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
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
|
(* -*- mode: coq; coq-prog-args: ("-nois" "-indices-matter" "-R" "." "Top" "-top" "bug_bad_univ_length_01") -*- *)
(* File reduced by coq-bug-finder from original input, then from 1199 lines to
430 lines, then from 444 lines to 430 lines, then from 964 lines to 255 lines,
then from 269 lines to 255 lines *)
(* coqc version 8.5 (January 2016) compiled on Jan 23 2016 16:15:22 with OCaml
4.01.0
coqtop version 8.5 (January 2016) *)
Declare ML Module "ltac_plugin".
Inductive False := .
Axiom proof_admitted : False.
Tactic Notation "admit" := case proof_admitted.
Require Coq.Init.Datatypes.
Import Coq.Init.Notations.
Global Set Universe Polymorphism.
Notation "A -> B" := (forall (_ : A), B) : type_scope.
Inductive True : Type :=
I : True.
Module Export Datatypes.
Set Implicit Arguments.
Notation nat := Coq.Init.Datatypes.nat.
Notation O := Coq.Init.Datatypes.O.
Notation S := Coq.Init.Datatypes.S.
Notation two := (S (S O)).
Record prod (A B : Type) := pair { fst : A ; snd : B }.
Notation "x * y" := (prod x y) : type_scope.
Open Scope nat_scope.
End Datatypes.
Module Export Specif.
Set Implicit Arguments.
Record sig {A} (P : A -> Type) := exist { proj1_sig : A ; proj2_sig : P
proj1_sig }.
Notation sigT := sig (only parsing).
Notation "{ x : A & P }" := (sigT (fun x:A => P)) : type_scope.
Notation projT1 := proj1_sig (only parsing).
Notation projT2 := proj2_sig (only parsing).
End Specif.
Definition Type1@{i} := Eval hnf in let gt := (Set : Type@{i}) in Type@{i}.
Definition Type2@{i j} := Eval hnf in let gt := (Type1@{j} : Type@{i}) in
Type@{i}.
Definition Type2le@{i j} := Eval hnf in let gt := (Set : Type@{i}) in
let ge := ((fun x => x) : Type1@{j} ->
Type@{i}) in Type@{i}.
Notation idmap := (fun x => x).
Delimit Scope function_scope with function.
Delimit Scope path_scope with path.
Delimit Scope fibration_scope with fibration.
Open Scope fibration_scope.
Open Scope function_scope.
Notation pr1 := projT1.
Notation pr2 := projT2.
Notation "x .1" := (pr1 x) (at level 3, format "x '.1'") : fibration_scope.
Notation "x .2" := (pr2 x) (at level 3, format "x '.2'") : fibration_scope.
Notation compose := (fun g f x => g (f x)).
Notation "g 'o' f" := (compose g%function f%function) (at level 40, left
associativity) : function_scope.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Definition inverse {A : Type} {x y : A} (p : x = y) : y = x
:= match p with idpath => idpath end.
Definition concat {A : Type} {x y z : A} (p : x = y) (q : y = z) : x = z :=
match p, q with idpath, idpath => idpath end.
Notation "p @ q" := (concat p%path q%path) (at level 20) : path_scope.
Notation "p ^" := (inverse p%path) (at level 3, format "p '^'") : path_scope.
Definition ap {A B:Type} (f:A -> B) {x y:A} (p:x = y) : f x = f y
:= match p with idpath => idpath end.
Definition pointwise_paths {A} {P:A->Type} (f g:forall x:A, P x)
:= forall x:A, f x = g x.
Notation "f == g" := (pointwise_paths f g) (at level 70, no associativity) :
type_scope.
Definition Sect {A B : Type} (s : A -> B) (r : B -> A) :=
forall x : A, r (s x) = x.
Class IsEquiv {A B : Type} (f : A -> B) := BuildIsEquiv {
equiv_inv : B -> A ;
eisretr : Sect equiv_inv f;
eissect : Sect f equiv_inv;
eisadj : forall x : A, eisretr (f x) = ap f (eissect x)
}.
Arguments eisretr {A B}%type_scope f%function_scope {_} _.
Arguments eissect {A B}%type_scope f%function_scope {_} _.
Notation "f ^-1" := (@equiv_inv _ _ f _) (at level 3, format "f '^-1'") :
function_scope.
Inductive Unit : Type1 :=
tt : Unit.
Local Open Scope path_scope.
Section EquivInverse.
Context {A B : Type} (f : A -> B) {feq : IsEquiv f}.
Theorem other_adj (b : B) : eissect f (f^-1 b) = ap f^-1 (eisretr f b).
admit.
Defined.
Global Instance isequiv_inverse : IsEquiv f^-1 | 10000
:= BuildIsEquiv B A f^-1 f (eissect f) (eisretr f) other_adj.
End EquivInverse.
Section Adjointify.
Context {A B : Type} (f : A -> B) (g : B -> A).
Context (isretr : Sect g f) (issect : Sect f g).
