X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matitaB%2Fmatita%2Fnlibrary%2Farithmetics%2FR.ma;fp=matitaB%2Fmatita%2Fnlibrary%2Farithmetics%2FR.ma;h=60de71dbd7a2eb6d5fd69a6851603f2bb72dc060;hb=cacbe3c6493ddce76c4c13379ade271d8dd172e8;hp=0000000000000000000000000000000000000000;hpb=f04a064bb34aabaf91dc0c48e3b08b37ecd7b0a2;p=helm.git

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+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).
+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.
+naxiom Qtimes_distr: ∀q1,q2,q3:Q.(q3 * q1 + q3 * q2) = q3 * (q1 + q2).
+naxiom Qdivides_distr: ∀q1,q2,q3:Q.(q1 / q3 + q2 / q3) = (q1 + q2) / q3.
+naxiom Qplus_comm: ∀q1,q2. q1 + q2 = q2 + q1.
+naxiom Qplus_assoc: ∀q1,q2,q3. q1 + q2 + q3 = q1 + (q2 + q3).
+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. *)
+
+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,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_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) ≝ ?.
+ 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) |
+        ##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 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 ⇒ 
+*)
+*)
+
+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_refl: 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); //;
+ #b; *; *; #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).
+
+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; #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.
+