X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Farithmetics%2FR.ma;h=e113fabf3c5b3572fc9e33229a14f58a8cc770e5;hb=130d99d356c82501fecb5f3aa1eb708b4d4c8b24;hp=5cbb571b6be886aaee5a7cb1b0f5033f02800987;hpb=a66dbbff290dd9b8e003e3694ec2cfbfa96e6d10;p=helm.git

diff --git a/helm/software/matita/nlibrary/arithmetics/R.ma b/helm/software/matita/nlibrary/arithmetics/R.ma
index 5cbb571b6..e113fabf3 100644
--- a/helm/software/matita/nlibrary/arithmetics/R.ma
+++ b/helm/software/matita/nlibrary/arithmetics/R.ma
@@ -21,10 +21,12 @@ naxiom Qplus: Q → Q → Q.
 naxiom Qtimes: Q → Q → Q.
 naxiom Qdivides: Q → Q → Q.
 naxiom Qle : Q → Q → Prop.
+naxiom Qlt: Q → Q → Prop.
 interpretation "Q plus" 'plus x y = (Qplus x y).
 interpretation "Q times" 'times x y = (Qtimes x y).
 interpretation "Q divides" 'divide x y = (Qdivides x y).
 interpretation "Q le" 'leq x y = (Qle x y).
+interpretation "Q lt" 'lt x y = (Qlt x y).
 naxiom Qtimes_plus: ∀n,m:nat.∀q:Q. (n * q + m * q) = (plus n m) * q.
 naxiom Qmult_one: ∀q:Q. 1 * q = q.
 naxiom Qdivides_mult: ∀q1,q2. (q1 * q2) / q1 = q2.
@@ -45,47 +47,33 @@ ntheorem lem1: ∀n:nat.∀q:Q. (n * q + q) = (S n) * q.
 #n; #q; ncut (plus n 1 = S n);##[//##]
 //; nqed.
 
-(*ndefinition aaa ≝ Qtimes_distr.
-ndefinition bbb ≝ Qmult_one.
-ndefinition ccc ≝ Qdivides_mult.*)
-
-naxiom disjoint: Q → Q → Q → Q → bool.
-
 ncoinductive locate : Q → Q → Prop ≝
    L: ∀l,l',u',u. l≤l' → u'≤((2 * l + u) / 3) → locate l' u' → locate l u
  | H: ∀l,l',u',u. ((l + 2 * u) / 3)≤l' → u'≤ u → locate l' u' → locate l u.
 
-ndefinition locate_inv_ind ≝ 
-λx1,x2:Q.λP:Q → Q → Prop.
- λH1: ∀l',u'.x1≤l' → u'≤((2 * x1 + x2) / 3) → locate l' u' → P x1 x2. 
- λH2: ∀l',u'. ((x1 + 2 * x2) / 3)≤l' → u'≤ x2 → locate l' u' → P x1 x2.
- λHterm:locate x1 x2.
-  (λHcut:x1=x1 → x2=x2 → P x1 x2. Hcut (refl Q x1) (refl Q x2))
-   match Hterm return λy1,y2.λp:locate y1 y2.
-    x1=y1 → x2=y2 →P x1 x2
-   with
-    [ L l l' u' u Hle1 Hle2 r ⇒ ?(*H1 l l' u' u ?*)
-    | H l l' u' u Hle1 Hle2 r ⇒ ?(*H2 l l' u' u ?*)].
-#a; #b; ##[ napply (H2 … r …) ##| napply (H1 … r …) ##] //.
+ndefinition locate_inv_ind':
+ ∀x1,x2:Q.∀P:Q → Q → Prop.
+  ∀H1: ∀l',u'.x1≤l' → u'≤((2 * x1 + x2) / 3) → locate l' u' → P x1 x2. 
+  ∀H2: ∀l',u'. ((x1 + 2 * x2) / 3)≤l' → u'≤ x2 → locate l' u' → P x1 x2.
+   locate x1 x2 → P x1 x2.
+ #x1; #x2; #P; #H1; #H2; #p; ninversion p; #l; #l'; #u'; #u; #Ha; #Hb; #E1;
+ #E2; #E3; ndestruct; /2/ width=5.
 nqed.
 
 ndefinition R ≝ ∃l,u:Q. locate l u.
 
-nlet corec Q_to_locate q : locate q q ≝ L q q q q ….
-  //;
- ncut (q = (2*q+q)/3)
-  [##2: #H; ncases H; //; (*NOT WORKING: nrewrite > H;*) napply Q_to_locate
-  | nrewrite < (Qdivides_mult 3 q) in ⊢ (? ? % ?);//
-  ]
+nlet corec Q_to_locate q : locate q q ≝ L q q q q … (Q_to_locate q).
+  //; nrewrite < (Qdivides_mult 3 q) in ⊢ (? % ?); //.
 nqed.
 
 ndefinition Q_to_R : Q → R.
  #q; @ q; @q; //.
 nqed.
 
