From 13767ef97ac9fc6914db580fc0bd44a4437fe284 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Thu, 9 Jun 2005 10:33:47 +0000 Subject: [PATCH] Updated to the new syntax for match. --- helm/matita/tests/andrea.ma | 281 +++++++++++++++++++++++++++++++++++- 1 file changed, 278 insertions(+), 3 deletions(-) diff --git a/helm/matita/tests/andrea.ma b/helm/matita/tests/andrea.ma index 242e374dd..ba4b2aaf9 100644 --- a/helm/matita/tests/andrea.ma +++ b/helm/matita/tests/andrea.ma @@ -1,3 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + + inductive True: Prop \def I : True. @@ -7,18 +22,18 @@ definition Not: Prop \to Prop \def \lambda A:Prop. (A \to False). theorem absurd : \forall A,C:Prop. A \to Not A \to C. -cut False.elim Hcut.apply H1.assumption. +intros.cut False.elim Hcut.apply H1.assumption. qed. 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. -intro. elim H. assumption. +intros. elim H. assumption. qed. theorem proj2: \forall A,B:Prop. (And A B) \to A. -intro. elim H. assumption. +intros. elim H. assumption. qed. inductive Or (A,B:Prop) : Prop \def @@ -33,3 +48,263 @@ inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def 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 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.apply refl_equal. +qed. + +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.apply refl_equal. +qed. + + +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. + +definition plus : nat \to nat \to nat \def +let rec plus (n,m:nat) \def + match n with + [ O \Rightarrow m + | (S p) \Rightarrow S (plus p m) ] +in +plus. + +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. + +definition times : nat \to nat \to nat \def +let rec times (n,m:nat) \def + match n with + [ O \Rightarrow O + | (S p) \Rightarrow (plus m (times p m)) ] +in +times. + +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. + +definition minus : nat \to nat \to nat \def +let rec minus (n,m:nat) \def + [\lambda n:nat.nat] match n with + [ O \Rightarrow O + | (S p) \Rightarrow + [\lambda n:nat.nat] match m with + [O \Rightarrow (S p) + | (S q) \Rightarrow minus p q ]] +in +minus. + +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 bool : Set \def + | true : bool + | false : bool. + +definition notn : 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]. + +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.intro. +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.intro.apply sym_eq.apply le_n_O_eq.assumption. +intros.whd.intro.apply f_equal.apply H2. +apply le_S_n.assumption. +apply le_S_n.assumption. +qed. + + +definition leb : nat \to nat \to bool \def +let rec leb (n,m:nat) \def + [\lambda n:nat.bool] match n with + [ O \Rightarrow true + | (S p) \Rightarrow + [\lambda n:nat.bool] match m with + [ O \Rightarrow false + | (S q) \Rightarrow leb p q]] +in leb. + +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.intro.apply H.apply le_S_n.assumption. + + -- 2.39.2