+(**************************************************************************)
+(* ___ *)
+(* ||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).
+\lambda A. (A \to False).
theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-intro.cut False.elim Hcut.apply H1.assumption.
+intros. elim (H1 H).
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
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.
+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.
-intro.elim H1.assumption.
+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).
-intro.elim H.apply refl_equal.
+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).
-intro.elim H1.elim H.apply refl_equal.
+intros.elim H1.elim H.apply refl_equal.
qed.
inductive nat : Set \def
theorem pred_Sn : \forall n:nat.
(eq nat n (pred (S n))).
-intro.apply refl_equal.
+intros.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))).
+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)).
-intro. simplify.intro.
+intros. simplify.intros.
apply H.apply injective_S.assumption.
qed.
| (S p) \Rightarrow True ].
theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-intro.simplify.intro.
+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)).
-intro.elim n.apply O_S.apply not_eq_S.assumption.
+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:nat with
+let rec plus n m \def
+ match n with
[ O \Rightarrow m
- | (S p) \Rightarrow S (plus p m) ]
-in
-plus.
+ | (S p) \Rightarrow S (plus p m) ].
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.
+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)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+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).
-intro.elim n.simplify.apply plus_n_O.
+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)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+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:nat with
+let rec times n m \def
+ match n with
[ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ]
-in
-times.
+ | (S p) \Rightarrow (plus m (times p m)) ].
theorem times_n_O: \forall n:nat. eq nat O (times n O).
-intro.elim n.simplify.apply refl_equal.simplify.assumption.
+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)).
-intro.elim n.simplify.apply refl_equal.
+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)).
theorem sym_times :
\forall n,m:nat. eq nat (times n m) (times m n).
-intro.elim n.simplify.apply times_n_O.
+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:nat with
+let rec minus n m \def
+ match n with
[ O \Rightarrow O
| (S p) \Rightarrow
- [\lambda n:nat.nat] match m:nat with
+ match m with
[O \Rightarrow (S p)
- | (S q) \Rightarrow minus p q ]]
-in
-minus.
+ | (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.
-intro.elim n.assumption.apply H1.
+intros.elim n.assumption.apply H1.
qed.
theorem nat_double_ind :
(\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.
+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
| 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.
+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).
-intro.elim H.
+intros.elim H.
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.
+intros.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.
+intros. 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.
+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.
-intro.elim H.exact I.exact I.
+intros.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.
+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).
-intro.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
+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.
-intro.cut le (pred (S n)) (pred (S m)).exact Hcut.
+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).
-intro.elim n.apply le_Sn_O.simplify.intro.
+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).
-intro.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
+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)).
-intro.whd.intro.
+intros.whd.intros.
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.
+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.
-definition leb : nat \to nat \to bool \def
-let rec leb (n,m:nat) \def
- [\lambda n:nat.bool] match n:nat with
+let rec leb n m \def
+ match n with
[ O \Rightarrow true
| (S p) \Rightarrow
- [\lambda n:nat.bool] match m:nat with
+ match m with
[ O \Rightarrow false
- | (S q) \Rightarrow leb p q]]
-in leb.
+ | (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.
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.
+simplify.intros.apply H.apply le_S_n.assumption.
qed.
+
+(*CSC: this requires too much time
+theorem prova :
+let three \def (S (S (S O))) in
+let nine \def (times three three) in
+let eightyone \def (times nine nine) in
+let square \def (times eightyone eightyone) in
+(eq nat square O).
+intro.
+intro.
+intro.intro.
+normalize goal at (? ? % ?).
+*)
\ No newline at end of file
projects / helm.git / blobdiff