X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Farithmetics%2FR.ma;h=60de71dbd7a2eb6d5fd69a6851603f2bb72dc060;hb=1092af803d3d1a9788008d8abf6c7470d68f22c7;hp=046bcbf88d8ec37c0fd6bc91e2b41f4d3c2d2a1b;hpb=c57405141d26ac2215a07b05d27a16a691dda50e;p=helm.git

diff --git a/helm/software/matita/nlibrary/arithmetics/R.ma b/helm/software/matita/nlibrary/arithmetics/R.ma
index 046bcbf88..60de71dbd 100644
--- a/helm/software/matita/nlibrary/arithmetics/R.ma
+++ b/helm/software/matita/nlibrary/arithmetics/R.ma
@@ -17,12 +17,18 @@ include "arithmetics/nat.ma".
 naxiom Q: Type[0].
 naxiom nat_to_Q: nat → Q.
 ncoercion nat_to_Q : ∀x:nat.Q ≝ nat_to_Q on _x:nat to Q.
+ndefinition bool_to_nat ≝ λb. match b with [ true ⇒ 1 | false ⇒ 0 ].
+ncoercion bool_to_nat : ∀b:bool.nat ≝ bool_to_nat on _b:bool to nat.
 naxiom Qplus: Q → Q → Q.
+naxiom Qminus: Q → Q → Q.
 naxiom Qtimes: Q → Q → Q.
 naxiom Qdivides: Q → Q → Q.
 naxiom Qle : Q → Q → Prop.
 naxiom Qlt: Q → Q → Prop.
+naxiom Qmin: Q → Q → Q.
+naxiom Qmax: Q → Q → Q.
 interpretation "Q plus" 'plus x y = (Qplus x y).
+interpretation "Q minus" 'minus x y = (Qminus 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).
@@ -38,8 +44,14 @@ ntheorem Qplus_assoc1: ∀q1,q2,q3. q1 + q2 + q3 = q3 + q2 + q1.
 #a; #b; #c; //; nqed.
 naxiom Qle_refl: ∀q1. q1≤q1.
 naxiom Qle_trans: ∀x,y,z. x≤y → y≤z → x≤z.
+naxiom Qlt_trans: ∀x,y,z. x < y → y < z → x < z.
+naxiom Qle_lt_trans1: ∀x,y,z. x ≤ y → y < z → x < z.
+naxiom Qle_lt_trans2: ∀x,y,z. x < y → y ≤ z → x < z.
 naxiom Qle_plus_compat: ∀x,y,z,t. x≤y → z≤t → x+z ≤ y+t.
+naxiom Qmult_zero: ∀q:Q. 0 * q = 0.
 
+naxiom phi: Q. (* the golden number *)
+naxiom golden: phi = phi * phi + phi * phi * phi.
 
 (* naxiom Ndivides_mult: ∀n:nat.∀q. (n * q) / n = q. *)
 
@@ -47,38 +59,65 @@ ntheorem lem1: ∀n:nat.∀q:Q. (n * q + q) = (S n) * q.
 #n; #q; ncut (plus n 1 = S n);##[//##]
 //; nqed.
 
+ntheorem Qplus_zero: ∀q:Q. 0 + q = q. //. nqed.
+
 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 …) ##] //.
+   L: ∀l,u. locate l ((1 - phi) * l + phi * u) → locate l u
+ | H: ∀l,u. locate (phi * l + (1 - phi) * u) u → locate l u.
+
+ndefinition locate_inv_ind':
+ ∀l,u:Q.∀P:Q → Q → Prop.
+  ∀H1: locate l ((1 - phi) * l + phi * u) → P l u. 
+  ∀H2: locate (phi * l + (1 - phi) * u) u → P l u.
+   locate l u → P l u.
+ #l; #u; #P; #H1; #H2; #p; ninversion p; #l; #u; #H; #E1; #E2;
+ ndestruct; /2/.
 nqed.
 
 ndefinition R ≝ ∃l,u:Q. locate l u.
 
-nlet corec Q_to_locate q : locate q q ≝ L q q q q … (Q_to_locate q).
+(*
+nlet corec Q_to_locate q : locate q q ≝ L q q … (Q_to_locate q).
   //; nrewrite < (Qdivides_mult 3 q) in ⊢ (? % ?); //.
 nqed.
 
 ndefinition Q_to_R : Q → R.
  #q; @ q; @q; //.
 nqed.
+*)
+
+nlemma help_auto1: ∀q:Q. false * q = 0. #q; nnormalize; //. nqed.
+
+(*
+nlet corec locate_add (l,u:?) (r1,r2: locate l u) (c1,c2:bool) :
+ locate (l + l + c1 * phi + c2 * phi * phi) (u + u + c1 * phi + c2 * phi * phi) ≝ ?.
+ napply (locate_inv_ind' … r1); napply (locate_inv_ind' … r2);
+ #r2'; #r1'; ncases c1; ncases c2
+  [ ##4: nnormalize; @1;
+    nlapply (locate_add … r1' r2' false false); nnormalize;
+    nrewrite > (Qmult_zero …); nrewrite > (Qmult_zero …); #K; nauto demod;
+     #K;
+    nnormalize in K; nrewrite > (Qmult_zero …) in K; nnormalize; #K;
+    napplyS K;
+     
+
+
+ 
+  [ ##1,4: ##[ @1 ? (l1'+l2') (u1'+u2') | @2 ? (l1'+l2') (u1'+u2') ]
+    ##[ ##1,5: /2/ | napplyS (Qle_plus_compat …leq1u leq2u) |
+        ##4: napplyS (Qle_plus_compat …leq1l leq2l)
+      |##*: /2/ ]
+ ##| ninversion r2; #l2''; #u2''; #leq2l'; #leq2u'; #r2';
+     ninversion r1; #l1''; #u1''; #leq1l'; #leq1u'; #r1';
+      ##[ @1 ? (l1''+l2'') (u1''+u2''); 
+      ##[ napply Qle_plus_compat; /3/;
+        ##| ##3: /2/;
+        ##| napplyS (Qle_plus_compat …leq1u' leq2u');
 
