big_op.v 21.6 KB
Newer Older
1
From iris.algebra Require Export monoid.
2
From stdpp Require Export functions gmap gmultiset.
3
4
5
Set Default Proof Using "Type*".
Local Existing Instances monoid_ne monoid_assoc monoid_comm
  monoid_left_id monoid_right_id monoid_proper
6
7
  monoid_homomorphism_rel_po monoid_homomorphism_rel_proper
  monoid_homomorphism_op_proper
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
  monoid_homomorphism_ne weak_monoid_homomorphism_proper.

(** We define the following big operators with binders build in:

- The operator [ [^o list] k ↦ x ∈ l, P ] folds over a list [l]. The binder [x]
  refers to each element at index [k].
- The operator [ [^o map] k ↦ x ∈ m, P ] folds over a map [m]. The binder [x]
  refers to each element at index [k].
- The operator [ [^o set] x ∈ X, P ] folds over a set [X]. The binder [x] refers
  to each element.

Since these big operators are like quantifiers, they have the same precedence as
[∀] and [∃]. *)

(** * Big ops over lists *)
Fixpoint big_opL `{Monoid M o} {A} (f : nat  A  M) (xs : list A) : M :=
  match xs with
  | [] => monoid_unit
  | x :: xs => o (f 0 x) (big_opL (λ n, f (S n)) xs)
  end.
Instance: Params (@big_opL) 4.
Arguments big_opL {M} o {_ A} _ !_ /.
30
Typeclasses Opaque big_opL.
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
Notation "'[^' o 'list]' k ↦ x ∈ l , P" := (big_opL o (λ k x, P) l)
  (at level 200, o at level 1, l at level 10, k, x at level 1, right associativity,
   format "[^ o  list]  k ↦ x  ∈  l ,  P") : C_scope.
Notation "'[^' o 'list]' x ∈ l , P" := (big_opL o (λ _ x, P) l)
  (at level 200, o at level 1, l at level 10, x at level 1, right associativity,
   format "[^ o  list]  x  ∈  l ,  P") : C_scope.

Definition big_opM `{Monoid M o} `{Countable K} {A} (f : K  A  M)
    (m : gmap K A) : M := big_opL o (λ _, curry f) (map_to_list m).
Instance: Params (@big_opM) 7.
Arguments big_opM {M} o {_ K _ _ A} _ _ : simpl never.
Typeclasses Opaque big_opM.
Notation "'[^' o 'map]' k ↦ x ∈ m , P" := (big_opM o (λ k x, P) m)
  (at level 200, o at level 1, m at level 10, k, x at level 1, right associativity,
   format "[^  o  map]  k ↦ x  ∈  m ,  P") : C_scope.
Notation "'[^' o 'map]' x ∈ m , P" := (big_opM o (λ _ x, P) m)
  (at level 200, o at level 1, m at level 10, x at level 1, right associativity,
   format "[^ o  map]  x  ∈  m ,  P") : C_scope.

Definition big_opS `{Monoid M o} `{Countable A} (f : A  M)
  (X : gset A) : M := big_opL o (λ _, f) (elements X).
Instance: Params (@big_opS) 6.
Arguments big_opS {M} o {_ A _ _} _ _ : simpl never.
Typeclasses Opaque big_opS.
Notation "'[^' o 'set]' x ∈ X , P" := (big_opS o (λ x, P) X)
  (at level 200, o at level 1, X at level 10, x at level 1, right associativity,
   format "[^ o  set]  x  ∈  X ,  P") : C_scope.

Definition big_opMS `{Monoid M o} `{Countable A} (f : A  M)
  (X : gmultiset A) : M := big_opL o (λ _, f) (elements X).
Instance: Params (@big_opMS) 7.
Arguments big_opMS {M} o {_ A _ _} _ _ : simpl never.
Typeclasses Opaque big_opMS.
Notation "'[^' o 'mset]' x ∈ X , P" := (big_opMS o (λ x, P) X)
  (at level 200, o at level 1, X at level 10, x at level 1, right associativity,
   format "[^ o  mset]  x  ∈  X ,  P") : C_scope.

