First attempt at using/simulating records with manifest types to encode

mathematical structures that form a DAG. So far it works quite well,
but the generation of the "coerced" projection should be automated.
Something to write a paper on.

diff --git a/matita/dama/integration_algebras.ma b/matita/dama/integration_algebras.ma
index ca43093807d3eceabebda8b75c0dfc8a068a5215..fdf082df31be4a1e3e91192c4a86aab3b3d43281 100644 (file)
--- a/matita/dama/integration_algebras.ma
+++ b/matita/dama/integration_algebras.ma
@@ -19,40 +19,66 @@ include "lattices.ma".
 
 (**************** Riesz Spaces ********************)
 
-record is_riesz_space (K:ordered_field_ch0) (V:vector_space K)
- (L:lattice V)
-: Prop
-\def
- { rs_compat_le_plus: ∀f,g,h:V. os_le ? L f g → os_le ? L (f+h) (g+h);
-   rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → os_le ? L (zero V) f → os_le ? L (zero V) (a*f)
+record pre_riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_vector_space:> vector_space K;
+   rs_lattice_: lattice;
+   rs_with: os_carrier rs_lattice_ = rs_vector_space
+ }.
+
+lemma rs_lattice: ∀K:ordered_field_ch0.pre_riesz_space K → lattice.
+ intros (K V);
+ apply mk_lattice;
+  [ apply (carrier V) 
+  | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
+    apply l_join
+  | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? (rs_with ? V));
+    apply l_meet
+  | apply 
+     (eq_rect' ? ?
+      (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
+       is_lattice a
+        (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+          (l_join (rs_lattice_ K V)) a H)
+        (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+          (l_meet (rs_lattice_ K V)) a H))
+      ? ? (rs_with ? V));
+    simplify;
+    apply l_lattice_properties
+  ].
+qed.
+
+coercion cic:/matita/integration_algebras/rs_lattice.con.
+
+record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
+ { rs_compat_le_plus: ∀f,g,h:V. f≤g → f+h≤g+h;
+   rs_compat_le_times: ∀a:K.∀f:V. zero K≤a → zero V≤f → zero V≤a*f
  }.
 
 record riesz_space (K:ordered_field_ch0) : Type \def
- { rs_vector_space:> vector_space K;
-   rs_lattice:> lattice rs_vector_space;
-   rs_riesz_space_properties: is_riesz_space ? rs_vector_space rs_lattice
+ { rs_pre_riesz_space:> pre_riesz_space K;
+   rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
  }.
 
 record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
- { positive: ∀u:V. os_le ? V 0 u → os_le ? K 0 (T u);
+ { positive: ∀u:V. os_le V 0 u → os_le K 0 (T u);
    linear1: ∀u,v:V. T (u+v) = T u + T v;
    linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
  }.
 
 record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
  { soc_incr:
-    ∀a:nat→V.∀l:V.is_increasing ? ? a → is_sup ? V a l →
-     is_increasing ? K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
+    ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l →
+     is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
  }.
 
-definition absolute_value \def λK.λS:riesz_space K.λf.join ? S f (-f).   
+definition absolute_value \def λK.λS:riesz_space K.λf.l_join S f (-f).   
 
 (**************** Normed Riesz spaces ****************************)
 
 definition is_riesz_norm ≝
  λR:real.λV:riesz_space R.λnorm:norm R V.
-  ∀f,g:V. os_le ? V (absolute_value ? V f) (absolute_value ? V g) →
-   os_le ? R (n_function R V norm f) (n_function R V norm g).
+  ∀f,g:V. os_le V (absolute_value ? V f) (absolute_value ? V g) →
+   os_le R (n_function R V norm f) (n_function R V norm g).
 
 record riesz_norm (R:real) (V:riesz_space R) : Type ≝
  { rn_norm:> norm R V;
@@ -78,7 +104,7 @@ record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
 record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
 \def
   { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
-     os_le ? S
+     os_le S
       (absolute_value ? S a)
       ((inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u) →
      a = 0
@@ -99,7 +125,7 @@ definition is_weak_unit ≝
   3. Fremlin proves u > 0 implies x /\ u > 0  > 0 for Archimedean spaces
    only. We pick this definition for now.
 *) λR:real.λV:archimedean_riesz_space R.λe:V.
-    ∀v:V. lt ? V 0 v → lt ? V 0 (meet ? V v e).
+    ∀v:V. lt V 0 v → lt V 0 (l_meet V v e).
 
 (* Here we are avoiding a construction (the quotient space to define
    f=g iff I(|f-g|)=0 *)
@@ -112,14 +138,14 @@ record integration_riesz_space (R:real) : Type \def
    irs_limit1:
     ∀f:irs_archimedean_riesz_space.
      tends_to ?
-      (λn.integral (meet ? irs_archimedean_riesz_space f
+      (λn.integral (l_meet irs_archimedean_riesz_space f
        ((sum_field R n)*irs_unit)))
        (integral f);
    irs_limit2:
     ∀f:irs_archimedean_riesz_space.
      tends_to ?
       (λn.
-        integral (meet ? irs_archimedean_riesz_space f
+        integral (l_meet irs_archimedean_riesz_space f
          ((inv ? (sum_field R (S n))
            (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
          ) * irs_unit))) 0;
@@ -250,10 +276,10 @@ record is_f_algebra (K) (S:archimedean_riesz_space K) (one: S)
 \def 
 { compat_mult_le:
    ∀f,g:S.
-    le ? S 0 f → le ? S 0 g → le ? S 0 (a_mult ? ? ? A f g);
+    os_le S 0 f → os_le S 0 g → os_le S 0 (a_mult ? ? ? A f g);
   compat_mult_meet:
    ∀f,g,h:S.
-    meet ? S f g = 0 → meet ? S (a_mult ? ? ? A h f) g = 0
+    l_meet S f g = 0 → l_meet S (a_mult ? ? ? A h f) g = 0
 }.
 
 record f_algebra (K:ordered_field_ch0) (R:archimedean_riesz_space K) (one:R) :

diff --git a/matita/dama/lattices.ma b/matita/dama/lattices.ma
index 0af365d00c662061440a2bc381b1fa646cc1c0c1..8f2aa763da3eb35aba1e8521c24f7bc06ec19f46 100644 (file)
--- a/matita/dama/lattices.ma
+++ b/matita/dama/lattices.ma
@@ -16,93 +16,62 @@ set "baseuri" "cic:/matita/lattices/".
 
 include "ordered_sets.ma".
 
-record pre_lattice (C:Type) : Type \def
- { join_: C → C → C;
-   meet_: C → C → C
+record is_lattice (C:Type) (join,meet:C→C→C) : Prop \def
+ { (* abelian semigroup properties *)
+   l_comm_j: symmetric ? join;
+   l_associative_j: associative ? join;
+   l_comm_m: symmetric ? meet;
+   l_associative_m: associative ? meet;
+   (* other properties *)
+   l_adsorb_j_m: ∀f,g:C. join f (meet f g) = f;
+   l_adsorb_m_j: ∀f,g:C. meet f (join f g) = f
  }.
 
