+++ /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/".
-
-include "equality.ma".
-include "logic.ma".
-include "bool.ma".
-include "compare.ma".
-
-inductive nat : Set \def
- | O : nat
- | S : nat \to nat.
-
-definition pred: nat \to nat \def
-\lambda n:nat. match n with
-[ O \Rightarrow O
-| (S u) \Rightarrow u ].
-
-theorem pred_Sn : \forall n:nat.
-(eq nat n (pred (S n))).
-intros; reflexivity.
-qed.
-
-theorem injective_S : \forall n,m:nat.
-(eq nat (S n) (S m)) \to (eq nat n m).
-intros;
-rewrite > pred_Sn;
-rewrite > pred_Sn m.
-apply f_equal; assumption.
-qed.
-
-theorem not_eq_S : \forall n,m:nat.
-Not (eq nat n m) \to Not (eq nat (S n) (S m)).
-intros; simplify; intros;
-apply H; apply injective_S; assumption.
-qed.
-
-definition not_zero : nat \to Prop \def
-\lambda n: nat.
- match n with
- [ O \Rightarrow False
- | (S p) \Rightarrow True ].
-
-theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-intros; simplify; intros;
-cut (not_zero O); [ exact Hcut | rewrite > H; exact I ].
-qed.
-
-theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
-intros.elim n.apply O_S.apply not_eq_S.assumption.
-qed.
-
-let rec plus n m \def
- match n with
- [ O \Rightarrow m
- | (S p) \Rightarrow S (plus p m) ].
-
-theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
-intros;elim n;
- [ simplify;reflexivity
- | simplify;apply f_equal;assumption ].
-qed.
-
-theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
-intros.elim n.simplify.reflexivity.
-simplify.apply f_equal.assumption.
-qed.
-
-theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
-intros.elim n.simplify.apply plus_n_O.
-simplify.rewrite > H.apply plus_n_Sm.
-qed.
-
-theorem assoc_plus:
-\forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p)).
-intros.elim n.simplify.reflexivity.
-simplify.apply f_equal.assumption.
-qed.
-
-let rec times n m \def
- match n with
- [ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ].
-
-theorem times_n_O: \forall n:nat. eq nat O (times n O).
-intros.elim n.simplify.reflexivity.
-simplify.assumption.
-qed.
-
-theorem times_n_Sm :
-\forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
-intros.elim n.simplify.reflexivity.
-simplify.apply f_equal.rewrite < H.
-transitivity (plus (plus e1 m) (times e1 m)).symmetry.
-apply assoc_plus.transitivity (plus (plus m e1) (times e1 m)).
-apply f_equal2.
-apply sym_plus.reflexivity.apply assoc_plus.
-qed.
-
-theorem sym_times :
-\forall n,m:nat. eq nat (times n m) (times m n).
-intros.elim n.simplify.apply times_n_O.
-simplify.rewrite > H.apply times_n_Sm.
-qed.
-
-let rec minus n m \def
- match n with
- [ O \Rightarrow O
- | (S p) \Rightarrow
- match m with
- [O \Rightarrow (S p)
- | (S q) \Rightarrow minus p q ]].
-
-theorem nat_case :
-\forall n:nat.\forall P:nat \to Prop.
-P O \to (\forall m:nat. P (S m)) \to P n.
-intros.elim n.assumption.apply H1.
-qed.
-
-theorem nat_double_ind :
-\forall R:nat \to nat \to Prop.
-(\forall n:nat. R O n) \to
-(\forall n:nat. R (S n) O) \to
-(\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
-intros 5.elim n.apply H.
-apply nat_case m.apply H1.intros.apply H2. apply H3.
-qed.
-
-inductive le (n:nat) : nat \to Prop \def
- | le_n : le n n
- | le_S : \forall m:nat. le n m \to le n (S m).
-
-theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p.
-intros.
-elim H1.assumption.
-apply le_S.assumption.
-qed.
-
-theorem le_n_S: \forall n,m:nat. le n m \to le (S n) (S m).
-intros.elim H.
-apply le_n.apply le_S.assumption.
-qed.
-
-theorem le_O_n : \forall n:nat. le O n.
-intros.elim n.apply le_n.apply le_S. assumption.
-qed.
-
-theorem le_n_Sn : \forall n:nat. le n (S n).
-intros. apply le_S.apply le_n.
-qed.
-
-theorem le_pred_n : \forall n:nat. le (pred n) n.
-intros.elim n.simplify.apply le_n.simplify.
-apply le_n_Sn.
-qed.
-
-theorem not_zero_le : \forall n,m:nat. (le (S n) m ) \to not_zero m.
-intros.elim H.exact I.exact I.
-qed.
-
-theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
-intros.simplify.intros.apply not_zero_le ? O H.
-qed.
-
-theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n).
-intros.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
-elim n.reflexivity.
-apply False_ind.apply (le_Sn_O ? H2).
-qed.
-
-theorem le_S_n : \forall n,m:nat. le (S n) (S m) \to le n m.
-intros.change with le (pred (S n)) (pred (S m)).
-elim H.apply le_n.apply trans_le ? (pred x).assumption.
-apply le_pred_n.
-qed.
-
-theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
-intros.elim n.apply le_Sn_O.simplify.intros.
-cut le (S e1) e1.apply H.assumption.apply le_S_n.assumption.
-qed.
-
-theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).
-intros.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
-apply nat_double_ind (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
-intros.whd.intros.
-apply le_n_O_eq.assumption.
-intros.symmetry.apply le_n_O_eq.assumption.
-intros.apply f_equal.apply H2.
-apply le_S_n.assumption.
-apply le_S_n.assumption.
-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 le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)).
-intros.
-apply (nat_double_ind
-(\lambda n,m:nat.if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m).
-simplify.intros.apply le_O_n.
-simplify.exact le_Sn_O.
-intros 2.simplify.elim (leb n1 m1).
-simplify.apply le_n_S.apply H.
-simplify.intros.apply H.apply le_S_n.assumption.
-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_invert: \forall n,m:nat.
-eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
-intros.
-apply nat_double_ind (\lambda n,m.eq compare (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.simplify.reflexivity.
-qed.
-
projects / helm.git / blobdiff