--- /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/".
+
+alias id "eq" = "cic:/matita/equality/eq.ind#xpointer(1/1)".
+alias id "refl_equal" = "cic:/matita/equality/eq.ind#xpointer(1/1/1)".
+alias id "sym_eq" = "cic:/matita/equality/sym_eq.con".
+alias id "f_equal" = "cic:/matita/equality/f_equal.con".
+alias id "Not" = "cic:/matita/logic/Not.con".
+alias id "False" = "cic:/matita/logic/False.ind#xpointer(1/1)".
+alias id "True" = "cic:/matita/logic/True.ind#xpointer(1/1)".
+alias id "trans_eq" = "cic:/matita/equality/trans_eq.con".
+alias id "I" = "cic:/matita/logic/True.ind#xpointer(1/1/1)".
+alias id "f_equal2" = "cic:/matita/equality/f_equal2.con".
+alias id "False_ind" = "cic:/matita/logic/False_ind.con".
+alias id "false" = "cic:/matita/bool/bool.ind#xpointer(1/1/2)".
+alias id "true" = "cic:/matita/bool/bool.ind#xpointer(1/1/1)".
+alias id "if_then_else" = "cic:/matita/bool/if_then_else.con".
+alias id "EQ" = "cic:/matita/compare/compare.ind#xpointer(1/1/2)".
+alias id "GT" = "cic:/matita/compare/compare.ind#xpointer(1/1/3)".
+alias id "LT" = "cic:/matita/compare/compare.ind#xpointer(1/1/1)".
+alias id "compare" = "cic:/matita/compare/compare.ind#xpointer(1/1)".
+alias id "compare_invert" = "cic:/matita/compare/compare_invert.con".
+
+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.
+apply refl_equal.
+qed.
+
+theorem injective_S : \forall n,m:nat.
+(eq nat (S n) (S m)) \to (eq nat n m).
+intros.
+(elim (sym_eq ? ? ? (pred_Sn n))).(elim (sym_eq ? ? ? (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.elim (sym_eq ? ? ? 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.apply refl_equal.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.apply refl_equal.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.elim (sym_eq ? ? ? 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.apply refl_equal.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.apply refl_equal.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.apply refl_equal.
+simplify.apply f_equal.elim H.
+apply trans_eq ? ? (plus (plus e m) (times e m)).apply sym_eq.
+apply assoc_plus.apply trans_eq ? ? (plus (plus m e) (times e m)).
+apply f_equal2.
+apply sym_plus.apply refl_equal.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.elim (sym_eq ? ? ? 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.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
+apply nat_case m1.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.apply refl_equal.
+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.cut le (pred (S n)) (pred (S m)).exact Hcut.
+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 e) e.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.whd.intros.apply sym_eq.apply le_n_O_eq.assumption.
+intros.whd.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.apply refl_equal.
+simplify.apply refl_equal.
+intro.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intros.simplify.elim H.apply refl_equal.
+qed.
+
projects / helm.git / blobdiff