+++ /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/Z/".
-
-include "nat.ma".
-
-inductive Z : Set \def
- OZ : Z
-| pos : nat \to Z
-| neg : nat \to Z.
-
-definition Z_of_nat \def
-\lambda n. match n with
-[ O \Rightarrow OZ
-| (S n)\Rightarrow pos n].
-
-coercion Z_of_nat.
-
-definition neg_Z_of_nat \def
-\lambda n. match n with
-[ O \Rightarrow OZ
-| (S n)\Rightarrow neg n].
-
-definition absZ \def
-\lambda z.
- match z with
-[ OZ \Rightarrow O
-| (pos n) \Rightarrow n
-| (neg n) \Rightarrow n].
-
-definition OZ_testb \def
-\lambda z.
-match z with
-[ OZ \Rightarrow true
-| (pos n) \Rightarrow false
-| (neg n) \Rightarrow false].
-
-theorem OZ_discr :
-\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)).
-intros.elim z.simplify.reflexivity.
-simplify.intros.
-cut match neg e1 with
-[ OZ \Rightarrow True
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut.rewrite > H.simplify.exact I.
-simplify.intros.
-cut match pos e2 with
-[ OZ \Rightarrow True
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut. rewrite > H.simplify.exact I.
-qed.
-
-definition Zsucc \def
-\lambda z. match z with
-[ OZ \Rightarrow pos O
-| (pos n) \Rightarrow pos (S n)
-| (neg n) \Rightarrow
- match n with
- [ O \Rightarrow OZ
- | (S p) \Rightarrow neg p]].
-
-definition Zpred \def
-\lambda z. match z with
-[ OZ \Rightarrow neg O
-| (pos n) \Rightarrow
- match n with
- [ O \Rightarrow OZ
- | (S p) \Rightarrow pos p]
-| (neg n) \Rightarrow neg (S n)].
-
-theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
-intros.elim z.reflexivity.
-elim e1.reflexivity.
-reflexivity.
-reflexivity.
-qed.
-
-theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
-intros.elim z.reflexivity.
-reflexivity.
-elim e2.reflexivity.
-reflexivity.
-qed.
-
-let rec Zplus x y : Z \def
- match x with
- [ OZ \Rightarrow y
- | (pos m) \Rightarrow
- match y with
- [ OZ \Rightarrow x
- | (pos n) \Rightarrow (pos (S (plus m n)))
- | (neg n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (neg (pred (minus n m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (pos (pred (minus m n)))]]
- | (neg m) \Rightarrow
- match y with
- [ OZ \Rightarrow x
- | (pos n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (pos (pred (minus n m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (neg (pred (minus m n)))]
- | (neg n) \Rightarrow (neg (S (plus m n)))]].
-
-theorem Zplus_z_O: \forall z:Z. eq Z (Zplus z OZ) z.
-intro.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
-intros.elim x.simplify.rewrite > Zplus_z_O.reflexivity.
-elim y.simplify.reflexivity.
-simplify.
-rewrite < sym_plus.reflexivity.
-simplify.
-rewrite > nat_compare_invert.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_invert.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify.elim (sym_plus ? ?).reflexivity.
-qed.
-
-theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
-intros.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-elim e2.simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
-intros.elim z.
-simplify.reflexivity.
-elim e1.simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_pp :
-\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.
-rewrite < plus_n_O.reflexivity.
-simplify.
-rewrite < plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_pn :
-\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
-intros.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_np :
-\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_nn:
-\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite < plus_n_Sm.reflexivity.
-simplify.rewrite > plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_succ_pred:
-\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
-intros.
-elim x. elim y.
-simplify.reflexivity.
-simplify.reflexivity.
-rewrite < Zsucc_pos.rewrite > Zsucc_pred.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_neg.rewrite > Zpred_succ.
-simplify.reflexivity.
-rewrite < Zplus_succ_pred_nn.reflexivity.
-apply Zplus_succ_pred_np.
-elim y.simplify.reflexivity.
-apply Zplus_succ_pred_pn.
-apply Zplus_succ_pred_pp.
-qed.
-
-theorem Zsucc_plus_pp :
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
-intros.reflexivity.
-qed.
-
-theorem Zsucc_plus_pn :
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_succ_pred_pn ? m1).
-elim H.reflexivity.
-qed.
-
-theorem Zsucc_plus_nn :
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_succ_pred_nn ? m1).
-reflexivity.
-qed.
-
-theorem Zsucc_plus_np :
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < H.
-rewrite < (Zplus_succ_pred_np ? (S m1)).
-reflexivity.
-qed.
-
-
-theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
-intros.elim x.elim y.
-simplify. reflexivity.
-rewrite < Zsucc_pos.reflexivity.
-simplify.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zsucc_plus_nn.
-apply Zsucc_plus_np.
-elim y.
