From de383fe5538ca5c5e53fae36229070ba5451c970 Mon Sep 17 00:00:00 2001
From: Enrico Zoli <??>
Date: Fri, 20 Oct 2006 09:59:39 +0000
Subject: [PATCH] 1. developed up to algebras 2. some lemmas commented out
 because of a new bug in rewrite

---
 matita/dama/integration_algebras.ma | 154 +++++++++++++++++-----------
 1 file changed, 92 insertions(+), 62 deletions(-)

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index 94bc002b3..66e47ce26 100644
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -22,6 +22,10 @@ definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
 
 definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
 
+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;
@@ -32,20 +36,64 @@ record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \d
     opp_inverse: left_inverse ? plus zero opp
  }.
 
-record is_field (C:Type) (plus:C→C→C) (mult:C→C→C) (zero,one:C) (opp:C→C)
- (inv:∀x:C.x ≠ zero →C)  : Prop
+record is_ring (C:Type) (plus:C→C→C) (mult:C→C→C) (zero:C) (opp:C→C) : Prop
 ≝
- {  (* abelian group properties *)
+ { (* abelian group properties *)
     abelian_group:> is_abelian_group ? plus zero opp;
-    (* abelian multiplicative semigroup properties *)
+    (* multiplicative semigroup properties *)
     mult_assoc: associative ? mult;
+    (* ring properties *)
+    mult_plus_distr_left: distributive ? mult plus;
+    mult_plus_distr_right: distributive_right C mult plus
+ }.
+ 
+record ring : Type \def
+ { r_carrier:> Type;
+   r_plus: r_carrier → r_carrier → r_carrier;
+   r_mult: r_carrier → r_carrier → r_carrier;
+   r_zero: r_carrier;
+   r_opp: r_carrier → r_carrier;
+   r_ring_properties:> is_ring ? r_plus r_mult r_zero r_opp
+ }.
+
+notation "0" with precedence 89
+for @{ 'zero }.
+
+interpretation "Ring zero" 'zero =
+ (cic:/matita/integration_algebras/r_zero.con _).
+
+interpretation "Ring plus" 'plus a b =
+ (cic:/matita/integration_algebras/r_plus.con _ a b).
+
+interpretation "Ring mult" 'times a b =
+ (cic:/matita/integration_algebras/r_mult.con _ a b).
+
+interpretation "Ring opp" 'uminus a =
+ (cic:/matita/integration_algebras/r_opp.con _ a).
+
+lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
+ intros;
+ generalize in match (zero_neutral ? ? ? ? R 0); intro;
+ generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
+ rewrite > (mult_plus_distr_right ? ? ? ? ? R) in H1;
+ generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
+ rewrite < (plus_assoc ? ? ? ? R) in H;
+ rewrite > (opp_inverse ? ? ? ? R) in H;
+ rewrite > (zero_neutral ? ? ? ? R) in H;
+ assumption.
+qed.
+
+record is_field (C:Type) (plus:C→C→C) (mult:C→C→C) (zero,one:C) (opp:C→C)
+ (inv:∀x:C.x ≠ zero →C)  : Prop
+≝
+ {  (* ring properties *)
+    ring_properties: is_ring ? plus mult zero opp;
+    (* multiplicative abelian properties *)
     mult_comm: symmetric ? mult;
     (* multiplicative monoid properties *)
     one_neutral: left_neutral ? mult one;
     (* multiplicative group properties *)
     inv_inverse: ∀x.∀p: x ≠ zero. mult (inv x p) x = one;
-    (* ring properties *)
-    mult_plus_distr: distributive ? mult plus;
     (* integral domain *)
     not_eq_zero_one: zero ≠ one
  }.
@@ -57,40 +105,23 @@ let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n  ≝
   ].
 
 record field : Type \def
- { carrier:> Type;
-   plus: carrier → carrier → carrier;
-   mult: carrier → carrier → carrier;
-   zero: carrier;
-   one: carrier;
-   opp: carrier → carrier;
-   inv: ∀x:carrier. x ≠ zero → carrier;
-   field_properties: is_field ? plus mult zero one opp inv
+ { f_ring:> ring;
+   one: f_ring;
+   inv: ∀x:f_ring. x ≠ 0 → f_ring;
+   field_properties:
+    is_field ? (r_plus f_ring) (r_mult f_ring) (r_zero f_ring) one
+     (r_opp f_ring) inv
  }.
 
 definition sum_field ≝
- λF:field. sum ? (plus F) (zero F) (one F).
+ λF:field. sum ? (r_plus F) (r_zero F) (one F).
  
-notation "0" with precedence 89
-for @{ 'zero }.
-
-interpretation "Field zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
-
 notation "1" with precedence 89
 for @{ 'one }.
 
 interpretation "Field one" 'one =
  (cic:/matita/integration_algebras/one.con _).
 
-interpretation "Field plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
-
-interpretation "Field mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
-
-interpretation "Field opp" 'uminus a =
- (cic:/matita/integration_algebras/opp.con _ a).
- 
 record is_ordered_field_ch0 (C:Type) (plus,mult:C→C→C) (zero,one:C) (opp:C→C)
  (inv:∀x:C.x ≠ zero → C) (le:C→C→Prop) : Prop \def
  { (* field properties *)
@@ -106,8 +137,8 @@ record ordered_field_ch0 : Type \def
  { of_field:> field;
    of_le: of_field → of_field → Prop;
    of_ordered_field_properties:>
-    is_ordered_field_ch0 ? (plus of_field) (mult of_field) (zero of_field)
-     (one of_field) (opp of_field) (inv of_field) of_le
+    is_ordered_field_ch0 ? (r_plus of_field) (r_mult of_field) (r_zero of_field)
+     (one of_field) (r_opp of_field) (inv of_field) of_le
  }.
 
 interpretation "Ordered field le" 'leq a b =
@@ -118,37 +149,23 @@ definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
 interpretation "Ordered field lt" 'lt a b =
  (cic:/matita/integration_algebras/lt.con _ a b).
 
