Every exception that used to have type string is now a string Lazy.t.

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

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index 194b38d84f255e3924a3273919f01bb96d320ca5..55e526c8fad0f80eade4775141c99326975efee8 100644 (file)
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -14,9 +14,9 @@
 
 set "baseuri" "cic:/matita/nat/compare".
 
-include "nat/orders.ma".
 include "datatypes/bool.ma".
 include "datatypes/compare.ma".
+include "nat/orders.ma".
 
 let rec eqb n m \def 
 match n with 
@@ -32,18 +32,18 @@ match n with
 theorem eqb_to_Prop: \forall n,m:nat. 
 match (eqb n m) with
 [ true  \Rightarrow n = m 
-| false \Rightarrow \lnot (n = m)].
+| false \Rightarrow n \neq m].
 intros.
 apply nat_elim2
 (\lambda n,m:nat.match (eqb n m) with
 [ true  \Rightarrow n = m 
-| false \Rightarrow \lnot (n = m)]).
+| false \Rightarrow n \neq m]).
 intro.elim n1.
 simplify.reflexivity.
 simplify.apply not_eq_O_S.
 intro.
 simplify.
-intro. apply not_eq_O_S n1 ?.apply sym_eq.assumption.
+intro. apply not_eq_O_S n1.apply sym_eq.assumption.
 intros.simplify.
 generalize in match H.
 elim (eqb n1 m1).
@@ -52,18 +52,37 @@ simplify.intro.apply H1.apply inj_S.assumption.
 qed.
 
 theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
-(n=m \to (P true)) \to (\lnot n=m \to (P false)) \to (P (eqb n m)). 
+(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)). 
 intros.
 cut 
 match (eqb n m) with
 [ true  \Rightarrow n = m
-| false \Rightarrow \lnot (n = m)] \to (P (eqb n m)).
+| false \Rightarrow n \neq m] \to (P (eqb n m)).
 apply Hcut.apply eqb_to_Prop.
 elim eqb n m.
 apply (H H2).
 apply (H1 H2).
 qed.
 
+theorem eqb_n_n: \forall n. eqb n n = true.
+intro.elim n.simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem eq_to_eqb_true: \forall n,m:nat.
+n = m \to eqb n m = true.
+intros.apply eqb_elim n m.
+intros. reflexivity.
+intros.apply False_ind.apply H1 H.
+qed.
+
+theorem not_eq_to_eqb_false: \forall n,m:nat.
+\lnot (n = m) \to eqb n m = false.
+intros.apply eqb_elim n m.
+intros. apply False_ind.apply H H1.
+intros.reflexivity.
+qed.
+
 let rec leb n m \def 
 match n with 
     [ O \Rightarrow true
@@ -75,12 +94,12 @@ match n with
 theorem leb_to_Prop: \forall n,m:nat. 
 match (leb n m) with
 [ true  \Rightarrow n \leq m 
-| false \Rightarrow \lnot (n \leq m)].
+| false \Rightarrow n \nleq m].
 intros.
 apply nat_elim2
 (\lambda n,m:nat.match (leb n m) with
 [ true  \Rightarrow n \leq m 
-| false \Rightarrow \lnot (n \leq m)]).
+| false \Rightarrow n \nleq m]).
 simplify.exact le_O_n.
 simplify.exact not_le_Sn_O.
 intros 2.simplify.elim (leb n1 m1).
@@ -89,13 +108,13 @@ simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
 theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
-(n \leq m \to (P true)) \to (\not (n \leq m) \to (P false)) \to
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
 P (leb n m).
 intros.
 cut 
 match (leb n m) with
 [ true  \Rightarrow n \leq m
-| false \Rightarrow \lnot (n \leq m)] \to (P (leb n m)).
+| false \Rightarrow n \nleq m] \to (P (leb n m)).
 apply Hcut.apply leb_to_Prop.
 elim leb n m.
 apply (H H2).
@@ -154,8 +173,8 @@ simplify.apply le_S_S.apply le_O_n.
 intro.simplify.apply le_S_S. apply le_O_n.
 intros 2.simplify.elim (nat_compare n1 m1).
 simplify. apply le_S_S.apply H.
-simplify. apply le_S_S.apply H.
 simplify. apply eq_f. apply H.
+simplify. apply le_S_S.apply H.
 qed.
 
 theorem nat_compare_n_m_m_n: \forall n,m:nat. 
@@ -181,6 +200,6 @@ cut match (nat_compare n m) with
 apply Hcut.apply nat_compare_to_Prop.
 elim (nat_compare n m).
 apply (H H3).
-apply (H2 H3).
 apply (H1 H3).
+apply (H2 H3).
 qed.