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
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: ClassicalFacts.v,v 1.6.2.1 2004/07/16 19:31:06 herbelin Exp $ i*)
(** Some facts and definitions about classical logic *)
(** [prop_degeneracy] (also referred as propositional completeness) *)
(* asserts (up to consistency) that there are only two distinct formulas *)
Definition prop_degeneracy := forall A:Prop, A = True \/ A = False.
(** [prop_extensionality] asserts equivalent formulas are equal *)
Definition prop_extensionality := forall A B:Prop, (A <-> B) -> A = B.
(** [excluded_middle] asserts we can reason by case on the truth *)
(* or falsity of any formula *)
Definition excluded_middle := forall A:Prop, A \/ ~ A.
(** [proof_irrelevance] asserts equality of all proofs of a given formula *)
Definition proof_irrelevance := forall (A:Prop) (a1 a2:A), a1 = a2.
(** We show [prop_degeneracy <-> (prop_extensionality /\ excluded_middle)] *)
Lemma prop_degen_ext : prop_degeneracy -> prop_extensionality.
Proof.
intros H A B [Hab Hba].
destruct (H A); destruct (H B).
rewrite H1; exact H0.
absurd B.
rewrite H1; exact (fun H => H).
apply Hab; rewrite H0; exact I.
absurd A.
rewrite H0; exact (fun H => H).
apply Hba; rewrite H1; exact I.
rewrite H1; exact H0.
Qed.
Lemma prop_degen_em : prop_degeneracy -> excluded_middle.
Proof.
intros H A.
destruct (H A).
left; rewrite H0; exact I.
right; rewrite H0; exact (fun x => x).
Qed.
Lemma prop_ext_em_degen :
prop_extensionality -> excluded_middle -> prop_degeneracy.
Proof.
intros Ext EM A.
destruct (EM A).
left; apply (Ext A True); split;
[ exact (fun _ => I) | exact (fun _ => H) ].
right; apply (Ext A False); split; [ exact H | apply False_ind ].
Qed.
(** We successively show that:
[prop_extensionality]
implies equality of [A] and [A->A] for inhabited [A], which
implies the existence of a (trivial) retract from [A->A] to [A]
(just take the identity), which
implies the existence of a fixpoint operator in [A]
(e.g. take the Y combinator of lambda-calculus)
*)
Definition inhabited (A:Prop) := A.
Lemma prop_ext_A_eq_A_imp_A :
prop_extensionality -> forall A:Prop, inhabited A -> (A -> A) = A.
Proof.
intros Ext A a.
apply (Ext (A -> A) A); split; [ exact (fun _ => a) | exact (fun _ _ => a) ].
Qed.
Record retract (A B:Prop) : Prop :=
{f1 : A -> B; f2 : B -> A; f1_o_f2 : forall x:B, f1 (f2 x) = x}.
Lemma prop_ext_retract_A_A_imp_A :
prop_extensionality -> forall A:Prop, inhabited A -> retract A (A -> A).
Proof.
intros Ext A a.
rewrite (prop_ext_A_eq_A_imp_A Ext A a).
exists (fun x:A => x) (fun x:A => x).
reflexivity.
Qed.
Record has_fixpoint (A:Prop) : Prop :=
{F : (A -> A) -> A; Fix : forall f:A -> A, F f = f (F f)}.
Lemma ext_prop_fixpoint :
prop_extensionality -> forall A:Prop, inhabited A -> has_fixpoint A.
Proof.
intros Ext A a.
case (prop_ext_retract_A_A_imp_A Ext A a); intros g1 g2 g1_o_g2.
exists (fun f => (fun x:A => f (g1 x x)) (g2 (fun x => f (g1 x x)))).
intro f.
pattern (g1 (g2 (fun x:A => f (g1 x x)))) at 1 in |- *.
rewrite (g1_o_g2 (fun x:A => f (g1 x x))).
reflexivity.
Qed.
(** Assume we have booleans with the property that there is at most 2
booleans (which is equivalent to dependent case analysis). Consider
the fixpoint of the negation function: it is either true or false by
dependent case analysis, but also the opposite by fixpoint. Hence
proof-irrelevance.
We then map bool proof-irrelevance to all propositions.
*)
Section Proof_irrelevance_gen.
Variable bool : Prop.
Variable true : bool.
Variable false : bool.
