eq_transitive ?? (-x + -y + x + y)); [2: apply (eq_symmetric ??? (plus_assoc ????))]
-apply (eq_transitive ?? (-y + -x + x + y)); [2: repeat apply feq_plusr; apply plus_comm]
-apply (eq_transitive ?? (-y + (-x + x) + y)); [2: apply feq_plusr; apply plus_assoc;]
-apply (eq_transitive ?? (-y + 0 + y)); 
-  [2: apply feq_plusr; apply feq_plusl; apply eq_symmetric; apply opp_inverse]
-apply (eq_transitive ?? (-y + y)); 
-  [2: apply feq_plusr; apply eq_symmetric; 
-      apply (eq_transitive ?? (0+-y)); [apply plus_comm|apply zero_neutral]]
-apply eq_symmetric; apply opp_inverse.
+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_transitive ?? (--x + -x)); [apply plus_comm]
-apply (eq_transitive ?? 0); [apply opp_inverse]
-apply eq_symmetric; apply opp_inverse;
+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_transitive ?? 0); [apply zero_neutral;]
-apply eq_symmetric; apply opp_inverse;
+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.