+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/nat/compare".
-
-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
- [ O \Rightarrow true
- | (S p) \Rightarrow
- match m with
- [ O \Rightarrow false
- | (S q) \Rightarrow leb p q]].
-
-theorem leb_to_Prop: \forall n,m:nat.
-match (leb n m) with
-[ true \Rightarrow 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 n \nleq m])).
-simplify.exact le_O_n.
-simplify.exact not_le_Sn_O.
-intros 2.simplify.elim ((leb n1 m1)).
-simplify.apply le_S_S.apply H.
-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.
-(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 n \nleq m] \to (P (leb n m))).
-apply Hcut.apply leb_to_Prop.
-elim (leb n m).
-apply ((H H2)).
-apply ((H1 H2)).
-qed.
-
-let rec nat_compare n m: compare \def
-match n with
-[ O \Rightarrow
- match m with
- [ O \Rightarrow EQ
- | (S q) \Rightarrow LT ]
-| (S p) \Rightarrow
- match m with
- [ O \Rightarrow GT
- | (S q) \Rightarrow nat_compare p q]].
-
-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.
-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 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 n < m
- | EQ \Rightarrow n=m
- | GT \Rightarrow m < n ])).
-intro.elim n1.simplify.reflexivity.
-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.
-nat_compare n m = compare_invert (nat_compare m n).
-intros.
-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.
-simplify.reflexivity.
-intros.simplify.elim H.reflexivity.
-qed.
-
-theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-(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 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 ((H1 H3)).
-apply ((H2 H3)).
-qed.
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