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
- intros;
+axiom le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
+(* intros;
  generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
  rewrite > (zero_neutral ? ? ? ? F) in H1;
  rewrite > (plus_comm ? ? ? ? F) in H1;
  rewrite > (opp_inverse ? ? ? ? F) in H1;
  assumption.
-qed.
+qed.*)
 
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
- intros;
+axiom le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
+(* intros;
  generalize in match (of_plus_compat ? ? ? ? ? ? ? ? F ? ? (-x) H); intro;
  rewrite > (zero_neutral ? ? ? ? F) in H1;
  rewrite > (plus_comm ? ? ? ? F) in H1;
  rewrite > (opp_inverse ? ? ? ? F) in H1;
  assumption.
-qed.
-
-(* To be proved for rings only *)
-lemma eq_mult_zero_x_zero: ∀F:ordered_field_ch0.∀x:F.0*x=0.
- intros;
- generalize in match (zero_neutral ? ? ? ? F 0); intro;
- generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
- rewrite > (mult_plus_distr ? ? ? ? ? ? ? F) in H1;
- generalize in match (eq_f ? ? (λy.-(x*0)+y) ? ? H1); intro; clear H1;
- rewrite < (plus_assoc ? ? ? ? F) in H;
- rewrite > (opp_inverse ? ? ? ? F) in H;
- rewrite > (zero_neutral ? ? ? ? F) in H;
- rewrite > (mult_comm ? ? ? ? ? ? ? F) in H;
- assumption.
-qed.
+qed.*)
 
 (*
 lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
@@ -165,15 +182,15 @@ lemma not_eq_x_zero_to_lt_zero_mult_x_x:
 axiom not_eq_sum_field_zero: ∀F,n. n > O → sum_field F n ≠ 0.
 
 record is_vector_space (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
- (mult:K→C→C) : Prop
+ (emult:K→C→C) : Prop
 ≝
  { (* abelian group properties *)
    vs_abelian_group: is_abelian_group ? plus zero opp;
    (* other properties *)
-   vs_nilpotent: ∀v. mult 0 v = zero;
-   vs_neutral: ∀v. mult 1 v = v;
-   vs_distributive: ∀a,b,v. mult (a + b) v = plus (mult a v) (mult b v);
-   vs_associative: ∀a,b,v. mult (a * b) v = mult a (mult b v)
+   vs_nilpotent: ∀v. emult 0 v = zero;
+   vs_neutral: ∀v. emult 1 v = v;
+   vs_distributive: ∀a,b,v. emult (a + b) v = plus (emult a v) (emult b v);
+   vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
  }.
 
 record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
@@ -190,23 +207,36 @@ record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
 definition le \def λC.λmeet:C→C→C.λf,g. meet f g = f.
 
 record is_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C) (zero:C)
- (opp:C→C) (mult:K→C→C) (join,meet:C→C→C) : Prop \def
+ (opp:C→C) (emult:K→C→C) (join,meet:C→C→C) : Prop \def
  { (* vector space properties *)
-   rs_vector_space: is_vector_space K C plus zero opp mult;
+   rs_vector_space: is_vector_space K C plus zero opp emult;
    (* lattice properties *)
    rs_lattice: is_lattice C join meet;
    (* other properties *)
    rs_compat_le_plus: ∀f,g,h. le ? meet f g →le ? meet (plus f h) (plus g h);
-   rs_compat_le_times: ∀a,f. 0≤a → le ? meet zero f → le ? meet zero (mult a f)  
+   rs_compat_le_times: ∀a,f. 0≤a → le ? meet zero f → le ? meet zero (emult a f)  
  }.
 
 definition absolute_value \def λC:Type.λopp.λjoin:C→C→C.λf.join f (opp f).   
 
 record is_archimedean_riesz_space (K:ordered_field_ch0) (C:Type) (plus:C→C→C)
- (zero:C) (opp:C→C) (mult:Type_OF_ordered_field_ch0 K→C→C) (join,meet:C→C→C) : Prop \def
+ (zero:C) (opp:C→C) (mult:Type_OF_ordered_field_ch0 K→C→C) (join,meet:C→C→C)
+ :Prop \def
   { ars_riesz_space: is_riesz_space ? ? plus zero opp mult join meet;
     ars_archimedean: ∃u.∀n,a.∀p:n > O.
      le C meet (absolute_value ? opp join a)
       (mult (inv K (sum_field K n) (not_eq_sum_field_zero K n p)) u) →
      a = zero
-  }.
\ No newline at end of file
+  }.
+
+record is_algebra (K: field) (C:Type) (plus:C→C→C) (zero:C) (opp:C→C)
+ (emult:K→C→C) (mult:C→C→C) : Prop
+≝
+ { (* vector space properties *)
+   a_vector_space_properties: is_vector_space ? ? plus zero opp emult;
+   (* ring properties *)
+   a_ring: is_ring ? plus mult zero opp;
+   (* algebra properties *)
+   a_associative_left: ∀a,f,g. emult a (mult f g) = mult (emult a f) g;
+   a_associative_right: ∀a,f,g. emult a (mult f g) = mult f (emult a g)
+ }.
\ No newline at end of file
-- 
2.39.2