-definition carrier_of_lattice ≝ λC:Type.λL:pre_lattice C.C.
-
-coercion cic:/matita/lattices/carrier_of_lattice.con.
-
-definition join : ∀C.∀L:pre_lattice C.L→L→L ≝ join_.
-definition meet : ∀C.∀L:pre_lattice C.L→L→L ≝ meet_.
+record lattice : Type \def
+ { l_carrier:> Type;
+   l_join: l_carrier→l_carrier→l_carrier;
+   l_meet: l_carrier→l_carrier→l_carrier;
+   l_lattice_properties:> is_lattice ? l_join l_meet
+ }.
 
 interpretation "Lattice meet" 'and a b =
- (cic:/matita/lattices/meet.con _ _ a b).
+ (cic:/matita/lattices/l_meet.con _ a b).
 
 interpretation "Lattice join" 'or a b =
- (cic:/matita/lattices/join.con _ _ a b).
-
-record is_lattice (C:Type) (L:pre_lattice C) : Prop \def
- { (* abelian semigroup properties *)
-   l_comm_j: symmetric ? (join ? L);
-   l_associative_j: associative ? (join ? L);
-   l_comm_m: symmetric ? (meet ? L);
-   l_associative_m: associative ? (meet ? L);
-   (* other properties *)
-   l_adsorb_j_m: ∀f,g:L. (f ∨ f ∧ g) = f;
-   l_adsorb_m_j: ∀f,g:L. (f ∧ (f ∨ g)) = f
- }.
-
-record lattice (C:Type) : Type \def
- { l_pre_lattice:> pre_lattice C;
-   l_lattice_properties:> is_lattice ? l_pre_lattice
- }.
+ (cic:/matita/lattices/l_join.con _ a b).
 
-definition le \def λC:Type.λL:lattice C.λf,g:L. (f ∧ g) = f.
+definition le \def λL:lattice.λf,g:L. (f ∧ g) = f.
 
-definition ordered_set_of_lattice: ∀C.lattice C → ordered_set C.
- intros;
+definition ordered_set_of_lattice: lattice → ordered_set.
+ intros (L);
  apply mk_ordered_set;
-  [ apply mk_pre_ordered_set;
-    apply (le ? l)
+  [2: apply (le L)
+  | skip
   | apply mk_is_order_relation;
      [ unfold reflexive;
        intros;
        unfold;
-       simplify;
-       unfold le;
-       change in x with (carrier_of_lattice ? l);
-       rewrite < (l_adsorb_j_m ? ? l ? x) in ⊢ (? ? (? ? ? ? %) ?);
+       rewrite < (l_adsorb_j_m ? ? ? L ? x) in ⊢ (? ? (? ? ? %) ?);
        rewrite > l_adsorb_m_j;
         [ reflexivity
-        | apply (l_lattice_properties ? l)
+        | apply (l_lattice_properties L)
         ]
      | intros;
        unfold transitive;
-       simplify;
        unfold le;
        intros;
        rewrite < H;
-       rewrite > (l_associative_m ? ? l);
+       rewrite > (l_associative_m ? ? ? L);
        rewrite > H1;
        reflexivity
      | unfold antisimmetric;
-       unfold os_le;
-       simplify;
        unfold le;
        intros;
        rewrite < H;
-       rewrite > (l_comm_m ? ? l);
+       rewrite > (l_comm_m ? ? ? L);
        assumption
      ]
   ]
 qed.
 
-coercion cic:/matita/lattices/ordered_set_of_lattice.con.
-
-(*
-interpretation "Lattice le" 'leq a b =
- (cic:/matita/integration_algebras/le.con _ _ a b).
-
-definition lt \def λC:Type.λL:lattice C.λf,g. le ? L f g ∧ f ≠ g.
-
-interpretation "Lattice lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ _ a b).
-*)
-
-(* The next coercion introduces a cycle in the coercion graph with
-   unexpected bad results
-coercion cic:/matita/integration_algebras/carrier_of_lattice.con.
-*)
+coercion cic:/matita/lattices/ordered_set_of_lattice.con.
\ No newline at end of file

diff --git a/matita/dama/ordered_fields_ch0.ma b/matita/dama/ordered_fields_ch0.ma
index 4a7b7a2662a0906dba90eb95b7f07e0e5d5e48fd..07f32fadef0d8eef0c5e3bb00101a197a5bb3910 100644 (file)
--- a/matita/dama/ordered_fields_ch0.ma
+++ b/matita/dama/ordered_fields_ch0.ma
@@ -15,71 +15,83 @@
 set "baseuri" "cic:/matita/ordered_fields_ch0/".
 
 include "fields.ma".
-include "ordered_sets.ma".
+include "ordered_sets2.ma".
 
 (*CSC: non capisco questi alias! Una volta non servivano*)
 alias id "plus" = "cic:/matita/groups/plus.con".
 alias symbol "plus" = "Abelian group plus".
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Type \def
- { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
-   of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
-   of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
-   (* 0 characteristics *)
-   (*CSC: qua c'era un ? al posto di F *)
-   of_char0: ∀n. n > O → sum F (plus F) 0 1 n ≠ 0
- }.
- 
-record ordered_field_ch0 : Type \def
+
+record pre_ordered_field_ch0: Type ≝
  { of_field:> field;
-   of_ordered_set:> ordered_set of_field;
-   of_reflexive: reflexive ? (os_le ? of_ordered_set);
-   of_antisimmetric: antisimmetric ? (os_le ? of_ordered_set);
-   of_cotransitive: cotransitive ? (os_le ? of_ordered_set);
-   (*CSC: qui c'era un ? al posto di of_field *)
-   of_ordered_field_properties:> is_ordered_field_ch0 of_field (os_le ? of_ordered_set)
+   of_cotransitively_ordered_set_: cotransitively_ordered_set;
+   of_with: os_carrier of_cotransitively_ordered_set_ = carrier of_field
  }.
 
-(*theorem ordered_set_of_ordered_field_ch0:
- ∀F:ordered_field_ch0.ordered_set F.
- intros;
- apply mk_ordered_set;
-  [ apply (mk_pre_ordered_set ? (of_le F))
-  | apply mk_is_order_relation;
-     [ apply (of_reflexive F)
-     | apply antisimmetric_to_cotransitive_to_transitive;
-       [ apply (of_antisimmetric F)
-       | apply (of_cotransitive F)
-       ]
-     | apply (of_antisimmetric F)
-     ]
-  ].
+(*CSC: the following lemma (that is also a coercion) should be automatically
+  generated *)
+lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set.
+ intro F;
+ apply mk_cotransitively_ordered_set;
+ [ apply mk_ordered_set;
+   [ apply (carrier F)
+   | apply
+      (eq_rect ? ? (λC:Type.C→C→Prop) (os_le (of_cotransitively_ordered_set_ F)) ? (of_with F))
+   | apply
+      (eq_rect' Type (os_carrier (of_cotransitively_ordered_set_ F))
+        (λa:Type.λH:os_carrier (of_cotransitively_ordered_set_ F)=a.
+          is_order_relation a (eq_rect Type (os_carrier (of_cotransitively_ordered_set_ F)) (λC:Type.C→C→Prop) (os_le (of_cotransitively_ordered_set_ F)) a H))
+        ? (Type_OF_pre_ordered_field_ch0 F) (of_with F));
+     simplify; 
+     apply (os_order_relation_properties (of_cotransitively_ordered_set_ F))
+   ]
+ | simplify;
+   apply
+    (eq_rect' ? ?
+      (λa:Type.λH:os_carrier (of_cotransitively_ordered_set_ F)=a.
+        cotransitive a
+         match H in eq
+          return
+           λa2:Type.λH1:os_carrier (of_cotransitively_ordered_set_ F)=a2.
+            a2→a2→Prop
+         with 
+          [refl_eq ⇒ os_le (of_cotransitively_ordered_set_ F)])
+      ? ? (of_with F));
+   simplify;
+   apply cos_cotransitive
+ ].
 qed.
 