 (*
 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) |
@@ -97,6 +136,7 @@ nlet corec apart (l1,u1) (r1: locate l1 u1) (l2,u2) (r2: locate l2 u2) : CProp[0
   [ true ⇒ True
   | false ⇒ 
 *)
+*)
 
 include "topology/igft.ma".
 include "datatypes/pairs.ma".
@@ -104,7 +144,7 @@ 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_refl: reflexive … pre_r;
    pre_trans: transitive … pre_r
  }.
 
@@ -162,7 +202,7 @@ 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); //;
+ #b; *; *; #b; *; #H1; #H2; #H3; napply (ftleqleft … b); //;
  napply H; napply H1 (*CSC: auto non va! *).
 nqed.
 
@@ -175,8 +215,71 @@ ncoinductive ftfish (A : Ax_pro) (F : Ω^A) : A → CProp[0] ≝
 
 interpretation "fish" 'fish a U = (ftfish ? U a).
 
+nlemma ftcoreflexivity: ∀A: Ax_pro.∀F.∀a:A. a ⋉ F → a ∈ F.
+ #A; #F; #a; #H; ncases H; //.
+nqed.
+
+nlemma ftcoleqinfinity:
+ ∀A: Ax_pro.∀F.∀a:A. a ⋉ F →
+  ∀b. (a ≤ b → ∀i. (∃x. x ∈ 𝐂 b i ↓ (singleton … a) ∧ x ⋉ F)).
+ #A; #F; #a; #H; ncases H; /2/.
+nqed.
+
+nlemma ftcoleqleft:
+ ∀A: Ax_pro.∀F.∀a:A. a ⋉ F →
+  (∀b. a ≤ b → b ⋉ F).
+ #A; #F; #a; #H; ncases H; /2/.
+nqed.
+
+alias symbol "I" (instance 7) = "I".
+alias symbol "I" (instance 18) = "I".
+alias symbol "I" (instance 18) = "I".
+alias symbol "I" (instance 18) = "I".
+nlet corec ftfish_coind
+ (A: Ax_pro) (F: Ω^A) (P: A → CProp[0])
+ (Hcorefl: ∀a. P a → a ∈ F)
+ (Hcoleqleft: ∀a. P a → ∀b. a ≤ b → P b)
+ (Hcoleqinfinity: ∀a. P a → ∀b. a ≤ b → ∀i:𝐈 b. ∃x. x ∈ 𝐂 b i ↓ (singleton … a) ∧ P x)
+: ∀a:A. P a → a ⋉ F ≝ ?.
+ #a; #H; @
+  [ /2/
+  | #b; #H; napply (ftfish_coind … Hcorefl Hcoleqleft Hcoleqinfinity); /2/
+  | #b; #H1; #i; ncases (Hcoleqinfinity a H ? H1 i); #x; *; #H2; #H3;
+    @ x; @; //; napply (ftfish_coind … Hcorefl Hcoleqleft Hcoleqinfinity); //]
+nqed.
+
+(*CSC: non serve manco questo (vedi sotto) *)
+nlemma auto_hint3: ∀A. S__o__AAx A = S (AAx A).
+ #A; //.
+nqed.
+
 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
+ #A; #F; #a; #H; #i; nlapply (ftcoleqinfinity … F … a … i); //; #H;
+ ncases H; #c; *; *; *; #b; *; #H1; #H2; #H3; #H4; @ b; @ [ napply H1 (*CSC: auto non va *)]
+ napply (ftcoleqleft … c); //.
+nqed.
+
+nrecord Pt (A: Ax_pro) : Type[1] ≝
+ { pt_set: Ω^A;
+   pt_inhabited: ∃a. a ∈ pt_set;
+   pt_filtering: ∀a,b. a ∈ pt_set → b ∈ pt_set → ∃c. c ∈ (singleton … a) ↓ (singleton … b) → c ∈ pt_set;
+   pt_closed: pt_set ⊆ {b | b ⋉ pt_set}   
+ }.
+
+ndefinition Rd ≝ Pt Rax.
+
+naxiom daemon: False.
+
+ndefinition Q_to_R: Q → Rd.
+ #q; @
+  [ napply { c | fst … c < q ∧ q < snd … c  }
+  | @ [ @ (Qminus q 1) (Qplus q 1) | ncases daemon ]
+##| #c; #d; #Hc; #Hd; @ [ @ (Qmin (fst … c) (fst … d)) (Qmax (snd … c) (snd … d)) | ncases daemon]
+##| #a; #H; napply (ftfish_coind Rax ? (λa. fst … a < q ∧ q < snd … a)); /2/
+    [ /5/ | #b; *; #H1; #H2; #c; *; #H3; #H4; #i; ncases i
+      [ #w; nnormalize;
+    ##| nnormalize;
+  ]
+nqed.
+