From: Andrea Asperti Date: Mon, 30 May 2005 15:57:28 +0000 (+0000) Subject: fix X-Git-Tag: PRE_INDEX_1~103 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=9e9de11063c35559135987c72328231b57e4a609;p=helm.git fix --- diff --git a/helm/matita/tests/match.ma b/helm/matita/tests/match.ma new file mode 100644 index 000000000..fb0c1b0c4 --- /dev/null +++ b/helm/matita/tests/match.ma @@ -0,0 +1,371 @@ +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. +intro.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. +qed. + +theorem proj2: \forall A,B:Prop. (And A B) \to A. +intro. 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. +intro. 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. +intro.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). +intro.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). +intro.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))). +intro.apply refl_equal. +qed. + +theorem injective_S : \forall n,m:nat. +(eq nat (S n) (S m)) \to (eq nat n m). +intro.(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)). +intro. simplify.intro. +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)). +intro.simplify.intro. +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)). +intro.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:nat 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). +intro.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)). +intro.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). +intro.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)). +intro.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:nat 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). +intro.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)). +intro.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). +intro.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:nat with + [ O \Rightarrow O + | (S p) \Rightarrow + [\lambda n:nat.nat] match m:nat 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. +intro.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. +intro.cut \forall m:nat.R n m.apply Hcut.elim n.apply H. +apply nat_case m1.apply H1.intro.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. +intro. +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). +intro.elim H.alias id "le_n" = "cic:/matita/andrea/le.ind#xpointer(1/1/1)". +apply le_n.apply le_S.assumption. +qed. + +theorem le_O_n : \forall n:nat. le O n. +intro.elim n.apply le_n.apply le_S. assumption. +qed. + +theorem le_n_Sn : \forall n:nat. le n (S n). +intro. apply le_S.apply le_n. +qed. + +theorem le_pred_n : \forall n:nat. le (pred n) n. +intro.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. +intro.elim H.exact I.exact I. +qed. + +theorem le_Sn_O: \forall n:nat. Not (le (S n) O). +intro.simplify.intro.apply not_zero_le ? O H. +qed. + +theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n). +intro.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. +intro.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). +intro.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). +intro.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)). +intro.whd.intro.alias id "le_n_O_eq" = "cic:/matita/andrea/le_n_O_eq.con". +apply le_n_O_eq.assumption. +intro.whd.intro.apply sym_eq.apply le_n_O_eq.assumption. +intro.whd.intro.apply f_equal.apply H2. +apply le_S_n.assumption. +apply le_S_n.assumption. +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 leb : nat \to nat \to bool \def +let rec leb (n,m:nat) \def + [\lambda n:nat.bool] match n:nat with + [ O \Rightarrow true + | (S p) \Rightarrow + [\lambda n:nat.bool] match m:nat with + [ O \Rightarrow false + | (S q) \Rightarrow leb p q]] +in leb. + +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]. + +theorem le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)). +intro. +apply (nat_double_ind +(\lambda n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m). +intro.whd.apply le_O_n.intro.whd.apply le_Sn_O. + +intros 2. +simplify. +cut (( + +\lambda b:bool. [\lambda x:bool.Prop] match b:bool with +[ true \Rightarrow (le n1 m1) +| false \Rightarrow (Not(le n1 m1))] + +\to + + [\lambda x:bool.Prop] match b:bool with +[ true \Rightarrow (le (S n1) (S m1)) +| false \Rightarrow (Not(le (S n1) (S m1)))] +) (leb n1 m1)). +goal 8. + +exact ( +[ +\lambda b:bool. [\lambda x:bool.Prop] match b:bool with +[ true \Rightarrow (le n1 m1) +| false \Rightarrow (Not(le n1 m1))] + +\to + + [\lambda x:bool.Prop] match b:bool with +[ true \Rightarrow (le (S n1) (S m1)) +| false \Rightarrow (Not(le (S n1) (S m1)))] + +] +match (leb n1 m1) : bool with +[ true \Rightarrow \lambda p:(le n1 m1). le_n_S n1 m1 p +| false \Rightarrow \lambda p:(Not(le n1 m1)). \lambda q:(le (S n1) (S m1)).p(le_S_n n1 m1 q)] +). +apply Hcut. + + +cut (eq bool (leb n1 m1) (leb (S n1) (S m1))). +goal 8. +simplify.apply refl_equal. +cut + (if_then_else + (leb (S n1) (S m1)) + (le (S n1) (S m1)) + (Not (le (S n1) ( S m1)) )). +apply Hcut1.elim Hcut.simplify. +check ([\lambda b:bool. [\lambda b:bool.Prop]match (b:bool) with + [ true \Rightarrow + (if_then_else b (le n1 m1) (Not (le n1 m1))) + | false \Rightarrow + (if_then_else b (le n1 m1) (Not (le n1 m1))) +] +] +match (leb n1 m1) : bool with +[ true \Rightarrow H | false \Rightarrow H ]). + + +exact ( +[\lambda b:bool. match (b:bool) with + [ true \Rightarrow + (if_then_else b (le (S n1 ) (S m1)) (Not ((le (S n1) (S m1))))) + | false \Rightarrow + (if_then_else b (le (S n1 ) (S m1)) (Not ((le (S n1) (S m1))))) +] +] +match (leb n1 m1) : bool with +[ true \Rightarrow le_n_S n1 m1 H +| false \Rightarrow + (\lambda p:(le (S n1) (S m1)).H (le_S_n n1 m1 p))]). + + + + +elim n.simplify.apply le_O_n.elim m. +simplify.apply le_Sn_O.simplify. +cut (match (leb e e1):bool with +[true \Rightarrow (eq bool (leb e e1) true) +| false \Rightarrow (eq bool (leb e e1) false)]). +goal 20. +exact [\lambda b:bool. match b with +[true \Rightarrow (eq bool b true) +| false \Rightarrow (eq bool b false)]] match (leb e e1):bool with +[true \Rightarrow refl_equal bool true +| false \Rightarrow refl_equal bool false]. \ No newline at end of file