-coercion cic:/matita/ordered_fields_ch0/ordered_set_of_ordered_field_ch0.con.
-*)
+coercion cic:/matita/ordered_fields_ch0/of_cotransitively_ordered_set.con.
 
-(*interpretation "Ordered field le" 'leq a b =
- (cic:/matita/ordered_fields_ch0/of_le.con _ a b).
- *)
+record is_ordered_field_ch0 (F:pre_ordered_field_ch0) : Type \def
+ { of_mult_compat: ∀a,b:F. 0≤a → 0≤b → 0≤a*b;
+   of_plus_compat: ∀a,b,c:F. a≤b → a+c≤b+c;
+   of_weak_tricotomy : ∀a,b:F. a≠b → a≤b ∨ b≤a;
+   (* 0 characteristics *)
+   of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
+ }.
+ 
+record ordered_field_ch0 : Type \def
+ { of_pre_ordered_field_ch0:> pre_ordered_field_ch0;
+   of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0
+ }.
 
-(*CSC: qua c'era uno zero*)
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F.(zero F) ≤ x → -x ≤ 0.
-intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
+lemma 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 in H1;
  rewrite > plus_comm in H1;
  rewrite > opp_inverse in H1;
- (*assumption*)apply H1.
+ assumption.
 qed.
 
-(*CSC: qua c'era uno zero*)
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → (zero F) ≤ -x.
+lemma 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;
+ generalize in match (of_plus_compat ? F ? ? (-x) H); intro;
  rewrite > zero_neutral in H1;
  rewrite > plus_comm in H1;
  rewrite > opp_inverse in H1;
- (*assumption.*) apply H1.
+ assumption.
 qed.
 
 (*
@@ -95,16 +107,16 @@ lemma not_eq_x_zero_to_lt_zero_mult_x_x:
 *)  
 
 axiom lt_zero_to_lt_inv_zero:
- ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt ? F 0 x → lt ? F 0 (inv ? x p).
+ ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt F 0 x → lt F 0 (inv ? x p).
 
 alias symbol "lt" = "natural 'less than'".
 
 (* The ordering is not necessary. *)
 axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
-axiom le_zero_sum_field: ∀F:ordered_field_ch0.∀n. O<n → lt ? F 0 (sum_field F n).
+axiom le_zero_sum_field: ∀F:ordered_field_ch0.∀n. O<n → lt F 0 (sum_field F n).
 
 axiom lt_zero_to_le_inv_zero:
- ∀F:ordered_field_ch0.∀n:nat.∀p:sum_field F n ≠ 0. os_le ? F 0 (inv ? (sum_field ? n) p).
+ ∀F:ordered_field_ch0.∀n:nat.∀p:sum_field F n ≠ 0. 0 ≤ inv ? (sum_field ? n) p.
 
 definition tends_to : ∀F:ordered_field_ch0.∀f:nat→F.∀l:F.Prop.
  apply
@@ -142,4 +154,4 @@ qed.
 definition is_complete ≝
  λF:ordered_field_ch0.
   ∀f:nat→F. is_cauchy_seq ? f →
-   ex F (λl:F. tends_to ? f l).
\ No newline at end of file
+   ex F (λl:F. tends_to ? f l).

diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
index bf76fa1de414809178683c5f6222ed1941940049..27b10aeefbac510bfc7d65a7d4ae70873a6483a0 100644 (file)
--- a/matita/dama/ordered_sets.ma
+++ b/matita/dama/ordered_sets.ma
@@ -18,35 +18,27 @@ include "higher_order_defs/relations.ma".
 include "nat/plus.ma".
 include "constructive_connectives.ma".
 
-record pre_ordered_set (C:Type) : Type ≝
- { le_:C→C→Prop }.
-
-definition carrier_of_pre_ordered_set ≝ λC:Type.λO:pre_ordered_set C.C.
-
-coercion cic:/matita/ordered_sets/carrier_of_pre_ordered_set.con.
-
-definition os_le: ∀C.∀O:pre_ordered_set C.O → O → Prop ≝ le_.
-
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ _ a b).
-
 definition cotransitive ≝
  λC:Type.λle:C→C→Prop.∀x,y,z:C. le x y → le x z ∨ le z y. 
 
 definition antisimmetric ≝
  λC:Type.λle:C→C→Prop.∀x,y:C. le x y → le y x → x=y.
 
-record is_order_relation (C) (O:pre_ordered_set C) : Type ≝
- { or_reflexive: reflexive ? (os_le ? O);
-   or_transitive: transitive ? (os_le ? O);
-   or_antisimmetric: antisimmetric ? (os_le ? O)
+record is_order_relation (C:Type) (le:C→C→Prop) : Type ≝
+ { or_reflexive: reflexive ? le;
+   or_transitive: transitive ? le;
+   or_antisimmetric: antisimmetric ? le
  }.
 
-record ordered_set (C:Type): Type ≝
- { os_pre_ordered_set:> pre_ordered_set C;
-   os_order_relation_properties:> is_order_relation ? os_pre_ordered_set
+record ordered_set: Type ≝
+ { os_carrier:> Type;
+   os_le: os_carrier → os_carrier → Prop;
+   os_order_relation_properties:> is_order_relation ? os_le
  }.
 
+interpretation "Ordered Sets le" 'leq a b =
+ (cic:/matita/ordered_sets/os_le.con _ a b).
+
 theorem antisimmetric_to_cotransitive_to_transitive:
  ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
   transitive ? le.
@@ -60,139 +52,139 @@ theorem antisimmetric_to_cotransitive_to_transitive:
   ].
 qed.
 
-definition is_increasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a n ≤ a (S n).
-definition is_decreasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a (S n) ≤ a n.
+definition is_increasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
+definition is_decreasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n.
 
-definition is_upper_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { sup_upper_bound: is_upper_bound ? O a u; 
-   sup_least_upper_bound: ∀v:O. is_upper_bound ? O a v → u≤v
+record is_sup (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+ { sup_upper_bound: is_upper_bound O a u; 
+   sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
  }.
 
-record is_inf (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { inf_lower_bound: is_lower_bound ? O a u; 
-   inf_greatest_lower_bound: ∀v:O. is_lower_bound ? O a v → v≤u
+record is_inf (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+ { inf_lower_bound: is_lower_bound O a u; 
+   inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
  }.
 
-record is_bounded_below (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_below (O:ordered_set) (a:nat→O) : Type ≝
  { ib_lower_bound: O;
-   ib_lower_bound_is_lower_bound: is_lower_bound ? ? a ib_lower_bound
+   ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
  }.
 
-record is_bounded_above (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_above (O:ordered_set) (a:nat→O) : Type ≝
  { ib_upper_bound: O;
-   ib_upper_bound_is_upper_bound: is_upper_bound ? ? a ib_upper_bound
+   ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
  }.
 
