From 0b57df2b9578e65c733df29ed6fa00c047a606e8 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Fri, 16 Nov 2007 14:12:48 +0000
Subject: [PATCH] more cleanup

---
 helm/software/matita/dama/excedence.ma      |  4 +-
 helm/software/matita/dama/groups.ma         | 47 ++++++++++++--------
 helm/software/matita/dama/ordered_groups.ma | 48 ++++++++++-----------
 3 files changed, 55 insertions(+), 44 deletions(-)

diff --git a/helm/software/matita/dama/excedence.ma b/helm/software/matita/dama/excedence.ma
index 84a033c3a..735fe2262 100644
--- a/helm/software/matita/dama/excedence.ma
+++ b/helm/software/matita/dama/excedence.ma
@@ -88,12 +88,14 @@ lemma eq_symmetric_:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_symmetric.
 
 coercion cic:/matita/excedence/eq_symmetric_.con.
 
-lemma eq_transitive: ∀E.transitive ? (eq E).
+lemma eq_transitive_: ∀E.transitive ? (eq E).
 (* bug. intros k deve fare whd quanto basta *)
 intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy); 
 [apply Exy|apply Eyz] assumption.
 qed.
 
+lemma eq_transitive:∀E:apartness.∀x,y,z:E.x ≈ y → y ≈ z → x ≈ z ≝ eq_transitive_.
+
 (* BUG: vedere se ricapita *)
 lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
 intros 5 (E x y Lxy Lyx); intro H;
diff --git a/helm/software/matita/dama/groups.ma b/helm/software/matita/dama/groups.ma
index f67625817..28f7858a0 100644
--- a/helm/software/matita/dama/groups.ma
+++ b/helm/software/matita/dama/groups.ma
@@ -36,10 +36,10 @@ 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)  
 }.
 
@@ -60,6 +60,11 @@ definition minus ≝
 interpretation "Abelian group minus" 'minus a b =
  (cic:/matita/groups/minus.con _ a b).
 
+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.
@@ -71,7 +76,7 @@ intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x));
 assumption;
 qed.  
 
-coercion cic:/matita/groups/feq_plusl.con.
+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;
@@ -85,8 +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.   
-   
+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_symmetric_ (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_symmetric_ (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));
@@ -95,25 +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 ????)); 
 |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 ??)));
+    |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]]
@@ -130,7 +143,7 @@ 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 ?? 0 ? (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;]
diff --git a/helm/software/matita/dama/ordered_groups.ma b/helm/software/matita/dama/ordered_groups.ma
index fb4b29f0d..cff205ce9 100644
--- a/helm/software/matita/dama/ordered_groups.ma
+++ b/helm/software/matita/dama/ordered_groups.ma
@@ -43,62 +43,58 @@ lemma plus_cancr_le:
 intros 5 (G x y z L);
 apply (le_rewl ??? (0+x) (zero_neutral ??));
 apply (le_rewl ??? (x+0) (plus_comm ???));
-apply (le_rewl ??? (x+(-z+z))); [apply feq_plusl;apply opp_inverse;]
-apply (le_rewl ??? (x+(z+ -z))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewl ??? (x+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
+apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
+apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
+apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
 apply (le_rewr ??? (0+y) (zero_neutral ??));
 apply (le_rewr ??? (y+0) (plus_comm ???));
-apply (le_rewr ??? (y+(-z+z))); [apply feq_plusl;apply opp_inverse;]
-apply (le_rewr ??? (y+(z+ -z))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewr ??? (y+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
-apply (fle_plusr ??? (-z));
-assumption;
+apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
+apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
+apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
+apply (fle_plusr ??? (-z) L);
 qed.
 
 lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
 intros (G f g h);
 apply (plus_cancr_le ??? (-h));
-apply (le_rewl ??? (f+h+ -h)); [apply feq_plusr;apply plus_comm;]
+apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
 apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
-apply (le_rewl ??? (f+(-h+h))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewl ??? (f+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
+apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
+apply (le_rewl ??? (f+0) (opp_inverse ??));
 apply (le_rewl ??? (0+f) (plus_comm ???));
-apply (le_rewl ??? (f) (eq_symmetric ??? (zero_neutral ??)));
-apply (le_rewr ??? (g+h+ -h)); [apply feq_plusr;apply plus_comm;]
+apply (le_rewl ??? (f) (zero_neutral ??));
+apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
 apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
-apply (le_rewr ??? (g+(-h+h))); [apply feq_plusl;apply plus_comm;]
-apply (le_rewr ??? (g+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
+apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
+apply (le_rewr ??? (g+0) (opp_inverse ??));
 apply (le_rewr ??? (0+g) (plus_comm ???));
-apply (le_rewr ??? (g) (eq_symmetric ??? (zero_neutral ??)));
-assumption;
+apply (le_rewr ??? (g) (zero_neutral ??) H);
 qed.
 
 lemma plus_cancl_le: 
   ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
 intros 5 (G x y z L);
 apply (le_rewl ??? (0+x) (zero_neutral ??));
-apply (le_rewl ??? ((-z+z)+x)); [apply feq_plusr;apply opp_inverse;]
+apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
 apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
 apply (le_rewr ??? (0+y) (zero_neutral ??));
-apply (le_rewr ??? ((-z+z)+y)); [apply feq_plusr;apply opp_inverse;]
+apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
 apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
-apply (fle_plusl ??? (-z));
-assumption;
+apply (fle_plusl ??? (-z) L);
 qed.
 
 
 lemma le_zero_x_to_le_opp_x_zero: 
   ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
 intros (G x Px); apply (plus_cancr_le ??? x);
-apply (le_rewl ??? 0 (eq_symmetric ??? (opp_inverse ??)));
-apply (le_rewr ??? x (eq_symmetric ??? (zero_neutral ??)));
-assumption;
+apply (le_rewl ??? 0 (opp_inverse ??));
+apply (le_rewr ??? x (zero_neutral ??) Px);
 qed.
 
 lemma le_x_zero_to_le_zero_opp_x: 
   ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
 intros (G x Lx0); apply (plus_cancr_le ??? x);
-apply (le_rewr ??? 0 (eq_symmetric ??? (opp_inverse ??)));
-apply (le_rewl ??? x (eq_symmetric ??? (zero_neutral ??)));
+apply (le_rewr ??? 0 (opp_inverse ??));
+apply (le_rewl ??? x (zero_neutral ??));
 assumption; 
 qed.
-- 
2.39.2