(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
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
+ [ O \Rightarrow
+ match m with
+ [ O \Rightarrow true
+ | (S q) \Rightarrow false]
+ | (S p) \Rightarrow
+ match m with
+ [ O \Rightarrow false
+ | (S q) \Rightarrow eqb p q]].
+
+theorem eqb_to_Prop: \forall n,m:nat.
+match (eqb n m) with
+[ true \Rightarrow 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 n \neq m])).
+intro.elim n1.
+simplify.reflexivity.
+simplify.apply not_eq_O_S.
+intro.
+simplify.unfold Not.
+intro. apply (not_eq_O_S n1).apply sym_eq.assumption.
+intros.simplify.
+generalize in match H.
+elim ((eqb n1 m1)).
+simplify.apply eq_f.apply H1.
+simplify.unfold Not.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 (n \neq m \to (P false)) \to (P (eqb n m)).
+intros.
+cut
+(match (eqb n m) with
+[ true \Rightarrow 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 eqb_true_to_eq: \forall n,m:nat.
+eqb n m = true \to n = m.
+intros.
+change with
+match true with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop.
+qed.
+
+theorem eqb_false_to_not_eq: \forall n,m:nat.
+eqb n m = false \to n \neq m.
+intros.
+change with
+match false with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop.
+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
theorem leb_to_Prop: \forall n,m:nat.
match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))].
+[ true \Rightarrow n \leq m
+| false \Rightarrow n \nleq m].
intros.
-apply nat_elim2
+apply (nat_elim2
(\lambda n,m:nat.match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))]).
+[ true \Rightarrow 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).
+intros 2.simplify.elim ((leb n1 m1)).
simplify.apply le_S_S.apply H.
-simplify.intros.apply H.apply le_S_S_to_le.assumption.
+simplify.unfold Not.intros.apply H.apply le_S_S_to_le.assumption.
qed.
theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
-((le n m) \to (P true)) \to ((Not (le n 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 (le n m)
-| false \Rightarrow (Not (le n m))] \to (P (leb n m)).
+(match (leb n m) with
+[ true \Rightarrow n \leq m
+| false \Rightarrow n \nleq m] \to (P (leb n m))).
apply Hcut.apply leb_to_Prop.
-elim leb n m.
-apply (H H2).
-apply (H1 H2).
+elim (leb n m).
+apply ((H H2)).
+apply ((H1 H2)).
qed.
let rec nat_compare n m: compare \def
[ O \Rightarrow GT
| (S q) \Rightarrow nat_compare p q]].
-theorem nat_compare_n_n: \forall n:nat.(eq compare (nat_compare n n) EQ).
+theorem nat_compare_n_n: \forall n:nat. nat_compare n n = EQ.
intro.elim n.
simplify.reflexivity.
simplify.assumption.
qed.
theorem nat_compare_S_S: \forall n,m:nat.
-eq compare (nat_compare n m) (nat_compare (S n) (S m)).
+nat_compare n m = nat_compare (S n) (S m).
intros.simplify.reflexivity.
qed.
+theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
+intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
+apply eq_f.apply pred_Sn.
+qed.
+
+theorem nat_compare_pred_pred:
+\forall n,m:nat.lt O n \to lt O m \to
+eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
+intros.
+apply (lt_O_n_elim n H).
+apply (lt_O_n_elim m H1).
+intros.
+simplify.reflexivity.
+qed.
+
theorem nat_compare_to_Prop: \forall n,m:nat.
match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
- | EQ \Rightarrow (eq nat n m)
- | GT \Rightarrow (lt m n) ].
+ [ LT \Rightarrow n < m
+ | EQ \Rightarrow n=m
+ | GT \Rightarrow m < n ].
intros.
-apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
- | EQ \Rightarrow (eq nat n m)
- | GT \Rightarrow (lt m n) ]).
+apply (nat_elim2 (\lambda n,m.match (nat_compare n m) with
+ [ LT \Rightarrow n < m
+ | EQ \Rightarrow n=m
+ | GT \Rightarrow m < n ])).
intro.elim n1.simplify.reflexivity.
-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.unfold lt.apply le_S_S.apply le_O_n.
+intro.simplify.unfold lt.apply le_S_S. apply le_O_n.
+intros 2.simplify.elim ((nat_compare n1 m1)).
+simplify. unfold lt. apply le_S_S.apply H.
simplify. apply eq_f. apply H.
+simplify. unfold lt.apply le_S_S.apply H.
qed.
theorem nat_compare_n_m_m_n: \forall n,m:nat.
-eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+nat_compare n m = compare_invert (nat_compare m n).
intros.
-apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+apply (nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n))).
intros.elim n1.simplify.reflexivity.
simplify.reflexivity.
intro.elim n1.simplify.reflexivity.
qed.
theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to
+(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to
(P (nat_compare n m)).
intros.
-cut match (nat_compare n m) with
-[ LT \Rightarrow (lt n m)
-| EQ \Rightarrow (eq nat n m)
-| GT \Rightarrow (lt m n)] \to
-(P (nat_compare n m)).
+cut (match (nat_compare n m) with
+[ LT \Rightarrow n < m
+| EQ \Rightarrow n=m
+| GT \Rightarrow m < n] \to
+(P (nat_compare n m))).
apply Hcut.apply nat_compare_to_Prop.
-elim (nat_compare n m).
-apply (H H3).
-apply (H2 H3).
-apply (H1 H3).
+elim ((nat_compare n m)).
+apply ((H H3)).
+apply ((H1 H3)).
+apply ((H2 H3)).
qed.
projects / helm.git / blobdiff