-record is_bounded (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
- { ib_bounded_below:> is_bounded_below ? ? a;
-   ib_bounded_above:> is_bounded_above ? ? a
+record is_bounded (O:ordered_set) (a:nat→O) : Type ≝
+ { ib_bounded_below:> is_bounded_below ? a;
+   ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_below_sequence (O:ordered_set) : Type ≝
  { bbs_seq:1> nat→O;
-   bbs_is_bounded_below:> is_bounded_below ? ? bbs_seq
+   bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_above_sequence (O:ordered_set) : Type ≝
  { bas_seq:1> nat→O;
-   bas_is_bounded_above:> is_bounded_above ? ? bas_seq
+   bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_sequence (O:ordered_set) : Type ≝
  { bs_seq:1> nat → O;
-   bs_is_bounded_below: is_bounded_below ? ? bs_seq;
-   bs_is_bounded_above: is_bounded_above ? ? bs_seq
+   bs_is_bounded_below: is_bounded_below ? bs_seq;
+   bs_is_bounded_above: is_bounded_above ? bs_seq
  }.
 
 definition bounded_below_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
-  mk_bounded_below_sequence ? ? b (bs_is_bounded_below ? ? b).
+ λO:ordered_set.λb:bounded_sequence O.
+  mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
 
 coercion cic:/matita/ordered_sets/bounded_below_sequence_of_bounded_sequence.con.
 
 definition bounded_above_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
-  mk_bounded_above_sequence ? ? b (bs_is_bounded_above ? ? b).
+ λO:ordered_set.λb:bounded_sequence O.
+  mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
 
 coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con.
 
 definition lower_bound ≝
- λC.λO:ordered_set C.λb:bounded_below_sequence ? O.
-  ib_lower_bound ? ? b (bbs_is_bounded_below ? ? b).
+ λO:ordered_set.λb:bounded_below_sequence O.
+  ib_lower_bound ? b (bbs_is_bounded_below ? b).
 
 lemma lower_bound_is_lower_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_below_sequence ? O.
-  is_lower_bound ? ? b (lower_bound ? ? b).
+ ∀O:ordered_set.∀b:bounded_below_sequence O.
+  is_lower_bound ? b (lower_bound ? b).
  intros;
  unfold lower_bound;
  apply ib_lower_bound_is_lower_bound.
 qed.
 
 definition upper_bound ≝
- λC.λO:ordered_set C.λb:bounded_above_sequence ? O.
-  ib_upper_bound ? ? b (bas_is_bounded_above ? ? b).
+ λO:ordered_set.λb:bounded_above_sequence O.
+  ib_upper_bound ? b (bas_is_bounded_above ? b).
 
 lemma upper_bound_is_upper_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_above_sequence ? O.
-  is_upper_bound ? ? b (upper_bound ? ? b).
+ ∀O:ordered_set.∀b:bounded_above_sequence O.
+  is_upper_bound ? b (upper_bound ? b).
  intros;
  unfold upper_bound;
  apply ib_upper_bound_is_upper_bound.
 qed.
 
-record is_dedekind_sigma_complete (C:Type) (O:ordered_set C) : Type ≝
- { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? ? a m → ex ? (λs:O.is_inf ? O a s);
+record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
+ { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s);
    dsc_inf_proof_irrelevant:
-    ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? ? a m.∀p':is_lower_bound ? ? a m'.
+    ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'.
      (match dsc_inf a m p with [ ex_intro s _ ⇒ s ]) =
      (match dsc_inf a m' p' with [ ex_intro s' _ ⇒ s' ]); 
-   dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? ? a m → ex ? (λs:O.is_sup ? O a s);
+   dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? a m → ex ? (λs:O.is_sup O a s);
    dsc_sup_proof_irrelevant:
-    ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? ? a m.∀p':is_upper_bound ? ? a m'.
+    ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? a m.∀p':is_upper_bound ? a m'.
      (match dsc_sup a m p with [ ex_intro s _ ⇒ s ]) =
      (match dsc_sup a m' p' with [ ex_intro s' _ ⇒ s' ])    
  }.
 
-record dedekind_sigma_complete_ordered_set (C:Type) : Type ≝
- { dscos_ordered_set:> ordered_set C;
+record dedekind_sigma_complete_ordered_set : Type ≝
+ { dscos_ordered_set:> ordered_set;
    dscos_dedekind_sigma_complete_properties:>
-    is_dedekind_sigma_complete ? dscos_ordered_set
+    is_dedekind_sigma_complete dscos_ordered_set
  }.
 
 definition inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_below_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_below_sequence O → O.
  intros;
  elim
-  (dsc_inf ? O (dscos_dedekind_sigma_complete_properties ? O) b);
+  (dsc_inf O (dscos_dedekind_sigma_complete_properties O) b);
   [ apply a
-  | apply (lower_bound ? ? b)
+  | apply (lower_bound ? b)
   | apply lower_bound_is_lower_bound
   ]
 qed.
 
 lemma inf_is_inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.
-   is_inf ? ? a (inf ? ? a).
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.
+   is_inf ? a (inf ? a).
  intros;
  unfold inf;
  simplify;
- elim (dsc_inf C O (dscos_dedekind_sigma_complete_properties C O) a
-(lower_bound C O a) (lower_bound_is_lower_bound C O a));
+ elim (dsc_inf O (dscos_dedekind_sigma_complete_properties O) a
+(lower_bound O a) (lower_bound_is_lower_bound O a));
  simplify;
  assumption.
 qed.
 
 lemma inf_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a,a':bounded_below_sequence ? O.
-   bbs_seq ? ? a = bbs_seq ? ? a' →
-    inf ? ? a = inf ? ? a'.
- intros 4;
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,a':bounded_below_sequence O.
+   bbs_seq ? a = bbs_seq ? a' →
+    inf ? a = inf ? a'.
+ intros 3;
  elim a 0;
  elim a';
  simplify in H;
@@ -201,43 +193,43 @@ lemma inf_proof_irrelevant:
  rewrite > H;
  intro;
  simplify;
- rewrite < (dsc_inf_proof_irrelevant C O O f (ib_lower_bound ? ? f i2)
-  (ib_lower_bound ? ? f i) (ib_lower_bound_is_lower_bound ? ? f i2)
-  (ib_lower_bound_is_lower_bound ? ? f i));
+ rewrite < (dsc_inf_proof_irrelevant O O f (ib_lower_bound ? f i)
+  (ib_lower_bound ? f i2) (ib_lower_bound_is_lower_bound ? f i)
+  (ib_lower_bound_is_lower_bound ? f i2));
  reflexivity.
 qed.
 
 definition sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_above_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_above_sequence O → O.
  intros;
  elim
-  (dsc_sup ? O (dscos_dedekind_sigma_complete_properties ? O) b);
+  (dsc_sup O (dscos_dedekind_sigma_complete_properties O) b);
   [ apply a
-  | apply (upper_bound ? ? b)
+  | apply (upper_bound ? b)
   | apply upper_bound_is_upper_bound
   ].
 qed.
 
 lemma sup_is_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_above_sequence ? O.
-   is_sup ? ? a (sup ? ? a).
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_above_sequence O.
+   is_sup ? a (sup ? a).
  intros;
  unfold sup;
  simplify;
- elim (dsc_sup C O (dscos_dedekind_sigma_complete_properties C O) a
-(upper_bound C O a) (upper_bound_is_upper_bound C O a));
+ elim (dsc_sup O (dscos_dedekind_sigma_complete_properties O) a
+(upper_bound O a) (upper_bound_is_upper_bound O a));
  simplify;
  assumption.
 qed.
 