(** * Properties about big ops *)
Section big_op.
Context `{Monoid M o}.
Implicit Types xs : list M.
Infix "`o`" := o (at level 50, left associativity).

(** ** Big ops over lists *)
Section list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types f g : nat  A  M.

  Lemma big_opL_nil f : ([^o list] ky  [], f k y) = monoid_unit.
  Proof. done. Qed.
  Lemma big_opL_cons f x l :
    ([^o list] ky  x :: l, f k y) = f 0 x `o` [^o list] ky  l, f (S k) y.
  Proof. done. Qed.
  Lemma big_opL_singleton f x : ([^o list] ky  [x], f k y)  f 0 x.
  Proof. by rewrite /= right_id. Qed.
  Lemma big_opL_app f l1 l2 :
    ([^o list] ky  l1 ++ l2, f k y)
     ([^o list] ky  l1, f k y) `o` ([^o list] ky  l2, f (length l1 + k) y).
  Proof.
    revert f. induction l1 as [|x l1 IH]=> f /=; first by rewrite left_id.
    by rewrite IH assoc.
  Qed.

95
96
97
  Lemma big_opL_unit l : ([^o list] ky  l, monoid_unit)  (monoid_unit : M).
  Proof. induction l; rewrite /= ?left_id //. Qed.

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
  Lemma big_opL_forall R f g l :
    Reflexive R 
    Proper (R ==> R ==> R) o 
    ( k y, l !! k = Some y  R (f k y) (g k y)) 
    R ([^o list] k  y  l, f k y) ([^o list] k  y  l, g k y).
  Proof.
    intros ??. revert f g. induction l as [|x l IH]=> f g ? //=; f_equiv; eauto.
  Qed.

  Lemma big_opL_ext f g l :
    ( k y, l !! k = Some y  f k y = g k y) 
    ([^o list] k  y  l, f k y) = [^o list] k  y  l, g k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_opL_proper f g l :
    ( k y, l !! k = Some y  f k y  g k y) 
    ([^o list] k  y  l, f k y)  ([^o list] k  y  l, g k y).
  Proof. apply big_opL_forall; apply _. Qed.

  Lemma big_opL_permutation (f : A  M) l1 l2 :
    l1  l2  ([^o list] x  l1, f x)  ([^o list] x  l2, f x).
  Proof.
    induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto.
    - by rewrite IH.
    - by rewrite !assoc (comm _ (f x)).
    - by etrans.
  Qed.
  Global Instance big_opL_permutation' (f : A  M) :
    Proper (() ==> ()) (big_opL o (λ _, f)).
  Proof. intros xs1 xs2. apply big_opL_permutation. Qed.

  Global Instance big_opL_ne n :
    Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==>
            eq ==> dist n) (big_opL o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_opL_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> eq ==> ())
           (big_opL o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.

  Lemma big_opL_consZ_l (f : Z  A  M) x l :
    ([^o list] ky  x :: l, f k y) = f 0 x `o` [^o list] ky  l, f (1 + k)%Z y.
  Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed.
  Lemma big_opL_consZ_r (f : Z  A  M) x l :
    ([^o list] ky  x :: l, f k y) = f 0 x `o` [^o list] ky  l, f (k + 1)%Z y.
  Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed.

  Lemma big_opL_fmap {B} (h : A  B) (f : nat  B  M) l :
    ([^o list] ky  h <$> l, f k y)  ([^o list] ky  l, f k (h y)).
  Proof. revert f. induction l as [|x l IH]=> f; csimpl=> //. by rewrite IH. Qed.

  Lemma big_opL_opL f g l :
    ([^o list] kx  l, f k x `o` g k x)
     ([^o list] kx  l, f k x) `o` ([^o list] kx  l, g k x).
  Proof.
    revert f g; induction l as [|x l IH]=> f g /=; first by rewrite left_id.
    by rewrite IH -!assoc (assoc _ (g _ _)) [(g _ _ `o` _)]comm -!assoc.
  Qed.
End list.

