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 interpretation "boolean not" 'not x = (notb x).
71 definition andb : bool \to bool \to bool\def
75 | false \Rightarrow false ].
77 interpretation "boolean and" 'and x y = (andb x y).
79 theorem andb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
81 [ true \Rightarrow P b2
82 | false \Rightarrow P false] \to P (b1 \land b2).
83 intros 3.elim b1.exact H. exact H.
86 theorem and_true: \forall a,b:bool.
87 andb a b =true \to a =true \land b= true.
90 [reflexivity|assumption]
92 apply not_eq_true_false.
98 theorem andb_true_true: \forall b1,b2. (b1 \land b2) = true \to b1 = true.
104 theorem andb_true_true_r: \forall b1,b2. (b1 \land b2) = true \to b2 = true.
107 |apply False_ind.apply not_eq_true_false.
108 apply sym_eq.assumption
112 definition orb : bool \to bool \to bool\def
115 [ true \Rightarrow true
116 | false \Rightarrow b2].
118 theorem orb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
120 [ true \Rightarrow P true
121 | false \Rightarrow P b2] \to P (orb b1 b2).
122 intros 3.elim b1.exact H. exact H.
125 interpretation "boolean or" 'or x y = (orb x y).
127 definition if_then_else : bool \to Prop \to Prop \to Prop \def
128 \lambda b:bool.\lambda P,Q:Prop.
131 | false \Rightarrow Q].
133 (*CSC: missing notation for if_then_else *)
135 theorem bool_to_decidable_eq:
136 \forall b1,b2:bool. decidable (b1=b2).
142 right. exact not_eq_true_false.
144 right. unfold Not. intro.
145 apply not_eq_true_false.
150 theorem P_x_to_P_x_to_eq:
151 \forall A:Set. \forall P: A \to bool.
152 \forall x:A. \forall p1,p2:P x = true. p1 = p2.
154 apply eq_to_eq_to_eq_p_q.
155 exact bool_to_decidable_eq.
159 (* some basic properties of and - or*)
160 theorem andb_sym: \forall A,B:bool.
161 (A \land B) = (B \land A).
169 theorem andb_assoc: \forall A,B,C:bool.
170 (A \land (B \land C)) = ((A \land B) \land C).
179 theorem orb_sym: \forall A,B:bool.
180 (A \lor B) = (B \lor A).
188 theorem true_to_true_to_andb_true: \forall A,B:bool.
189 A = true \to B = true \to (A \land B) = true.