+(*
 nlet corec locate_add (l1,u1:?) (r1: locate l1 u1) (l2,u2:?) (r2: locate l2 u2) :
  locate (l1 + l2) (u1 + u2) ≝ ?.
- ninversion r1; ninversion r2; #l2'; #u2'; #leq2l; #leq2u; #r2;
+ napply (locate_inv_ind' … r1); napply (locate_inv_ind' … r2); #l2'; #u2'; #leq2l; #leq2u; #r2;
  #l1'; #u1'; #leq1l; #leq1u; #r1
   [ ##1,4: ##[ @1 ? (l1'+l2') (u1'+u2') | @2 ? (l1'+l2') (u1'+u2') ]
     ##[ ##1,5: /2/ | napplyS (Qle_plus_compat …leq1u leq2u) |
@@ -103,4 +91,87 @@ nlet corec apart (l1,u1) (r1: locate l1 u1) (l2,u2) (r2: locate l2 u2) : CProp[0
  match disjoint l1 u1 l2 u2 with
   [ true ⇒ True
   | false ⇒ 
-*) 
\ No newline at end of file
+*)
+
+include "topology/igft.ma".
+include "datatypes/pairs.ma".
+include "datatypes/sums.ma".
+
+nrecord pre_order (A: Type[0]) : Type[1] ≝
+ { pre_r :2> A → A → CProp[0];
+   pre_sym: reflexive … pre_r;
+   pre_trans: transitive … pre_r
+ }.
+
+nrecord Ax_pro : Type[1] ≝
+ { AAx :> Ax;
+   Aleq: pre_order AAx
+ }.
+
+interpretation "Ax_pro leq" 'leq x y = (pre_r ? (Aleq ?) x y).
+
+(*CSC: per auto per sotto, ma non sembra aiutare *)
+nlemma And_elim1: ∀A,B. A ∧ B → A.
+ #A; #B; *; //.
+nqed.
+
+nlemma And_elim2: ∀A,B. A ∧ B → B.
+ #A; #B; *; //.
+nqed.
+(*CSC: /fine per auto per sotto *)
+
+ndefinition Rax : Ax_pro.
+ @
+  [ @ (Q × Q)
+    [ #p; napply (unit + sigma … (λc. fst … p < fst … c ∧ fst … c < snd … c ∧ snd … c < snd … p))
+    | #c; *
+      [ #_; napply {c' | fst … c < fst … c' ∧ snd … c' < snd … c}
+      | *; #c'; #_; napply {d' | fst … d' = fst … c  ∧ snd … d' = fst … c'
+                               ∨ fst … d' = snd … c' ∧ snd … d' = snd … c } ]##]
+##| @ (λc,d. fst … d ≤ fst … c ∧ snd … c ≤ snd … d)
+     [ /2/
+     | nnormalize; #z; #x; #y; *; #H1; #H2; *; /3/; (*CSC: perche' non va? *) ##]
+nqed.
+
+ndefinition downarrow: ∀S:Ax_pro. Ω \sup S → Ω \sup S ≝
+ λS:Ax_pro.λU:Ω ^S.{a | ∃b:S. b ∈ U ∧ a ≤ b}.
+
+interpretation "downarrow" 'downarrow a = (downarrow ? a).
+
+ndefinition fintersects: ∀S:Ax_pro. Ω \sup S → Ω \sup S → Ω \sup S ≝
+ λS.λU,V. ↓U ∩ ↓V.
+
+interpretation "fintersects" 'fintersects U V = (fintersects ? U V).
+
+ndefinition singleton ≝ λA.λa:A.{b | b=a}.
+
+interpretation "singleton" 'singl a = (singleton ? a).
+
+ninductive ftcover (A : Ax_pro) (U : Ω^A) : A → CProp[0] ≝
+| ftreflexivity : ∀a. a ∈ U → ftcover A U a
+| ftleqinfinity : ∀a,b. a ≤ b → ∀i. (∀x. x ∈ 𝐂 b i ↓ (singleton … a) → ftcover A U x) → ftcover A U a
+| ftleqleft     : ∀a,b. a ≤ b → ftcover A U b → ftcover A U a.
+
+interpretation "ftcovers" 'covers a U = (ftcover ? U a).
+
+ntheorem ftinfinity: ∀A: Ax_pro. ∀U: Ω^A. ∀a. ∀i. (∀x. x ∈ 𝐂 a i → x ◃ U) → a ◃ U.
+ #A; #U; #a; #i; #H;
+ napply (ftleqinfinity … a … i); //;
+ #x; *; *; #b; *; #H1; #H2; #H3; napply (ftleqleft … b); //;
+ napply H; napply H1 (*CSC: auto non va! *).
+nqed.
+
+ncoinductive ftfish (A : Ax_pro) (F : Ω^A) : A → CProp[0] ≝
+| ftfish : ∀a.
+    a ∈ F →
+    (∀b. a ≤ b → ftfish A F b) →
+    (∀b. a ≤ b → ∀i:𝐈 b. ∃x.  x ∈ 𝐂 b i ↓ (singleton … a) ∧ ftfish A F x) →
+    ftfish A F a.
+
+interpretation "fish" 'fish a U = (ftfish ? U a).
+
+alias symbol "I" (instance 6) = "I".
+ntheorem ftcoinfinity: ∀A: Ax_pro. ∀F: Ω^A. ∀a. a ⋉ F → (∀i: 𝐈 a. ∃b. b ∈ 𝐂 a i ∧ b ⋉ F).
+ #A; #F; #a; #H; ncases H; #b; #_; #_; #H2; #i; ncases (H2 … i); //;
+ #x; *; *; *; #y; *; #K2; #K3; #_; #K5; @y; @ K2; ncases K5 in K3; /2/.
+nqed.
\ No newline at end of file