 lemma sup_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a,a':bounded_above_sequence ? O.
-   bas_seq ? ? a = bas_seq ? ? a' →
-    sup ? ? a = sup ? ? a'.
- intros 4;
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,a':bounded_above_sequence O.
+   bas_seq ? a = bas_seq ? a' →
+    sup ? a = sup ? a'.
+ intros 3;
  elim a 0;
  elim a';
  simplify in H;
@@ -246,18 +238,18 @@ lemma sup_proof_irrelevant:
  rewrite > H;
  intro;
  simplify;
- rewrite < (dsc_sup_proof_irrelevant C O O f (ib_upper_bound ? ? f i2)
-  (ib_upper_bound ? ? f i) (ib_upper_bound_is_upper_bound ? ? f i2)
-  (ib_upper_bound_is_upper_bound ? ? f i));
+ rewrite < (dsc_sup_proof_irrelevant O O f (ib_upper_bound ? f i2)
+  (ib_upper_bound ? f i) (ib_upper_bound_is_upper_bound ? f i2)
+  (ib_upper_bound_is_upper_bound ? f i));
  reflexivity.
 qed.
 
 axiom daemon: False.
 
 theorem inf_le_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_sequence ? O. inf ? ? a ≤ sup ? ? a.
- intros (C O');
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_sequence O. inf ? a ≤ sup ? a.
+ intros (O');
  apply (or_transitive ? ? O' ? (a O));
   [ elim daemon (*apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? a))*)
   | elim daemon (*apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? a))*)
@@ -265,75 +257,75 @@ theorem inf_le_sup:
 qed.
 
 lemma inf_respects_le:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.∀m:O.
-   is_upper_bound ? ? a m → inf ? ? a ≤ m.
- intros (C O');
- apply (or_transitive ? ? O' ? (sup ? ? (mk_bounded_sequence ? ? a ? ?)));
-  [ apply (bbs_is_bounded_below ? ? a)
-  | apply (mk_is_bounded_above ? ? ? m H)
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.∀m:O.
+   is_upper_bound ? a m → inf ? a ≤ m.
+ intros (O');
+ apply (or_transitive ? ? O' ? (sup ? (mk_bounded_sequence ? a ? ?)));
+  [ apply (bbs_is_bounded_below ? a)
+  | apply (mk_is_bounded_above ? ? m H)
   | apply inf_le_sup 
   | apply
-     (sup_least_upper_bound ? ? ? ?
-      (sup_is_sup ? ? (mk_bounded_sequence C O' a a
-        (mk_is_bounded_above C O' a m H))));
+     (sup_least_upper_bound ? ? ?
+      (sup_is_sup ? (mk_bounded_sequence O' a a
+        (mk_is_bounded_above O' a m H))));
     assumption
   ].
 qed. 
 
 definition is_sequentially_monotone ≝
- λC.λO:ordered_set C.λf:O→O.
-  ∀a:nat→O.∀p:is_increasing ? ? a.
-   is_increasing ? ? (λi.f (a i)).
+ λO:ordered_set.λf:O→O.
+  ∀a:nat→O.∀p:is_increasing ? a.
+   is_increasing ? (λi.f (a i)).
 
-record is_order_continuous (C)
- (O:dedekind_sigma_complete_ordered_set C) (f:O→O) : Prop
+record is_order_continuous
+ (O:dedekind_sigma_complete_ordered_set) (f:O→O) : Prop
 ≝
- { ioc_is_sequentially_monotone: is_sequentially_monotone ? ? f;
+ { ioc_is_sequentially_monotone: is_sequentially_monotone ? f;
    ioc_is_upper_bound_f_sup:
-    ∀a:bounded_above_sequence ? O.
-     is_upper_bound ? ? (λi.f (a i)) (f (sup ? ? a));
+    ∀a:bounded_above_sequence O.
+     is_upper_bound ? (λi.f (a i)) (f (sup ? a));
    ioc_respects_sup:
-    ∀a:bounded_above_sequence ? O.
-     is_increasing ? ? a →
-      f (sup ? ? a) =
-       sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i))
-        (mk_is_bounded_above ? ? ? (f (sup ? ? a))
+    ∀a:bounded_above_sequence O.
+     is_increasing ? a →
+      f (sup ? a) =
+       sup ? (mk_bounded_above_sequence ? (λi.f (a i))
+        (mk_is_bounded_above ? ? (f (sup ? a))
          (ioc_is_upper_bound_f_sup a)));
    ioc_is_lower_bound_f_inf:
-    ∀a:bounded_below_sequence ? O.
-     is_lower_bound ? ? (λi.f (a i)) (f (inf ? ? a));
+    ∀a:bounded_below_sequence O.
+     is_lower_bound ? (λi.f (a i)) (f (inf ? a));
    ioc_respects_inf:
-    ∀a:bounded_below_sequence ? O.
-     is_decreasing ? ? a →
-      f (inf ? ? a) =
-       inf ? ? (mk_bounded_below_sequence ? ? (λi.f (a i))
-        (mk_is_bounded_below ? ? ? (f (inf ? ? a))
+    ∀a:bounded_below_sequence O.
+     is_decreasing ? a →
+      f (inf ? a) =
+       inf ? (mk_bounded_below_sequence ? (λi.f (a i))
+        (mk_is_bounded_below ? ? (f (inf ? a))
          (ioc_is_lower_bound_f_inf a)))   
  }.
 
 theorem tail_inf_increasing:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.
-   let y ≝ λi.mk_bounded_below_sequence ? ? (λj.a (i+j)) ? in
-   let x ≝ λi.inf ? ? (y i) in
-    is_increasing ? ? x.
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.
+   let y ≝ λi.mk_bounded_below_sequence ? (λj.a (i+j)) ? in
+   let x ≝ λi.inf ? (y i) in
+    is_increasing ? x.
+ [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a));
    simplify;
    intro;
-   apply (ib_lower_bound_is_lower_bound ? ? a a)
+   apply (ib_lower_bound_is_lower_bound ? a a)
  | intros;
    unfold is_increasing;
    intro;
-   unfold x in ⊢ (? ? ? ? %);
-   apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? (y (S n))));
-   change with (is_lower_bound ? ? (y (S n)) (inf ? ? (y n)));
+   unfold x in ⊢ (? ? ? %);
+   apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? (y (S n))));
+   change with (is_lower_bound ? (y (S n)) (inf ? (y n)));
    unfold is_lower_bound;
    intro;
-   generalize in match (inf_lower_bound ? ? ? ? (inf_is_inf ? ? (y n)) (S n1));
+   generalize in match (inf_lower_bound ? ? ? (inf_is_inf ? (y n)) (S n1));
    (*CSC: coercion per FunClass inserita a mano*)
-   suppose (inf ? ? (y n) ≤ bbs_seq ? ? (y n) (S n1)) (H);
-   cut (bbs_seq ? ? (y n) (S n1) = bbs_seq ? ? (y (S n)) n1);
+   suppose (inf ? (y n) ≤ bbs_seq ? (y n) (S n1)) (H);
+   cut (bbs_seq ? (y n) (S n1) = bbs_seq ? (y (S n)) n1);
     [ rewrite < Hcut;
       assumption
     | unfold y;
@@ -344,52 +336,52 @@ theorem tail_inf_increasing:
 qed.
 
 definition is_liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_below_sequence ? O → O → Prop.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_below_sequence O → O → Prop.
  intros;
  apply
-  (is_sup ? ? (λi.inf ? ? (mk_bounded_below_sequence ? ? (λj.b (i+j)) ?)) t);
- apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
+  (is_sup ? (λi.inf ? (mk_bounded_below_sequence ? (λj.b (i+j)) ?)) t);
+ apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b));
  simplify;
  intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b).
+ apply (ib_lower_bound_is_lower_bound ? b b).
 qed.  
 