(** ** Big ops over finite maps *)
Section gmap.
  Context `{Countable K} {A : Type}.
  Implicit Types m : gmap K A.
  Implicit Types f g : K  A  M.

  Lemma big_opM_forall R f g m :
    Reflexive R  Proper (R ==> R ==> R) o 
    ( k x, m !! k = Some x  R (f k x) (g k x)) 
    R ([^o map] k  x  m, f k x) ([^o map] k  x  m, g k x).
  Proof.
    intros ?? Hf. apply (big_opL_forall R); auto.
    intros k [i x] ?%elem_of_list_lookup_2. by apply Hf, elem_of_map_to_list.
  Qed.

  Lemma big_opM_ext f g m :
    ( k x, m !! k = Some x  f k x = g k x) 
    ([^o map] k  x  m, f k x) = ([^o map] k  x  m, g k x).
  Proof. apply big_opM_forall; apply _. Qed.
  Lemma big_opM_proper f g m :
    ( k x, m !! k = Some x  f k x  g k x) 
    ([^o map] k  x  m, f k x)  ([^o map] k  x  m, g k x).
  Proof. apply big_opM_forall; apply _. Qed.

  Global Instance big_opM_ne n :
    Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> eq ==> dist n)
           (big_opM o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opM_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_opM_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> eq ==> ())
           (big_opM o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opM_forall; apply _ || intros; apply Hf. Qed.

  Lemma big_opM_empty f : ([^o map] kx  , f k x) = monoid_unit.
  Proof. by rewrite /big_opM map_to_list_empty. Qed.

  Lemma big_opM_insert f m i x :
    m !! i = None 
    ([^o map] ky  <[i:=x]> m, f k y)  f i x `o` [^o map] ky  m, f k y.
  Proof. intros ?. by rewrite /big_opM map_to_list_insert. Qed.

  Lemma big_opM_delete f m i x :
    m !! i = Some x 
    ([^o map] ky  m, f k y)  f i x `o` [^o map] ky  delete i m, f k y.
  Proof.
    intros. rewrite -big_opM_insert ?lookup_delete //.
    by rewrite insert_delete insert_id.
  Qed.

  Lemma big_opM_singleton f i x : ([^o map] ky  {[i:=x]}, f k y)  f i x.
  Proof.
    rewrite -insert_empty big_opM_insert/=; last auto using lookup_empty.
    by rewrite big_opM_empty right_id.
  Qed.

212
213
214
  Lemma big_opM_unit m : ([^o map] ky  m, monoid_unit)  (monoid_unit : M).
  Proof. induction m using map_ind; rewrite /= ?big_opM_insert ?left_id //. Qed.

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
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
  Lemma big_opM_fmap {B} (h : A  B) (f : K  B  M) m :
    ([^o map] ky  h <$> m, f k y)  ([^o map] ky  m, f k (h y)).
  Proof.
    rewrite /big_opM map_to_list_fmap big_opL_fmap.
    by apply big_opL_proper=> ? [??].
  Qed.

  Lemma big_opM_insert_override (f : K  A  M) m i x x' :
    m !! i = Some x  f i x  f i x' 
    ([^o map] ky  <[i:=x']> m, f k y)  ([^o map] ky  m, f k y).
  Proof.
    intros ? Hx. rewrite -insert_delete big_opM_insert ?lookup_delete //.
    by rewrite -Hx -big_opM_delete.
  Qed.

  Lemma big_opM_fn_insert {B} (g : K  A  B  M) (f : K  B) m i (x : A) b :
    m !! i = None 
    ([^o map] ky  <[i:=x]> m, g k y (<[i:=b]> f k))
     g i x b `o` [^o map] ky  m, g k y (f k).
  Proof.
    intros. rewrite big_opM_insert // fn_lookup_insert.
    f_equiv; apply big_opM_proper; auto=> k y ?.
    by rewrite fn_lookup_insert_ne; last set_solver.
  Qed.
  Lemma big_opM_fn_insert' (f : K  M) m i x P :
    m !! i = None 
    ([^o map] ky  <[i:=x]> m, <[i:=P]> f k)  (P `o` [^o map] ky  m, f k).
  Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed.

