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 set "baseuri" "cic:/matita/datatypes/bool/".
17 include "logic/equality.ma".
19 inductive bool : Set \def
23 theorem bool_elim: \forall P:bool \to Prop. \forall b:bool.
25 \to (b = false \to P false)
29 [ apply H; reflexivity
30 | apply H1; reflexivity
34 theorem not_eq_true_false : true \neq false.
38 [ true \Rightarrow False
39 | flase \Rightarrow True].
40 rewrite > H.simplify.exact I.
43 definition notb : bool \to bool \def
46 [ true \Rightarrow false
47 | false \Rightarrow true ].
49 theorem notb_elim: \forall b:bool.\forall P:bool \to Prop.
51 [ true \Rightarrow P false
52 | false \Rightarrow P true] \to P (notb b).
53 intros 2.elim b.exact H. exact H.
56 (*CSC: the URI must disappear: there is a bug now *)
57 interpretation "boolean not" 'not x = (cic:/matita/datatypes/bool/notb.con x).
59 definition andb : bool \to bool \to bool\def
63 | false \Rightarrow false ].
65 theorem andb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
67 [ true \Rightarrow P b2
68 | false \Rightarrow P false] \to P (andb b1 b2).
69 intros 3.elim b1.exact H. exact H.
72 (*CSC: the URI must disappear: there is a bug now *)
73 interpretation "boolean and" 'and x y = (cic:/matita/datatypes/bool/andb.con x y).
75 definition orb : bool \to bool \to bool\def
78 [ true \Rightarrow true
79 | false \Rightarrow b2].
81 theorem orb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
83 [ true \Rightarrow P true
84 | false \Rightarrow P b2] \to P (orb b1 b2).
85 intros 3.elim b1.exact H. exact H.
88 (*CSC: the URI must disappear: there is a bug now *)
89 interpretation "boolean or" 'or x y = (cic:/matita/datatypes/bool/orb.con x y).
91 definition if_then_else : bool \to Prop \to Prop \to Prop \def
92 \lambda b:bool.\lambda P,Q:Prop.
95 | false \Rightarrow Q].
97 (*CSC: missing notation for if_then_else *)
99 theorem bool_to_decidable_eq:
100 \forall b1,b2:bool. decidable (b1=b2).
106 right. exact not_eq_true_false.
108 right. unfold Not. intro.
109 apply not_eq_true_false.
114 theorem P_x_to_P_x_to_eq:
115 \forall A:Set. \forall P: A \to bool.
116 \forall x:A. \forall p1,p2:P x = true. p1 = p2.
118 apply eq_to_eq_to_eq_p_q.
119 exact bool_to_decidable_eq.