 definition liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_sequence O → O.
  intros;
  apply
-  (sup ? ?
-   (mk_bounded_above_sequence ? ?
-     (λi.inf ? ?
-       (mk_bounded_below_sequence ? ?
+  (sup ?
+   (mk_bounded_above_sequence ?
+     (λi.inf ?
+       (mk_bounded_below_sequence ?
          (λj.b (i+j)) ?)) ?));
-  [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
+  [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b));
     simplify;
     intros;
-    apply (ib_lower_bound_is_lower_bound ? ? b b)
-  | apply (mk_is_bounded_above ? ? ? (ib_upper_bound ? ? b b));
+    apply (ib_lower_bound_is_lower_bound ? b b)
+  | apply (mk_is_bounded_above ? ? (ib_upper_bound ? b b));
     unfold is_upper_bound;
     intro;
     change with
-     (inf C O
-  (mk_bounded_below_sequence C O (\lambda j:nat.b (n+j))
-   (mk_is_bounded_below C O (\lambda j:nat.b (n+j)) (ib_lower_bound C O b b)
-    (\lambda j:nat.ib_lower_bound_is_lower_bound C O b b (n+j))))
-\leq ib_upper_bound C O b b);
-    apply (inf_respects_le ? O);
+     (inf O
+  (mk_bounded_below_sequence O (\lambda j:nat.b (n+j))
+   (mk_is_bounded_below O (\lambda j:nat.b (n+j)) (ib_lower_bound O b b)
+    (\lambda j:nat.ib_lower_bound_is_lower_bound O b b (n+j))))
+\leq ib_upper_bound O b b);
+    apply (inf_respects_le O);
     simplify;
     intro;
-    apply (ib_upper_bound_is_upper_bound ? ? b b)
+    apply (ib_upper_bound_is_upper_bound ? b b)
   ].
 qed.
 
-definition reverse_ordered_set: ∀C.ordered_set C → ordered_set C.
+definition reverse_ordered_set: ordered_set → ordered_set.
  intros;
  apply mk_ordered_set;
-  [ apply mk_pre_ordered_set;
-    apply (λx,y:o.y ≤ x)
+  [2:apply (λx,y:o.y ≤ x)
+  | skip
   | apply mk_is_order_relation;
      [ simplify;
        intros;
@@ -415,8 +407,8 @@ interpretation "Ordered set ge" 'geq a b =
    (cic:/matita/ordered_sets/reverse_ordered_set.con _ _)) a b).
 
 lemma is_lower_bound_reverse_is_upper_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
-  is_lower_bound ? O a l → is_upper_bound ? (reverse_ordered_set ? O) a l.
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_lower_bound O a l → is_upper_bound (reverse_ordered_set O) a l.
  intros;
  unfold;
  intro;
@@ -427,8 +419,8 @@ lemma is_lower_bound_reverse_is_upper_bound:
 qed.
 
 lemma is_upper_bound_reverse_is_lower_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
-  is_upper_bound ? O a l → is_lower_bound ? (reverse_ordered_set ? O) a l.
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_upper_bound O a l → is_lower_bound (reverse_ordered_set O) a l.
  intros;
  unfold;
  intro;
@@ -439,8 +431,8 @@ lemma is_upper_bound_reverse_is_lower_bound:
 qed.
 
 lemma reverse_is_lower_bound_is_upper_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
-  is_lower_bound ? (reverse_ordered_set ? O) a l → is_upper_bound ? O a l.
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_lower_bound (reverse_ordered_set O) a l → is_upper_bound O a l.
  intros;
  unfold in H;
  unfold reverse_ordered_set in H;
@@ -448,8 +440,8 @@ lemma reverse_is_lower_bound_is_upper_bound:
 qed.
 
 lemma reverse_is_upper_bound_is_lower_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
-  is_upper_bound ? (reverse_ordered_set ? O) a l → is_lower_bound ? O a l.
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_upper_bound (reverse_ordered_set O) a l → is_lower_bound O a l.
  intros;
  unfold in H;
  unfold reverse_ordered_set in H;
@@ -458,73 +450,73 @@ qed.
 
 
 lemma is_inf_to_reverse_is_sup:
- ∀C.∀O:ordered_set C.∀a:bounded_below_sequence ? O.∀l:O.
-  is_inf ? O a l → is_sup ? (reverse_ordered_set ? O) a l.
+ ∀O:ordered_set.∀a:bounded_below_sequence O.∀l:O.
+  is_inf O a l → is_sup (reverse_ordered_set O) a l.
  intros;
- apply (mk_is_sup C (reverse_ordered_set ? ?));
+ apply (mk_is_sup (reverse_ordered_set O));
   [ apply is_lower_bound_reverse_is_upper_bound;
     apply inf_lower_bound;
     assumption
   | intros;
-    change in v with (Type_OF_ordered_set ? O);
+    change in v with (os_carrier O);
     change with (v ≤ l);
-    apply (inf_greatest_lower_bound ? ? ? ? H);
+    apply (inf_greatest_lower_bound ? ? ? H);
     apply reverse_is_upper_bound_is_lower_bound;
     assumption
   ].
 qed.
  
 lemma is_sup_to_reverse_is_inf:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
-  is_sup ? O a l → is_inf ? (reverse_ordered_set ? O) a l.
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_sup O a l → is_inf (reverse_ordered_set O) a l.
  intros;
- apply (mk_is_inf C (reverse_ordered_set ? ?));
+ apply (mk_is_inf (reverse_ordered_set O));
   [ apply is_upper_bound_reverse_is_lower_bound;
     apply sup_upper_bound;
     assumption
   | intros;
-    change in v with (Type_OF_ordered_set ? O);
+    change in v with (os_carrier O);
     change with (l ≤ v);
-    apply (sup_least_upper_bound ? ? ? ? H);
+    apply (sup_least_upper_bound ? ? ? H);
     apply reverse_is_lower_bound_is_upper_bound;
     assumption
   ].
 qed.
 
 lemma reverse_is_sup_to_is_inf:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
-  is_sup ? (reverse_ordered_set ? O) a l → is_inf ? O a l.
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_sup (reverse_ordered_set O) a l → is_inf O a l.
  intros;
  apply mk_is_inf;
   [ apply reverse_is_upper_bound_is_lower_bound;
-    change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
+    change in l with (os_carrier (reverse_ordered_set O));
     apply sup_upper_bound;
     assumption
   | intros;
-    change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
-    change in v with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
-    change with (os_le ? (reverse_ordered_set ? O) l v);
-    apply (sup_least_upper_bound ? ? ? ? H);
-    change in v with (Type_OF_ordered_set ? O);
+    change in l with (os_carrier (reverse_ordered_set O));
+    change in v with (os_carrier (reverse_ordered_set O));
+    change with (os_le (reverse_ordered_set O) l v);
+    apply (sup_least_upper_bound ? ? ? H);
+    change in v with (os_carrier O);
     apply is_lower_bound_reverse_is_upper_bound;
     assumption
   ].
 qed.
 
