From b0b85f3cad753caba19e785e09cc10ff8a6c00d9 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Tue, 13 Nov 2007 16:40:43 +0000
Subject: [PATCH] snapshot

---
 .../constructive_higher_order_relations.ma    |   6 +
 matita/dama/excedence.ma                      |  61 ++++----
 matita/dama/groups.ma                         | 133 ++++++++++--------
 matita/dama/ordered_groups.ma                 |  37 ++---
 matita/dama/ordered_sets.ma                   |   4 +
 5 files changed, 125 insertions(+), 116 deletions(-)

diff --git a/matita/dama/constructive_higher_order_relations.ma b/matita/dama/constructive_higher_order_relations.ma
index b66ba6843..cf58e2640 100644
--- a/matita/dama/constructive_higher_order_relations.ma
+++ b/matita/dama/constructive_higher_order_relations.ma
@@ -29,3 +29,9 @@ definition symmetric ≝
 
 definition transitive ≝
  λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
+
+definition associative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z).
+
+definition commutative ≝
+ λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).
diff --git a/matita/dama/excedence.ma b/matita/dama/excedence.ma
index a78d61184..d91c61170 100644
--- a/matita/dama/excedence.ma
+++ b/matita/dama/excedence.ma
@@ -43,51 +43,55 @@ intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2);
 cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)] 
 qed.
 
-definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+record apartness : Type ≝ {
+  ap_carr:> Type;
+  ap_apart: ap_carr → ap_carr → Type;
+  ap_coreflexive: coreflexive ? ap_apart;
+  ap_symmetric: symmetric ? ap_apart;
+  ap_cotransitive: cotransitive ? ap_apart
+}.
 
 notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ a b). 
+interpretation "axiomatic apartness" 'apart x y = 
+  (cic:/matita/excedence/ap_apart.con _ x y).
 
-lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
-intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
-intros (H1); apply (H x); cases H1; assumption;
-qed.
+definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
 
-lemma apart_symmetric: ∀E.symmetric ? (apart E).
-intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption; 
+definition apart_of_excedence: excedence → apartness.
+intros (E); apply (mk_apartness E (apart E));  
+[1: unfold; cases E; simplify; clear E; intros (x); unfold;
+    intros (H1); apply (H x); cases H1; assumption;
+|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
+    cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+    [left; left|right; left|right; right|left; right] assumption]
 qed.
 
-lemma apart_cotrans: ∀E. cotransitive ? (apart E).
-intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
-cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
-[left; left|right; left|right; right|left; right] assumption.
-qed.
+coercion cic:/matita/excedence/apart_of_excedence.con.
 
-definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
+definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
 
 notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
 interpretation "alikeness" 'napart a b =
   (cic:/matita/excedence/eq.con _ a b). 
 
 lemma eq_reflexive:∀E. reflexive ? (eq E).
-intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
-apply (cRf x); cases H; assumption;
+intros (E); unfold; intros (x); apply ap_coreflexive; 
 qed.
 
 lemma eq_symmetric:∀E.symmetric ? (eq E).
-intros (E); unfold; unfold eq; unfold Not;
-intros (x y H1 H2); apply H1; cases H2; [right|left] assumption; 
+intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
+apply ap_symmetric; assumption; 
 qed.
 
 lemma eq_transitive: ∀E.transitive ? (eq E).
-intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not; 
-intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
-cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
-[1,3:left|*:right] assumption;
+(* 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 le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
-intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
+(* BUG: vedere se ricapita *)
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
+intros 5 (E x y Lxy Lyx); intro H;
 cases H; [apply Lxy;|apply Lyx] assumption;
 qed.
 
@@ -97,10 +101,11 @@ interpretation "ordered sets less than" 'lt a b =
  (cic:/matita/excedence/lt.con _ a b).
 
 lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
-intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS); 
-apply (apart_coreflexive ? x ABS);
+intros 2 (E x); intro H; cases H (_ ABS); 
+apply (ap_coreflexive ? x ABS);
 qed.
 
+(* 
 lemma lt_transitive: ∀E.transitive ? (lt E).
 intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); 
 split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
@@ -115,3 +120,5 @@ theorem lt_to_excede: ∀E:excedence.∀a,b:E. (a < b) → (b ≰ a).
 intros (E a b Lab); cases Lab (LEab Aab);
 cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)  
 qed.
+
+*)
\ No newline at end of file
diff --git a/matita/dama/groups.ma b/matita/dama/groups.ma
index 699bd73fc..a35b43d3b 100644
--- a/matita/dama/groups.ma
+++ b/matita/dama/groups.ma
@@ -14,47 +14,36 @@
 
 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
- }.
+ 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 }.
+notation "0" with precedence 89 for @{ 'zero }.
 
 interpretation "Abelian group zero" 'zero =
  (cic:/matita/groups/zero.con _).
@@ -71,37 +60,57 @@ definition minus ≝
 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 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.
-
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
+  
+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.
 