Hypothesis bool_elim : forall C:Prop, C -> C -> bool -> C.
Hypothesis
bool_elim_redl : forall (C:Prop) (c1 c2:C), c1 = bool_elim C c1 c2 true.
Hypothesis
bool_elim_redr : forall (C:Prop) (c1 c2:C), c2 = bool_elim C c1 c2 false.
Let bool_dep_induction :=
forall P:bool -> Prop, P true -> P false -> forall b:bool, P b.
Lemma aux : prop_extensionality -> bool_dep_induction -> true = false.
Proof.
intros Ext Ind.
case (ext_prop_fixpoint Ext bool true); intros G Gfix.
set (neg := fun b:bool => bool_elim bool false true b).
generalize (refl_equal (G neg)).
pattern (G neg) at 1 in |- *.
apply Ind with (b := G neg); intro Heq.
rewrite (bool_elim_redl bool false true).
change (true = neg true) in |- *; rewrite Heq; apply Gfix.
rewrite (bool_elim_redr bool false true).
change (neg false = false) in |- *; rewrite Heq; symmetry in |- *;
apply Gfix.
Qed.
Lemma ext_prop_dep_proof_irrel_gen :
prop_extensionality -> bool_dep_induction -> proof_irrelevance.
Proof.
intros Ext Ind A a1 a2.
set (f := fun b:bool => bool_elim A a1 a2 b).
rewrite (bool_elim_redl A a1 a2).
change (f true = a2) in |- *.
rewrite (bool_elim_redr A a1 a2).
change (f true = f false) in |- *.
rewrite (aux Ext Ind).
reflexivity.
Qed.
End Proof_irrelevance_gen.
(** In the pure Calculus of Constructions, we can define the boolean
proposition bool = (C:Prop)C->C->C but we cannot prove that it has at
most 2 elements.
*)
Section Proof_irrelevance_CC.
Definition BoolP := forall C:Prop, C -> C -> C.
Definition TrueP : BoolP := fun C c1 c2 => c1.
Definition FalseP : BoolP := fun C c1 c2 => c2.
Definition BoolP_elim C c1 c2 (b:BoolP) := b C c1 c2.
Definition BoolP_elim_redl (C:Prop) (c1 c2:C) :
c1 = BoolP_elim C c1 c2 TrueP := refl_equal c1.
Definition BoolP_elim_redr (C:Prop) (c1 c2:C) :
c2 = BoolP_elim C c1 c2 FalseP := refl_equal c2.
Definition BoolP_dep_induction :=
forall P:BoolP -> Prop, P TrueP -> P FalseP -> forall b:BoolP, P b.
Lemma ext_prop_dep_proof_irrel_cc :
prop_extensionality -> BoolP_dep_induction -> proof_irrelevance.
Proof
ext_prop_dep_proof_irrel_gen BoolP TrueP FalseP BoolP_elim BoolP_elim_redl
BoolP_elim_redr.
End Proof_irrelevance_CC.
(** In the Calculus of Inductive Constructions, inductively defined booleans
enjoy dependent case analysis, hence directly proof-irrelevance from
propositional extensionality.
*)
Section Proof_irrelevance_CIC.
Inductive boolP : Prop :=
| trueP : boolP
| falseP : boolP.
Definition boolP_elim_redl (C:Prop) (c1 c2:C) :
c1 = boolP_ind C c1 c2 trueP := refl_equal c1.
Definition boolP_elim_redr (C:Prop) (c1 c2:C) :
c2 = boolP_ind C c1 c2 falseP := refl_equal c2.
Scheme boolP_indd := Induction for boolP Sort Prop.
Lemma ext_prop_dep_proof_irrel_cic : prop_extensionality -> proof_irrelevance.
Proof
fun pe =>
ext_prop_dep_proof_irrel_gen boolP trueP falseP boolP_ind boolP_elim_redl
boolP_elim_redr pe boolP_indd.
End Proof_irrelevance_CIC.
(** Can we state proof irrelevance from propositional degeneracy
(i.e. propositional extensionality + excluded middle) without
dependent case analysis ?
Conjecture: it seems possible to build a model of CC interpreting
all non-empty types by the set of all lambda-terms. Such a model would
satisfy propositional degeneracy without satisfying proof-irrelevance
(nor dependent case analysis). This would imply that the previous
results cannot be refined.
*)
|