 lemma reverse_is_inf_to_is_sup:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
-  is_inf ? (reverse_ordered_set ? O) a l → is_sup ? O a l.
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_inf (reverse_ordered_set O) a l → is_sup O a l.
  intros;
  apply mk_is_sup;
   [ apply reverse_is_lower_bound_is_upper_bound;
-    change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
-    apply (inf_lower_bound ? ? ? ? H)
+    change in l with (os_carrier (reverse_ordered_set O));
+    apply (inf_lower_bound ? ? ? H)
   | intros;
-    change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
-    change in v with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
-    change with (os_le ? (reverse_ordered_set ? O) v l);
-    apply (inf_greatest_lower_bound ? ? ? ? H);
-    change in v with (Type_OF_ordered_set ? O);
+    change in l with (os_carrier (reverse_ordered_set O));
+    change in v with (os_carrier (reverse_ordered_set O));
+    change with (os_le (reverse_ordered_set O) v l);
+    apply (inf_greatest_lower_bound ? ? ? H);
+    change in v with (os_carrier O);
     apply is_upper_bound_reverse_is_lower_bound;
     assumption
   ].
@@ -532,11 +524,10 @@ qed.
 
 
 definition reverse_dedekind_sigma_complete_ordered_set:
- ∀C.
-  dedekind_sigma_complete_ordered_set C → dedekind_sigma_complete_ordered_set C.
+ dedekind_sigma_complete_ordered_set → dedekind_sigma_complete_ordered_set.
  intros;
  apply mk_dedekind_sigma_complete_ordered_set;
-  [ apply (reverse_ordered_set ? d)
+  [ apply (reverse_ordered_set d)
   | elim daemon
     (*apply mk_is_dedekind_sigma_complete;
      [ intros;
@@ -568,9 +559,9 @@ definition reverse_dedekind_sigma_complete_ordered_set:
 qed.
 
 definition reverse_bounded_sequence:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_sequence ? O →
-   bounded_sequence ? (reverse_dedekind_sigma_complete_ordered_set ? O).
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_sequence O →
+   bounded_sequence (reverse_dedekind_sigma_complete_ordered_set O).
  intros;
  apply mk_bounded_sequence;
   [ apply bs_seq;
@@ -583,10 +574,10 @@ definition reverse_bounded_sequence:
 qed.
 
 definition limsup ≝
- λC:Type.λO:dedekind_sigma_complete_ordered_set C.
-  λa:bounded_sequence ? O.
-   liminf ? (reverse_dedekind_sigma_complete_ordered_set ? O)
-    (reverse_bounded_sequence ? O a).
+ λO:dedekind_sigma_complete_ordered_set.
+  λa:bounded_sequence O.
+   liminf (reverse_dedekind_sigma_complete_ordered_set O)
+    (reverse_bounded_sequence O a).
 
 notation "hvbox(〈a〉)"
  non associative with precedence 45
@@ -599,12 +590,12 @@ interpretation "mk_bounded_below_sequence" 'hide_everything_but a
 = (cic:/matita/ordered_sets/bounded_below_sequence.ind#xpointer(1/1/1) _ _ a _).
 
 theorem eq_f_sup_sup_f:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_above_sequence ? O'.
-    ∀p:is_increasing ? ? a.
-     f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) ?).
- [ apply (mk_is_bounded_above ? ? ? (f (sup ? ? a))); 
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_above_sequence O'.
+    ∀p:is_increasing ? a.
+     f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) ?).
+ [ apply (mk_is_bounded_above ? ? (f (sup ? a))); 
    apply ioc_is_upper_bound_f_sup;
    assumption
  | intros;
@@ -614,83 +605,83 @@ theorem eq_f_sup_sup_f:
 qed.
 
 theorem eq_f_sup_sup_f':
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_above_sequence ? O'.
-    ∀p:is_increasing ? ? a.
-     ∀p':is_bounded_above ? ? (λi.f (a i)).
-      f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) p').
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_above_sequence O'.
+    ∀p:is_increasing ? a.
+     ∀p':is_bounded_above ? (λi.f (a i)).
+      f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) p').
  intros;
- rewrite > (eq_f_sup_sup_f ? ? f H a H1);
+ rewrite > (eq_f_sup_sup_f ? f H a H1);
  apply sup_proof_irrelevant;
  reflexivity.
 qed.
 
 theorem eq_f_liminf_sup_f_inf:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_sequence ? O'.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_sequence O'.
    let p1 := ? in
-    f (liminf ? ? a) =
-     sup ? ?
-      (mk_bounded_above_sequence ? ?
-        (λi.f (inf ? ?
-          (mk_bounded_below_sequence ? ?
+    f (liminf ? a) =
+     sup ?
+      (mk_bounded_above_sequence ?
+        (λi.f (inf ?
+          (mk_bounded_below_sequence ?
             (λj.a (i+j))
             ?)))
         p1).
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
+ [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a));
    simplify;
    intro;
-   apply (ib_lower_bound_is_lower_bound ? ? a a)
- | apply (mk_is_bounded_above ? ? ? (f (sup ? ? a)));
+   apply (ib_lower_bound_is_lower_bound ? a a)
+ | apply (mk_is_bounded_above ? ? (f (sup ? a)));
    unfold is_upper_bound;
    intro;
    apply (or_transitive ? ? O' ? (f (a n)));
-    [ generalize in match (ioc_is_lower_bound_f_inf ? ? ? H);
+    [ generalize in match (ioc_is_lower_bound_f_inf ? ? H);
       intro H1;
       simplify in H1;
-      rewrite > (plus_n_O n) in ⊢ (? ? ? ? (? (? ? ? ? %)));
-      apply (H1 (mk_bounded_below_sequence C O' (\lambda j:nat.a (n+j))
-(mk_is_bounded_below C O' (\lambda j:nat.a (n+j)) (ib_lower_bound C O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (n+j)))) O);
+      rewrite > (plus_n_O n) in ⊢ (? ? ? (? (? ? ? %)));
+      apply (H1 (mk_bounded_below_sequence O' (\lambda j:nat.a (n+j))
+(mk_is_bounded_below O' (\lambda j:nat.a (n+j)) (ib_lower_bound O' a a)
+ (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (n+j)))) O);
     | elim daemon (*apply (ioc_is_upper_bound_f_sup ? ? ? H)*)
     ]
  | intros;
    unfold liminf;
    clearbody p1;
    generalize in match (\lambda n:nat
-.inf_respects_le C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus n j))
-  (mk_is_bounded_below C O' (\lambda j:nat.a (plus n j))
-   (ib_lower_bound C O' a a)
-   (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (plus n j))))
- (ib_upper_bound C O' a a)
- (\lambda n1:nat.ib_upper_bound_is_upper_bound C O' a a (plus n n1)));
+.inf_respects_le O'
+ (mk_bounded_below_sequence O' (\lambda j:nat.a (plus n j))
+  (mk_is_bounded_below O' (\lambda j:nat.a (plus n j))
+   (ib_lower_bound O' a a)
+   (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (plus n j))))
+ (ib_upper_bound O' a a)
+ (\lambda n1:nat.ib_upper_bound_is_upper_bound O' a a (plus n n1)));
    intro p2;
-   apply (eq_f_sup_sup_f' ? ? f H (mk_bounded_above_sequence C O'
+   apply (eq_f_sup_sup_f' ? f H (mk_bounded_above_sequence O'
 (\lambda i:nat
- .inf C O'
-  (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-   (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-    (ib_lower_bound C O' a a)
-    (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
-(mk_is_bounded_above C O'
+ .inf O'
+  (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+   (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+    (ib_lower_bound O' a a)
+    (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
+(mk_is_bounded_above O'
  (\lambda i:nat
-  .inf C O'
-   (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-    (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-     (ib_lower_bound C O' a a)
-     (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
- (ib_upper_bound C O' a a) p2)));
+  .inf O'
+   (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+    (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+     (ib_lower_bound O' a a)
+     (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
+ (ib_upper_bound O' a a) p2)));
    unfold bas_seq;
    change with
-    (is_increasing ? ? (\lambda i:nat
-.inf C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-  (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-   (ib_lower_bound C O' a a)
-   (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n))))));
+    (is_increasing ? (\lambda i:nat
+.inf O'
+ (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+  (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+   (ib_lower_bound O' a a)
+   (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))));
    apply tail_inf_increasing
  ].
 qed.
@@ -698,7 +689,7 @@ qed.
 
