From 349a0e23813a7f33853e1f8fe48230276ac22934 Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Thu, 9 Jun 2005 10:43:44 +0000 Subject: [PATCH] andrea.ma removed (superseded by match.ma) --- helm/matita/tests/andrea.ma | 310 ------------------------------------ helm/matita/tests/match.ma | 27 +++- 2 files changed, 21 insertions(+), 316 deletions(-) delete mode 100644 helm/matita/tests/andrea.ma diff --git a/helm/matita/tests/andrea.ma b/helm/matita/tests/andrea.ma deleted file mode 100644 index ba4b2aaf9..000000000 --- a/helm/matita/tests/andrea.ma +++ /dev/null @@ -1,310 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. - -inductive False: Prop \def . - -definition Not: Prop \to Prop \def -\lambda A:Prop. (A \to False). - -theorem absurd : \forall A,C:Prop. A \to Not A \to C. -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. -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. - -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. - - diff --git a/helm/matita/tests/match.ma b/helm/matita/tests/match.ma index c952d0a92..3dbb580d5 100644 --- a/helm/matita/tests/match.ma +++ b/helm/matita/tests/match.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. @@ -97,7 +112,7 @@ intros.elim n.apply O_S.apply not_eq_S.assumption. qed. -let rec plus n m : nat \def +let rec plus n m \def match n with [ O \Rightarrow m | (S p) \Rightarrow S (plus p m) ]. @@ -120,7 +135,7 @@ theorem assoc_plus: intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption. qed. -let rec times n m : nat \def +let rec times n m \def match n with [ O \Rightarrow O | (S p) \Rightarrow (plus m (times p m)) ]. @@ -145,8 +160,8 @@ intros.elim n.simplify.apply times_n_O. simplify.elim (sym_eq ? ? ? H).apply times_n_Sm. qed. -let rec minus n m : nat \def - [\lambda n:nat.nat] match n with +let rec minus n m \def + match n with [ O \Rightarrow O | (S p) \Rightarrow [\lambda n:nat.nat] match m with @@ -261,8 +276,8 @@ apply le_S_n.assumption. apply le_S_n.assumption. qed. -let rec leb n m : bool \def - [\lambda n:nat.bool] match n with +let rec leb n m \def + match n with [ O \Rightarrow true | (S p) \Rightarrow [\lambda n:nat.bool] match m with -- 2.39.2