  Lemma big_opM_opM f g m :
    ([^o map] kx  m, f k x `o` g k x)
     ([^o map] kx  m, f k x) `o` ([^o map] kx  m, g k x).
  Proof. rewrite /big_opM -big_opL_opL. by apply big_opL_proper=> ? [??]. Qed.
End gmap.


(** ** Big ops over finite sets *)
Section gset.
  Context `{Countable A}.
  Implicit Types X : gset A.
  Implicit Types f : A  M.

  Lemma big_opS_forall R f g X :
    Reflexive R  Proper (R ==> R ==> R) o 
    ( x, x  X  R (f x) (g x)) 
    R ([^o set] x  X, f x) ([^o set] x  X, g x).
  Proof.
    intros ?? Hf. apply (big_opL_forall R); auto.
    intros k x ?%elem_of_list_lookup_2. by apply Hf, elem_of_elements.
  Qed.

  Lemma big_opS_ext f g X :
    ( x, x  X  f x = g x) 
    ([^o set] x  X, f x) = ([^o set] x  X, g x).
  Proof. apply big_opS_forall; apply _. Qed.
  Lemma big_opS_proper f g X :
    ( x, x  X  f x  g x) 
    ([^o set] x  X, f x)  ([^o set] x  X, g x).
  Proof. apply big_opS_forall; apply _. Qed.

  Global Instance big_opS_ne n :
    Proper (pointwise_relation _ (dist n) ==> eq ==> dist n) (big_opS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opS_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_opS_proper' :
    Proper (pointwise_relation _ () ==> eq ==> ()) (big_opS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opS_forall; apply _ || intros; apply Hf. Qed.

  Lemma big_opS_empty f : ([^o set] x  , f x) = monoid_unit.
  Proof. by rewrite /big_opS elements_empty. Qed.

  Lemma big_opS_insert f X x :
    x  X  ([^o set] y  {[ x ]}  X, f y)  (f x `o` [^o set] y  X, f y).
  Proof. intros. by rewrite /big_opS elements_union_singleton. Qed.
  Lemma big_opS_fn_insert {B} (f : A  B  M) h X x b :
    x  X 
    ([^o set] y  {[ x ]}  X, f y (<[x:=b]> h y))
     f x b `o` [^o set] y  X, f y (h y).
  Proof.
    intros. rewrite big_opS_insert // fn_lookup_insert.
    f_equiv; apply big_opS_proper; auto=> y ?.
    by rewrite fn_lookup_insert_ne; last set_solver.
  Qed.
  Lemma big_opS_fn_insert' f X x P :
    x  X  ([^o set] y  {[ x ]}  X, <[x:=P]> f y)  (P `o` [^o set] y  X, f y).
  Proof. apply (big_opS_fn_insert (λ y, id)). Qed.

  Lemma big_opS_union f X Y :
    X  Y 
    ([^o set] y  X  Y, f y)  ([^o set] y  X, f y) `o` ([^o set] y  Y, f y).
  Proof.
    intros. induction X as [|x X ? IH] using collection_ind_L.
    { by rewrite left_id_L big_opS_empty left_id. }
    rewrite -assoc_L !big_opS_insert; [|set_solver..].
    by rewrite -assoc IH; last set_solver.
  Qed.

  Lemma big_opS_delete f X x :
    x  X  ([^o set] y  X, f y)  f x `o` [^o set] y  X  {[ x ]}, f y.
  Proof.
    intros. rewrite -big_opS_insert; last set_solver.
    by rewrite -union_difference_L; last set_solver.
  Qed.

