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[helm.git] / helm / software / matita / nlibrary / arithmetics / R.ma

diff --git a/helm/software/matita/nlibrary/arithmetics/R.ma b/helm/software/matita/nlibrary/arithmetics/R.ma
index d58f69a05fdc1aafe5b4894eedd2ce7a05ad639e..60de71dbd7a2eb6d5fd69a6851603f2bb72dc060 100644 (file)
--- a/helm/software/matita/nlibrary/arithmetics/R.ma
+++ b/helm/software/matita/nlibrary/arithmetics/R.ma
@@ -25,6 +25,8 @@ 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).
@@ -42,6 +44,9 @@ 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.
 
@@ -139,7 +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
  }.
 
@@ -226,9 +231,55 @@ nlemma ftcoleqleft:
  #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.
\ No newline at end of file
+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.
+