 
 
-definition lt ≝ λC.λO:ordered_set C.λa,b:O.a ≤ b ∧ a ≠ b.
+definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
 
 interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ _ a b).
\ No newline at end of file
+ (cic:/matita/ordered_sets/lt.con _ _ a b).

diff --git a/matita/dama/ordered_sets2.ma b/matita/dama/ordered_sets2.ma
index 26e1cdb3f30767cf273cdce5d6b423abf70311ba..a8050ec210282e424f0aef5d9e15b1d3c5753b76 100644 (file)
--- a/matita/dama/ordered_sets2.ma
+++ b/matita/dama/ordered_sets2.ma
@@ -17,11 +17,11 @@ set "baseuri" "cic:/matita/ordered_sets2".
 include "ordered_sets.ma".
 
 theorem le_f_inf_inf_f:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_below_sequence ? O'.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_below_sequence O'.
     let p := ? in
-     f (inf ? ? a) ≤ inf ? ? (mk_bounded_below_sequence ? ? (λi. f (a i)) p).
+     f (inf ? a) ≤ inf ? (mk_bounded_below_sequence ? (λi. f (a i)) p).
  [ apply mk_is_bounded_below;
     [2: apply ioc_is_lower_bound_f_inf;
         assumption
@@ -29,34 +29,34 @@ theorem le_f_inf_inf_f:
     ] 
  | intros;
    clearbody p;
-   apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? ?));
+   apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? ?));
    simplify;
    intro;
-   letin b := (λi.match i with [ O ⇒ inf ? ? a | S _ ⇒ a n]);
+   letin b := (λi.match i with [ O ⇒ inf ? a | S _ ⇒ a n]);
    change with (f (b O) ≤ f (b (S O)));
-   apply (ioc_is_sequentially_monotone ? ? ? H);
+   apply (ioc_is_sequentially_monotone ? ? H);
    simplify;
    clear b;
    intro;
    elim n1; simplify;
-    [ apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? ?));
-    | apply (or_reflexive ? O' (dscos_ordered_set ? O'))
+    [ apply (inf_lower_bound ? ? ? (inf_is_inf ? ?));
+    | apply (or_reflexive O' ? (dscos_ordered_set O'))
     ]
  ].
 qed.
 
 theorem le_to_le_sup_sup:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀a,b:bounded_above_sequence ? O'.
-   (∀i.a i ≤ b i) → sup ? ? a ≤ sup ? ? b.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀a,b:bounded_above_sequence O'.
+   (∀i.a i ≤ b i) → sup ? a ≤ sup ? b.
  intros;
- apply (sup_least_upper_bound ? ? ? ? (sup_is_sup ? ? a));
+ apply (sup_least_upper_bound ? ? ? (sup_is_sup ? a));
  unfold;
  intro;
  apply (or_transitive ? ? O');
   [2: apply H
   | skip
-  | apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? b))
+  | apply (sup_upper_bound ? ? ? (sup_is_sup ? b))
   ].
 qed. 
 
@@ -64,8 +64,8 @@ interpretation "mk_bounded_sequence" 'hide_everything_but a
 = (cic:/matita/ordered_sets/bounded_sequence.ind#xpointer(1/1/1) _ _ a _ _).
 
 lemma reduce_bas_seq:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀p.∀i.
-  bas_seq ? ? (mk_bounded_above_sequence ? ? a p) i = a i.
+ ∀O:ordered_set.∀a:nat→O.∀p.∀i.
+  bas_seq ? (mk_bounded_above_sequence ? a p) i = a i.
  intros;
  reflexivity.
 qed.
@@ -78,41 +78,41 @@ qed.
 qed.*)
 
 axiom inf_extensional:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a,b:bounded_below_sequence ? O.
-   (∀i.a i = b i) → inf ? ? a = inf ? O b.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,b:bounded_below_sequence O.
+   (∀i.a i = b i) → inf ? a = inf O b.
    
-lemma eq_to_le: ∀C.∀O:ordered_set C.∀x,y:O.x=y → x ≤ y.
+lemma eq_to_le: ∀O:ordered_set.∀x,y:O.x=y → x ≤ y.
  intros;
  rewrite > H;
  apply (or_reflexive ? ? O).
 qed.
 
 theorem fatou:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_sequence ? O'.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_sequence O'.
     let pb := ? in
     let pa := ? in
-     f (liminf ? ? a) ≤ liminf ? ? (mk_bounded_sequence ? ? (λi. f (a i)) pb pa).
- [ letin bas ≝ (bounded_above_sequence_of_bounded_sequence ? ? a);
+     f (liminf ? a) ≤ liminf ? (mk_bounded_sequence ? (λi. f (a i)) pb pa).
+ [ letin bas ≝ (bounded_above_sequence_of_bounded_sequence ? a);
    apply mk_is_bounded_above;
-    [2: apply (ioc_is_upper_bound_f_sup ? ? ? H bas)
+    [2: apply (ioc_is_upper_bound_f_sup ? ? H bas)
     | skip
     ]
- | letin bbs ≝ (bounded_below_sequence_of_bounded_sequence ? ? a);
+ | letin bbs ≝ (bounded_below_sequence_of_bounded_sequence ? a);
    apply mk_is_bounded_below;
-    [2: apply (ioc_is_lower_bound_f_inf ? ? ? H bbs)
+    [2: apply (ioc_is_lower_bound_f_inf ? ? H bbs)
     | skip
     ] 
  | intros;
-   rewrite > eq_f_liminf_sup_f_inf in ⊢ (? ? ? % ?);
+   rewrite > eq_f_liminf_sup_f_inf in ⊢ (? ? % ?);
     [ unfold liminf;
       apply le_to_le_sup_sup;
       intro;
       rewrite > reduce_bas_seq;
       rewrite > reduce_bas_seq;
-      apply (or_transitive ? O' O');
+      apply (or_transitive ? ? O');
        [2: apply le_f_inf_inf_f;
            assumption
        | skip
@@ -125,3 +125,8 @@ theorem fatou:
     ]
  ].
 qed.
+
+record cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+   cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.

diff --git a/matita/dama/reals.ma b/matita/dama/reals.ma
index 9e458e29102df18e4d84b35767c7095fd110f85e..c03fdf26bf6c4d6ef3a6eefecb57f5bdc4f41116 100644 (file)
--- a/matita/dama/reals.ma
+++ b/matita/dama/reals.ma
@@ -35,7 +35,7 @@ qed.
 
 definition max_seq: ∀R:real.∀x,y:R. nat → R.
  intros (R x y);
- elim (of_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y));
+ elim (cos_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y));
   [ apply x
   | apply not_eq_sum_field_zero ;
     unfold;
@@ -157,7 +157,7 @@ lemma abs_x_ge_O: \forall R: real. \forall x:R. (zero R) ≤ abs R x.
  unfold to_zero;
  unfold max_seq;
  elim
-     (of_cotransitive R 0
+     (cos_cotransitive R 0
 (inv R (sum_field R (S n))
  (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))) (x--x)
 (lt_zero_to_le_inv_zero R (S n)