  Lemma big_opS_singleton f x : ([^o set] y  {[ x ]}, f y)  f x.
  Proof. intros. by rewrite /big_opS elements_singleton /= right_id. Qed.

321
322
323
324
325
  Lemma big_opS_unit X : ([^o set] y  X, monoid_unit)  (monoid_unit : M).
  Proof.
    induction X using collection_ind_L; rewrite /= ?big_opS_insert ?left_id //.
  Qed.

326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
  Lemma big_opS_opS f g X :
    ([^o set] y  X, f y `o` g y)  ([^o set] y  X, f y) `o` ([^o set] y  X, g y).
  Proof. by rewrite /big_opS -big_opL_opL. Qed.
End gset.

Lemma big_opM_dom `{Countable K} {A} (f : K  M) (m : gmap K A) :
  ([^o map] k_  m, f k)  ([^o set] k  dom _ m, f k).
Proof.
  induction m as [|i x ?? IH] using map_ind; [by rewrite dom_empty_L|].
  by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom.
Qed.

(** ** Big ops over finite msets *)
Section gmultiset.
  Context `{Countable A}.
  Implicit Types X : gmultiset A.
  Implicit Types f : A  M.

  Lemma big_opMS_forall R f g X :
    Reflexive R  Proper (R ==> R ==> R) o 
    ( x, x  X  R (f x) (g x)) 
    R ([^o mset] x  X, f x) ([^o mset] x  X, g x).
  Proof.
    intros ?? Hf. apply (big_opL_forall R); auto.
    intros k x ?%elem_of_list_lookup_2. by apply Hf, gmultiset_elem_of_elements.
  Qed.

  Lemma big_opMS_ext f g X :
    ( x, x  X  f x = g x) 
    ([^o mset] x  X, f x) = ([^o mset] x  X, g x).
  Proof. apply big_opMS_forall; apply _. Qed.
  Lemma big_opMS_proper f g X :
    ( x, x  X  f x  g x) 
    ([^o mset] x  X, f x)  ([^o mset] x  X, g x).
  Proof. apply big_opMS_forall; apply _. Qed.

  Global Instance big_opMS_ne n :
    Proper (pointwise_relation _ (dist n) ==> eq ==> dist n) (big_opMS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opMS_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_opMS_proper' :
    Proper (pointwise_relation _ () ==> eq ==> ()) (big_opMS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opMS_forall; apply _ || intros; apply Hf. Qed.

  Lemma big_opMS_empty f : ([^o mset] x  , f x) = monoid_unit.
  Proof. by rewrite /big_opMS gmultiset_elements_empty. Qed.

  Lemma big_opMS_union f X Y :
    ([^o mset] y  X  Y, f y)  ([^o mset] y  X, f y) `o` [^o mset] y  Y, f y.
  Proof. by rewrite /big_opMS gmultiset_elements_union big_opL_app. Qed.

  Lemma big_opMS_singleton f x : ([^o mset] y  {[ x ]}, f y)  f x.
  Proof.
    intros. by rewrite /big_opMS gmultiset_elements_singleton /= right_id.
  Qed.

  Lemma big_opMS_delete f X x :
    x  X  ([^o mset] y  X, f y)  f x `o` [^o mset] y  X  {[ x ]}, f y.
  Proof.
    intros. rewrite -big_opMS_singleton -big_opMS_union.
    by rewrite -gmultiset_union_difference'.
  Qed.

388
389
390
391
392
393
  Lemma big_opMS_unit X : ([^o mset] y  X, monoid_unit)  (monoid_unit : M).
  Proof.
    induction X using gmultiset_ind;
      rewrite /= ?big_opMS_union ?big_opMS_singleton ?left_id //.
  Qed.

394
395
396
397
398
399
400
401
402
403
  Lemma big_opMS_opMS f g X :
    ([^o mset] y  X, f y `o` g y)  ([^o mset] y  X, f y) `o` ([^o mset] y  X, g y).
  Proof. by rewrite /big_opMS -big_opL_opL. Qed.
End gmultiset.
End big_op.

