- renamed ocaml/ to components/

[helm.git] / helm / matita / library / Z / compare.ma

diff --git a/helm/matita/library/Z/compare.ma b/helm/matita/library/Z/compare.ma
deleted file mode 100644 (file)
index 4a50259..0000000
--- a/helm/matita/library/Z/compare.ma
+++ /dev/null
@@ -1,143 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
-(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU Lesser General Public License Version 2.1         *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Z/compare".
-
-include "Z/orders.ma".
-include "nat/compare.ma".
-
-(* boolean equality *)
-definition eqZb : Z \to Z \to bool \def
-\lambda x,y:Z.
-  match x with
-  [ OZ \Rightarrow 
-      match y with
-        [ OZ \Rightarrow true
-        | (pos q) \Rightarrow false
-        | (neg q) \Rightarrow false]
-  | (pos p) \Rightarrow 
-      match y with
-        [ OZ \Rightarrow false
-        | (pos q) \Rightarrow eqb p q
-        | (neg q) \Rightarrow false]     
-  | (neg p) \Rightarrow 
-      match y with
-        [ OZ \Rightarrow false
-        | (pos q) \Rightarrow false
-        | (neg q) \Rightarrow eqb p q]].
-
-theorem eqZb_to_Prop: 
-\forall x,y:Z. 
-match eqZb x y with
-[ true \Rightarrow x=y
-| false \Rightarrow x \neq y].
-intros.
-elim x.
-  elim y.
-    simplify.reflexivity.
-    simplify.apply not_eq_OZ_pos.
-    simplify.apply not_eq_OZ_neg.
-  elim y.
-    simplify.unfold Not.intro.apply (not_eq_OZ_pos n).apply sym_eq.assumption.
-    simplify.apply eqb_elim.
-      intro.simplify.apply eq_f.assumption.
-      intro.simplify.unfold Not.intro.apply H.apply inj_pos.assumption.
-    simplify.apply not_eq_pos_neg.
-  elim y.
-    simplify.unfold Not.intro.apply (not_eq_OZ_neg n).apply sym_eq.assumption.
-    simplify.unfold Not.intro.apply (not_eq_pos_neg n1 n).apply sym_eq.assumption.
-    simplify.apply eqb_elim.
-      intro.simplify.apply eq_f.assumption.
-      intro.simplify.unfold Not.intro.apply H.apply inj_neg.assumption.
-qed.
-
-theorem eqZb_elim: \forall x,y:Z.\forall P:bool \to Prop.
-(x=y \to (P true)) \to (x \neq y \to (P false)) \to P (eqZb x y).
-intros.
-cut 
-(match (eqZb x y) with
-[ true \Rightarrow x=y
-| false \Rightarrow x \neq y] \to P (eqZb x y)).
-apply Hcut.
-apply eqZb_to_Prop.
-elim (eqZb).
-apply (H H2).
-apply (H1 H2).
-qed.
-
-definition Z_compare : Z \to Z \to compare \def
-\lambda x,y:Z.
-  match x with
-  [ OZ \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow EQ
-    | (pos m) \Rightarrow LT
-    | (neg m) \Rightarrow GT ]
-  | (pos n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow GT
-    | (pos m) \Rightarrow (nat_compare n m)
-    | (neg m) \Rightarrow GT]
-  | (neg n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow LT
-    | (pos m) \Rightarrow LT
-    | (neg m) \Rightarrow nat_compare m n ]].
-
-theorem Z_compare_to_Prop : 
-\forall x,y:Z. match (Z_compare x y) with
-[ LT \Rightarrow x < y
-| EQ \Rightarrow x=y
-| GT \Rightarrow y < x]. 
-intros.
-elim x. 
-  elim y.
-    simplify.apply refl_eq.
-    simplify.exact I.
-    simplify.exact I.
-  elim y.
-    simplify.exact I.
-    simplify.
-      cut (match (nat_compare n n1) with
-      [ LT \Rightarrow n<n1
-      | EQ \Rightarrow n=n1
-      | GT \Rightarrow n1<n] \to 
-      match (nat_compare n n1) with
-      [ LT \Rightarrow (S n) \leq n1
-      | EQ \Rightarrow pos n = pos n1
-      | GT \Rightarrow (S n1) \leq n]). 
-        apply Hcut.apply nat_compare_to_Prop. 
-        elim (nat_compare n n1).
-          simplify.exact H.
-          simplify.apply eq_f.exact H.
-          simplify.exact H.
-    simplify.exact I.    
-  elim y. 
-    simplify.exact I.
-    simplify.exact I.
-    simplify. 
-      cut (match (nat_compare n1 n) with
-      [ LT \Rightarrow n1<n
-      | EQ \Rightarrow n1=n
-      | GT \Rightarrow n<n1] \to 
-      match (nat_compare n1 n) with
-      [ LT \Rightarrow (S n1) \leq n
-      | EQ \Rightarrow neg n = neg n1
-      | GT \Rightarrow (S n) \leq n1]). 
-        apply Hcut. apply nat_compare_to_Prop. 
-        elim (nat_compare n1 n).
-          simplify.exact H.
-          simplify.apply eq_f.apply sym_eq.exact H.
-          simplify.exact H.
-qed.