-rewrite < sym_Zplus OZ.reflexivity.
-apply Zsucc_plus_pn.
-apply Zsucc_plus_pp.
-qed.
-
-theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
-intros.
-cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
-rewrite > Hcut.
-rewrite > Zsucc_plus.
-rewrite > Zpred_succ.
-reflexivity.
-rewrite > Zsucc_pred.
-reflexivity.
-qed.
-
-theorem assoc_Zplus :
-\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
-intros.elim x.simplify.reflexivity.
-elim e1.rewrite < (Zpred_neg (Zplus y z)).
-rewrite < (Zpred_neg y).
-rewrite < Zpred_plus.
-reflexivity.
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (Zplus (neg e) y).
-apply f_equal.assumption.
-elim e2.rewrite < Zsucc_pos.
-rewrite < Zsucc_pos.
-rewrite > Zsucc_plus.
-reflexivity.
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (Zplus (pos e1) y).
-apply f_equal.assumption.
-qed.
+++ /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/bool/".
-
-inductive bool : Set \def
- | true : bool
- | false : bool.
-
-definition notb : bool \to bool \def
-\lambda b:bool.
- match b with
- [ true \Rightarrow false
- | false \Rightarrow true ].
-
-definition andb : bool \to bool \to bool\def
-\lambda b1,b2:bool.
- match b1 with
- [ true \Rightarrow
- match b2 with [true \Rightarrow true | false \Rightarrow false]
- | false \Rightarrow false ].
-
-definition orb : bool \to bool \to bool\def
-\lambda b1,b2:bool.
- match b1 with
- [ true \Rightarrow
- match b2 with [true \Rightarrow true | false \Rightarrow false]
- | false \Rightarrow false ].
-
-definition if_then_else : bool \to Prop \to Prop \to Prop \def
-\lambda b:bool.\lambda P,Q:Prop.
-match b with
-[ true \Rightarrow P
-| false \Rightarrow Q].
+++ /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/compare/".
-
-inductive compare :Set \def
-| LT : compare
-| EQ : compare
-| GT : compare.
-
-definition compare_invert: compare \to compare \def
- \lambda c.
- match c with
- [ LT \Rightarrow GT
- | EQ \Rightarrow EQ
- | GT \Rightarrow LT ].
+++ /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/equality/".
-
-inductive eq (A:Type) (x:A) : A \to Prop \def
- refl_equal : eq A x x.
-
-theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y \to eq A y x.
-intros. elim H. apply refl_equal.
-qed.
-
-theorem trans_eq : \forall A:Type.
-\forall x,y,z:A. eq A x y \to eq A y z \to eq A x z.
-intros.elim H1.assumption.
-qed.
-
-theorem eq_ind_r :
- \forall A:Type.\forall x:A. \forall P: A \to Prop.
- P x \to \forall y:A. eq A y x \to P y.
-intros.letin H1' \def sym_eq ? ? ? H1.clearbody H1'.
-elim H1'.assumption.
-qed.
-
-theorem f_equal: \forall A,B:Type.\forall f:A\to B.
-\forall x,y:A. eq A x y \to eq B (f x) (f y).
-intros.elim H.reflexivity.
-qed.
-
-default "equality"
- cic:/matita/equality/eq.ind
- cic:/matita/equality/sym_eq.con
- cic:/matita/equality/trans_eq.con
- cic:/matita/equality/eq_ind.con
- cic:/matita/equality/eq_ind_r.con.
-
-theorem f_equal2: \forall A,B,C:Type.\forall f:A\to B \to C.
-\forall x1,x2:A. \forall y1,y2:B.
-eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
-intros.elim H1.elim H.reflexivity.
-qed.
+++ /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/logic/".
-
-
-inductive True: Prop \def
-I : True.
-
-default "true" cic:/matita/logic/True.ind.
-
-inductive False: Prop \def .
-
-default "false" cic:/matita/logic/False.ind.
-
-definition Not: Prop \to Prop \def
-\lambda A. (A \to False).
-
-theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-intros. elim (H1 H).
-qed.
-
-default "absurd" cic:/matita/logic/absurd.ind.
-
-inductive And (A,B:Prop) : Prop \def
- conj : A \to B \to (And A B).
-
-theorem proj1: \forall A,B:Prop. (And A B) \to A.
-intros. elim H. assumption.
-qed.
-
-theorem proj2: \forall A,B:Prop. (And A B) \to A.
-intros. elim H. assumption.
-qed.
-
-inductive Or (A,B:Prop) : Prop \def
- or_introl : A \to (Or A B)
- | or_intror : B \to (Or A B).
-
-inductive ex (A:Type) (P:A \to Prop) : Prop \def
- ex_intro: \forall x:A. P x \to ex A P.
-
-inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def
- ex_intro2: \forall x:A. P x \to Q x \to ex2 A P Q.
+++ /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 / commitdiff