Section homomorphisms.
  Context `{Monoid M1 o1, Monoid M2 o2}.
  Infix "`o1`" := o1 (at level 50, left associativity).
  Infix "`o2`" := o2 (at level 50, left associativity).
404
  Instance foo {A} (R : relation A) : RewriteRelation R.
405

406
  Lemma big_opL_commute {A} (h : M1  M2) `{!MonoidHomomorphism o1 o2 R h}
407
      (f : nat  A  M1) l :
408
    R (h ([^o1 list] kx  l, f k x)) ([^o2 list] kx  l, h (f k x)).
409
410
  Proof.
    revert f. induction l as [|x l IH]=> f /=.
411
412
    - apply monoid_homomorphism_unit.
    - by rewrite monoid_homomorphism IH.
413
  Qed.
414
  Lemma big_opL_commute1 {A} (h : M1  M2) `{!WeakMonoidHomomorphism o1 o2 R h}
415
      (f : nat  A  M1) l :
416
    l  []  R (h ([^o1 list] kx  l, f k x)) ([^o2 list] kx  l, h (f k x)).
417
418
419
  Proof.
    intros ?. revert f. induction l as [|x [|x' l'] IH]=> f //.
    - by rewrite !big_opL_singleton.
420
    - by rewrite !(big_opL_cons _ x) monoid_homomorphism IH.
421
422
423
  Qed.

  Lemma big_opM_commute `{Countable K} {A} (h : M1  M2)
424
425
      `{!MonoidHomomorphism o1 o2 R h} (f : K  A  M1) m :
    R (h ([^o1 map] kx  m, f k x)) ([^o2 map] kx  m, h (f k x)).
426
427
428
429
430
431
  Proof.
    intros. induction m as [|i x m ? IH] using map_ind.
    - by rewrite !big_opM_empty monoid_homomorphism_unit.
    - by rewrite !big_opM_insert // monoid_homomorphism -IH.
  Qed.
  Lemma big_opM_commute1 `{Countable K} {A} (h : M1  M2)
432
433
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : K  A  M1) m :
    m    R (h ([^o1 map] kx  m, f k x)) ([^o2 map] kx  m, h (f k x)).
434
435
436
437
438
439
440
441
  Proof.
    intros. induction m as [|i x m ? IH] using map_ind; [done|].
    destruct (decide (m = )) as [->|].
    - by rewrite !big_opM_insert // !big_opM_empty !right_id.
    - by rewrite !big_opM_insert // monoid_homomorphism -IH //.
  Qed.

  Lemma big_opS_commute `{Countable A} (h : M1  M2)
442
443
      `{!MonoidHomomorphism o1 o2 R h} (f : A  M1) X :
    R (h ([^o1 set] x  X, f x)) ([^o2 set] x  X, h (f x)).
444
445
446
447
448
449
  Proof.
    intros. induction X as [|x X ? IH] using collection_ind_L.
    - by rewrite !big_opS_empty monoid_homomorphism_unit.
    - by rewrite !big_opS_insert // monoid_homomorphism -IH.
  Qed.
  Lemma big_opS_commute1 `{Countable A} (h : M1  M2)
450
451
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : A  M1) X :
    X    R (h ([^o1 set] x  X, f x)) ([^o2 set] x  X, h (f x)).
452
453
454
455
456
457
458
459
  Proof.
    intros. induction X as [|x X ? IH] using collection_ind_L; [done|].
    destruct (decide (X = )) as [->|].
    - by rewrite !big_opS_insert // !big_opS_empty !right_id.
    - by rewrite !big_opS_insert // monoid_homomorphism -IH //.
  Qed.

