1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 include "logic/equality.ma".
16 include "higher_order_defs/functions.ma".
18 inductive bool : Set \def
22 theorem bool_elim: \forall P:bool \to Prop. \forall b:bool.
24 \to (b = false \to P false)
28 [ apply H; reflexivity
29 | apply H1; reflexivity
33 theorem not_eq_true_false : true \neq false.
37 [ true \Rightarrow False
38 | false \Rightarrow True].
39 rewrite > H.simplify.exact I.
42 definition notb : bool \to bool \def
45 [ true \Rightarrow false
46 | false \Rightarrow true ].
48 theorem notb_elim: \forall b:bool.\forall P:bool \to Prop.
50 [ true \Rightarrow P false
51 | false \Rightarrow P true] \to P (notb b).
52 intros 2.elim b.exact H. exact H.
55 theorem notb_notb: \forall b:bool. notb (notb b) = b.
60 theorem injective_notb: injective bool bool notb.
64 rewrite < (notb_notb y).
69 (*CSC: the URI must disappear: there is a bug now *)
70 interpretation "boolean not" 'not x = (cic:/matita/datatypes/bool/notb.con x).
72 definition andb : bool \to bool \to bool\def
76 | false \Rightarrow false ].
78 (*CSC: the URI must disappear: there is a bug now *)
79 interpretation "boolean and" 'and x y = (cic:/matita/datatypes/bool/andb.con x y).
81 theorem andb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
83 [ true \Rightarrow P b2
84 | false \Rightarrow P false] \to P (b1 \land b2).
85 intros 3.elim b1.exact H. exact H.
88 theorem and_true: \forall a,b:bool.
89 andb a b =true \to a =true \land b= true.
92 [reflexivity|assumption]
94 apply not_eq_true_false.
100 theorem andb_true_true: \forall b1,b2. (b1 \land b2) = true \to b1 = true.
106 theorem andb_true_true_r: \forall b1,b2. (b1 \land b2) = true \to b2 = true.
109 |apply False_ind.apply not_eq_true_false.
110 apply sym_eq.assumption
114 definition orb : bool \to bool \to bool\def
117 [ true \Rightarrow true
118 | false \Rightarrow b2].
120 theorem orb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
122 [ true \Rightarrow P true
123 | false \Rightarrow P b2] \to P (orb b1 b2).
124 intros 3.elim b1.exact H. exact H.
127 (*CSC: the URI must disappear: there is a bug now *)
128 interpretation "boolean or" 'or x y = (cic:/matita/datatypes/bool/orb.con x y).
130 definition if_then_else : bool \to Prop \to Prop \to Prop \def
131 \lambda b:bool.\lambda P,Q:Prop.
134 | false \Rightarrow Q].
136 (*CSC: missing notation for if_then_else *)
138 theorem bool_to_decidable_eq:
139 \forall b1,b2:bool. decidable (b1=b2).
145 right. exact not_eq_true_false.
147 right. unfold Not. intro.
148 apply not_eq_true_false.
153 theorem P_x_to_P_x_to_eq:
154 \forall A:Set. \forall P: A \to bool.
155 \forall x:A. \forall p1,p2:P x = true. p1 = p2.
157 apply eq_to_eq_to_eq_p_q.
158 exact bool_to_decidable_eq.
162 (* some basic properties of and - or*)
163 theorem andb_sym: \forall A,B:bool.
164 (A \land B) = (B \land A).
172 theorem andb_assoc: \forall A,B,C:bool.
173 (A \land (B \land C)) = ((A \land B) \land C).
182 theorem orb_sym: \forall A,B:bool.
183 (A \lor B) = (B \lor A).
191 theorem true_to_true_to_andb_true: \forall A,B:bool.
192 A = true \to B = true \to (A \land B) = true.