Let issect' := fun x =>
ap g (ap f (issect x)^) @ ap g (isretr (f x)) @ issect x.
Let is_adjoint' (a : A) : isretr (f a) = ap f (issect' a).
admit.
Defined.
Definition isequiv_adjointify : IsEquiv f
:= BuildIsEquiv A B f g isretr issect' is_adjoint'.
End Adjointify.
Definition ExtensionAlong {A B : Type} (f : A -> B)
(P : B -> Type) (d : forall x:A, P (f x))
:= { s : forall y:B, P y & forall x:A, s (f x) = d x }.
Fixpoint ExtendableAlong@{i j k l}
(n : nat) {A : Type@{i}} {B : Type@{j}}
(f : A -> B) (C : B -> Type@{k}) : Type@{l}
:= match n with
| O => Unit@{l}
| S n => (forall (g : forall a, C (f a)),
ExtensionAlong@{i j k l l} f C g) *
forall (h k : forall b, C b),
ExtendableAlong n f (fun b => h b = k b)
end.
Definition ooExtendableAlong@{i j k l}
{A : Type@{i}} {B : Type@{j}}
(f : A -> B) (C : B -> Type@{k}) : Type@{l}
:= forall n, ExtendableAlong@{i j k l} n f C.
Module Type ReflectiveSubuniverses.
Parameter ReflectiveSubuniverse@{u a} : Type2@{u a}.
Parameter O_reflector@{u a i} : forall (O : ReflectiveSubuniverse@{u a}),
Type2le@{i a} -> Type2le@{i a}.
Parameter In@{u a i} : forall (O : ReflectiveSubuniverse@{u a}),
Type2le@{i a} -> Type2le@{i a}.
Parameter O_inO@{u a i} : forall (O : ReflectiveSubuniverse@{u a}) (T :
Type@{i}),
In@{u a i} O (O_reflector@{u a i} O T).
Parameter to@{u a i} : forall (O : ReflectiveSubuniverse@{u a}) (T :
Type@{i}),
T -> O_reflector@{u a i} O T.
Parameter inO_equiv_inO@{u a i j k} :
forall (O : ReflectiveSubuniverse@{u a}) (T : Type@{i}) (U : Type@{j})
(T_inO : In@{u a i} O T) (f : T -> U) (feq : IsEquiv f),
let gei := ((fun x => x) : Type@{i} -> Type@{k}) in
let gej := ((fun x => x) : Type@{j} -> Type@{k}) in
In@{u a j} O U.
Parameter extendable_to_O@{u a i j k}
: forall (O : ReflectiveSubuniverse@{u a}) {P : Type2le@{i a}} {Q :
Type2le@{j a}} {Q_inO : In@{u a j} O Q},
ooExtendableAlong@{i i j k} (to O P) (fun _ => Q).
End ReflectiveSubuniverses.
Module ReflectiveSubuniverses_Theory (Os : ReflectiveSubuniverses).
Export Os.
Existing Class In.
Coercion O_reflector : ReflectiveSubuniverse >-> Funclass.
Arguments inO_equiv_inO {O} T {U} {_} f {_}.
Global Existing Instance O_inO.
Section ORecursion.
Context {O : ReflectiveSubuniverse}.
Definition O_indpaths {P Q : Type} {Q_inO : In O Q}
(g h : O P -> Q) (p : g o to O P == h o to O P)
: g == h
:= (fst (snd (extendable_to_O O two) g h) p).1.
Definition O_indpaths_beta {P Q : Type} {Q_inO : In O Q}
(g h : O P -> Q) (p : g o (to O P) == h o (to O P)) (x : P)
: O_indpaths g h p (to O P x) = p x
:= (fst (snd (extendable_to_O O two) g h) p).2 x.
End ORecursion.
Section Reflective_Subuniverse.
Universes Ou Oa.
Context (O : ReflectiveSubuniverse@{Ou Oa}).
Definition inO_isequiv_to_O (T:Type)
: IsEquiv (to O T) -> In O T
:= fun _ => inO_equiv_inO (O T) (to O T)^-1.
Definition inO_to_O_retract (T:Type) (mu : O T -> T)
: Sect (to O T) mu -> In O T.
Proof.
unfold Sect; intros H.
apply inO_isequiv_to_O.
apply isequiv_adjointify with (g:=mu).
-
refine (O_indpaths (to O T o mu) idmap _).
intros x; exact (ap (to O T) (H x)).
-
exact H.
Defined.
Definition inO_paths@{i} (S : Type@{i}) {S_inO : In@{Ou Oa i} O S} (x y :
S) : In@{Ou Oa i} O (x=y).
Proof.
simple refine (inO_to_O_retract@{i} _ _ _); intro u.
-
assert (p : (fun _ : O (x=y) => x) == (fun _=> y)).
{
refine (O_indpaths _ _ _); simpl.
intro v; exact v.
}
exact (p u).
-
hnf.
rewrite O_indpaths_beta; reflexivity.
Qed.
Check inO_paths@{Type}.
|
