1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Properties of decidable propositions *)
Definition decidable (P:Prop) := P \/ ~ P.
Theorem dec_not_not : forall P:Prop, decidable P -> (~ P -> False) -> P.
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_True : decidable True.
Proof.
unfold decidable; auto.
Qed.
Theorem dec_False : decidable False.
Proof.
unfold decidable, not; auto.
Qed.
Theorem dec_or :
forall A B:Prop, decidable A -> decidable B -> decidable (A \/ B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_and :
forall A B:Prop, decidable A -> decidable B -> decidable (A /\ B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_not : forall A:Prop, decidable A -> decidable (~ A).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_imp :
forall A B:Prop, decidable A -> decidable B -> decidable (A -> B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_iff :
forall A B:Prop, decidable A -> decidable B -> decidable (A<->B).
Proof.
unfold decidable; tauto.
Qed.
Theorem not_not : forall P:Prop, decidable P -> ~ ~ P -> P.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_or : forall A B:Prop, ~ (A \/ B) -> ~ A /\ ~ B.
Proof.
tauto.
Qed.
Theorem not_and : forall A B:Prop, decidable A -> ~ (A /\ B) -> ~ A \/ ~ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_imp : forall A B:Prop, decidable A -> ~ (A -> B) -> A /\ ~ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem imp_simp : forall A B:Prop, decidable A -> (A -> B) -> ~ A \/ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_iff :
forall A B:Prop, decidable A -> decidable B ->
~ (A <-> B) -> (A /\ ~ B) \/ (~ A /\ B).
Proof.
unfold decidable; tauto.
Qed.
(** Results formulated with iff, used in FSetDecide.
Negation are expanded since it is unclear whether setoid rewrite
will always perform conversion. *)
(** We begin with lemmas that, when read from left to right,
can be understood as ways to eliminate uses of [not]. *)
Theorem not_true_iff : (True -> False) <-> False.
Proof.
tauto.
Qed.
Theorem not_false_iff : (False -> False) <-> True.
Proof.
tauto.
Qed.
Theorem not_not_iff : forall A:Prop, decidable A ->
(((A -> False) -> False) <-> A).
Proof.
unfold decidable; tauto.
Qed.
Theorem contrapositive : forall A B:Prop, decidable A ->
(((A -> False) -> (B -> False)) <-> (B -> A)).
Proof.
unfold decidable; tauto.
Qed.
Lemma or_not_l_iff_1 : forall A B: Prop, decidable A ->
((A -> False) \/ B <-> (A -> B)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_l_iff_2 : forall A B: Prop, decidable B ->
((A -> False) \/ B <-> (A -> B)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_r_iff_1 : forall A B: Prop, decidable A ->
(A \/ (B -> False) <-> (B -> A)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_r_iff_2 : forall A B: Prop, decidable B ->
(A \/ (B -> False) <-> (B -> A)).
Proof.
unfold decidable. tauto.
Qed.
Lemma imp_not_l : forall A B: Prop, decidable A ->
(((A -> False) -> B) <-> (A \/ B)).
Proof.
unfold decidable. tauto.
Qed.
(** Moving Negations Around:
We have four lemmas that, when read from left to right,
describe how to push negations toward the leaves of a
proposition and, when read from right to left, describe
how to pull negations toward the top of a proposition. *)
Theorem not_or_iff : forall A B:Prop,
(A \/ B -> False) <-> (A -> False) /\ (B -> False).
Proof.
tauto.
Qed.
Lemma not_and_iff : forall A B:Prop,
(A /\ B -> False) <-> (A -> B -> False).
Proof.
tauto.
Qed.
Lemma not_imp_iff : forall A B:Prop, decidable A ->
(((A -> B) -> False) <-> A /\ (B -> False)).
Proof.
unfold decidable. tauto.
Qed.
Lemma not_imp_rev_iff : forall A B : Prop, decidable A ->
(((A -> B) -> False) <-> (B -> False) /\ A).
Proof.
unfold decidable. tauto.
Qed.
(** With the following hint database, we can leverage [auto] to check
decidability of propositions. *)
Hint Resolve dec_True dec_False dec_or dec_and dec_imp dec_not dec_iff
: decidable_prop.
(** [solve_decidable using lib] will solve goals about the
decidability of a proposition, assisted by an auxiliary
database of lemmas. The database is intended to contain
lemmas stating the decidability of base propositions,
(e.g., the decidability of equality on a particular
inductive type). *)
Tactic Notation "solve_decidable" "using" ident(db) :=
match goal with
| |- decidable _ =>
solve [ auto 100 with decidable_prop db ]
end.
Tactic Notation "solve_decidable" :=
solve_decidable using core.
|