  Lemma big_opMS_commute `{Countable A} (h : M1  M2)
460
461
      `{!MonoidHomomorphism o1 o2 R h} (f : A  M1) X :
    R (h ([^o1 mset] x  X, f x)) ([^o2 mset] x  X, h (f x)).
462
463
464
465
466
467
  Proof.
    intros. induction X as [|x X IH] using gmultiset_ind.
    - by rewrite !big_opMS_empty monoid_homomorphism_unit.
    - by rewrite !big_opMS_union !big_opMS_singleton monoid_homomorphism -IH.
  Qed.
  Lemma big_opMS_commute1 `{Countable A} (h : M1  M2)
468
469
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : A  M1) X :
    X    R (h ([^o1 mset] x  X, f x)) ([^o2 mset] x  X, h (f x)).
470
471
472
473
474
475
476
477
478
479
  Proof.
    intros. induction X as [|x X IH] using gmultiset_ind; [done|].
    destruct (decide (X = )) as [->|].
    - by rewrite !big_opMS_union !big_opMS_singleton !big_opMS_empty !right_id.
    - by rewrite !big_opMS_union !big_opMS_singleton monoid_homomorphism -IH //.
  Qed.

  Context `{!LeibnizEquiv M2}.

  Lemma big_opL_commute_L {A} (h : M1  M2)
480
      `{!MonoidHomomorphism o1 o2 () h} (f : nat  A  M1) l :
481
482
483
    h ([^o1 list] kx  l, f k x) = ([^o2 list] kx  l, h (f k x)).
  Proof. unfold_leibniz. by apply big_opL_commute. Qed.
  Lemma big_opL_commute1_L {A} (h : M1  M2)
484
      `{!WeakMonoidHomomorphism o1 o2 () h} (f : nat  A  M1) l :
485
486
487
488
    l  []  h ([^o1 list] kx  l, f k x) = ([^o2 list] kx  l, h (f k x)).
  Proof. unfold_leibniz. by apply big_opL_commute1. Qed.

  Lemma big_opM_commute_L `{Countable K} {A} (h : M1  M2)
489
      `{!MonoidHomomorphism o1 o2 () h} (f : K  A  M1) m :
490
491
492
    h ([^o1 map] kx  m, f k x) = ([^o2 map] kx  m, h (f k x)).
  Proof. unfold_leibniz. by apply big_opM_commute. Qed.
  Lemma big_opM_commute1_L `{Countable K} {A} (h : M1  M2)
493
      `{!WeakMonoidHomomorphism o1 o2 () h} (f : K  A  M1) m :
494
495
496
497
    m    h ([^o1 map] kx  m, f k x) = ([^o2 map] kx  m, h (f k x)).
  Proof. unfold_leibniz. by apply big_opM_commute1. Qed.

  Lemma big_opS_commute_L `{Countable A} (h : M1  M2)
498
      `{!MonoidHomomorphism o1 o2 () h} (f : A  M1) X :
499
500
501
    h ([^o1 set] x  X, f x) = ([^o2 set] x  X, h (f x)).
  Proof. unfold_leibniz. by apply big_opS_commute. Qed.
  Lemma big_opS_commute1_L `{ Countable A} (h : M1  M2)
502
      `{!WeakMonoidHomomorphism o1 o2 () h} (f : A  M1) X :
503
504
505
506
    X    h ([^o1 set] x  X, f x) = ([^o2 set] x  X, h (f x)).
  Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opS_commute1. Qed.

  Lemma big_opMS_commute_L `{Countable A} (h : M1  M2)
507
      `{!MonoidHomomorphism o1 o2 () h} (f : A  M1) X :
508
509
510
    h ([^o1 mset] x  X, f x) = ([^o2 mset] x  X, h (f x)).
  Proof. unfold_leibniz. by apply big_opMS_commute. Qed.
  Lemma big_opMS_commute1_L `{Countable A} (h : M1  M2)
511
      `{!WeakMonoidHomomorphism o1 o2 () h} (f : A  M1) X :
512
513
514
    X    h ([^o1 mset] x  X, f x) = ([^o2 mset] x  X, h (f x)).
  Proof. intros. rewrite <-leibniz_equiv_iff. by apply big_opMS_commute1. Qed.
End homomorphisms.