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
+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 f_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.  
+   
+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.
-
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
+   
+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.   
+   
+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));
+            [1: apply (feq_plusr ???? (opp_inverse ??)); 
+            |2: apply (ap_rewr ???? (zero_neutral ? z)); assumption;]]]]
 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.
-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. 
+
+(*
 
 theorem eq_opp_plus_plus_opp_opp: ∀G:abelian_group.∀x,y:G. -(x+y) = -x + -y.
  intros;
@@ -137,4 +146,6 @@ theorem eq_zero_opp_zero: ∀G:abelian_group.0=-0.
    rewrite > zero_neutral;
    reflexivity
  ].
-qed.
\ No newline at end of file
+qed.
+
+*)
\ No newline at end of file
diff --git a/matita/dama/ordered_groups.ma b/matita/dama/ordered_groups.ma
index c9cab27f8..4d2e18e28 100644
--- a/matita/dama/ordered_groups.ma
+++ b/matita/dama/ordered_groups.ma
@@ -18,38 +18,19 @@ include "groups.ma".
 include "ordered_sets.ma".
 
 record pre_ordered_abelian_group : Type ≝
- { og_abelian_group:> abelian_group;
-   og_tordered_set_: tordered_set;
-   og_with: exc_carr og_tordered_set_ = og_abelian_group
+ { og_abelian_group_: abelian_group;
+   og_tordered_set:> tordered_set;
+   og_with: carr og_abelian_group_ = og_tordered_set
  }.
 
-lemma og_tordered_set: pre_ordered_abelian_group → tordered_set.
-intro G; apply mk_tordered_set;
-[1: apply mk_pordered_set;
-    [1: apply (mk_excedence G); 
-        [1: cases G; clear G; simplify; rewrite < H; clear H;
-            cases og_tordered_set_; clear og_tordered_set_; simplify;
-            cases tos_poset; simplify; cases pos_carr; simplify; assumption;
-        |2: cases G; simplify; cases H; simplify; clear H; 
-            cases og_tordered_set_; simplify; clear og_tordered_set_;
-            cases tos_poset; simplify; cases pos_carr; simplify;
-            intros; apply H;
-        |3: cases G; simplify; cases H; simplify; cases og_tordered_set_; simplify;
-            cases tos_poset; simplify; cases pos_carr; simplify; 
-            intros; apply c; assumption]
-    |2: cases G; simplify;
-        cases H; simplify; clear H; cases og_tordered_set_; simplify;
-        cases tos_poset; simplify; assumption;]
-|2: simplify; (* SLOW, senza la simplify il widget muore *)
-    cases G; simplify; 
-    generalize in match (tos_totality og_tordered_set_);
-    unfold total_order_property;
-    cases H; simplify;  cases og_tordered_set_; simplify;
-    cases tos_poset; simplify; cases pos_carr; simplify;
-    intros; apply f; assumption;]
+lemma og_abelian_group: pre_ordered_abelian_group → abelian_group.
+intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
+[apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
+unfold apartness_OF_pre_ordered_abelian_group; cases (og_with G); simplify;
+[apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
 qed.
 
-coercion cic:/matita/ordered_groups/og_tordered_set.con.
+coercion cic:/matita/ordered_groups/og_abelian_group.con.
 
 definition is_ordered_abelian_group ≝
  λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
index a7f772f34..9cd9f58a9 100644
--- a/matita/dama/ordered_sets.ma
+++ b/matita/dama/ordered_sets.ma
@@ -132,6 +132,8 @@ cases E (T f cRf cTf); simplify;
 |2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
 qed. 
 
+(*
+
 definition reverse_pordered_set: pordered_set → pordered_set.
 intros (p); apply (mk_pordered_set (reverse_excedence p));
 generalize in match (reverse_excedence p); intros (E); cases E (T f cRf cTf);
@@ -212,6 +214,8 @@ record cotransitively_ordered_set: Type :=
  }.
 *)
 
+*)
+
 definition total_order_property : ∀E:excedence. Type ≝
   λE:excedence. ∀a,b:E. a ≰ b → a < b.
 
-- 
2.39.2