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
|
import Mathlib
import Nominal.Support
import Nominal.Equivariance
/-!
# Basic Nominal Sets
This file contains common instances of nominal sets and their smallest supports.
-/
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
/-!
## Atoms
The set 𝔸 of atoms itself is a nominal set, where supp a = {a}.
-/
section Atoms
private lemma NominalSet.supp_atom' {a : 𝔸} : PermSet.HasSupport 𝔸 a {a} := by
simp [PermSet.HasSupport, PermSet.fix, PermSet.fix']
instance : NominalSet 𝔸 𝔸 where
fin_supp := by
intro a; use {a}
exact ⟨NominalSet.supp_atom' 𝔸, Set.finite_singleton a⟩
@[simp] lemma NominalSet.supp_atom {a : 𝔸} : supp 𝔸 a = {a} := by
ext b; constructor
· apply support_subset_supp 𝔸
· exact NominalSet.supp_atom' 𝔸
· exact Set.finite_singleton a
· suffices a ∈ supp 𝔸 a by
simp only [Set.mem_singleton_iff]
intro h
rwa [h]
intro S ⟨hsuppS, hfinS⟩
by_contra ha
-- Show that we can find a fresh c with (a c) ∈ fix S
obtain ⟨c, hc⟩ := Infinite.exists_notMem_finset ({a} ∪ hfinS.toFinset)
simp only [Finset.singleton_union, Finset.mem_insert, Set.Finite.mem_toFinset, not_or] at hc
have hfix : ⸨a, c⸩ ∈ PermSet.fix 𝔸 S := by
intro d hd
apply FPerm.swap_apply_of_ne_of_ne (ne_of_mem_of_not_mem hd ha) (ne_of_mem_of_not_mem hd hc.right)
have h₁ : ⸨a, c⸩ • a = c := by rw [PermSet.atom_swap, Equiv.swap_apply_left]
have h₂ : ⸨a, c⸩ • a = a := by apply hsuppS; assumption
rw [h₁] at h₂
exact hc.left h₂
@[simp] lemma NominalSet.fresh_atom {a b : 𝔸} : a # b ↔ a ≠ b := by
simp [IsFresh]
end Atoms
/-!
## Discrete Nominal Sets
Every set can be made into a discrete nominal set, where supp x = ∅.
-/
section Discrete
def NominalSet.discrete (X : Type) : NominalSet.Discrete 𝔸 X where
smul π x := x
one_smul := by intro; rfl
mul_smul := by intro _ _ _; rfl
is_discrete := by intro _ _; rfl
fin_supp := by
intro x; use {}
constructor
· intro π hπ; simp only [PermSet.fix', Set.mem_setOf_eq]; rfl
· exact Set.finite_empty
@[simp] lemma NominalSet.Discrete.supp {X : Type} [NominalSet.Discrete 𝔸 X] {x : X} : supp 𝔸 x = ∅ := by
rw [← Set.subset_empty_iff]
apply Set.sInter_subset_of_mem
constructor
· intro π hπ; simp [PermSet.fix', is_discrete]
· exact Set.finite_empty
@[simp] lemma NominalSet.Discrete.fresh {X : Type} [NominalSet.Discrete 𝔸 X] {a : 𝔸} {x : X} : a # x := by
simp [IsFresh]
/-- We consider Unit a discrete nominal set. -/
instance : NominalSet.Discrete 𝔸 Unit := NominalSet.discrete 𝔸 Unit
/-- We consider Bool a discrete nominal set. -/
instance : NominalSet.Discrete 𝔸 Bool := NominalSet.discrete 𝔸 Bool
end Discrete
/-!
## Products
Pairs of two nominal sets also form a nominal set, where supp (x, y) = supp x ∪ supp y.
-/
section Products
namespace Prod
instance {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] : PermSet 𝔸 (X × Y) where
variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
private lemma supp' {x : X} {y : Y} : PermSet.HasSupport 𝔸 (x, y) (NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y) := by
intro π hπ
obtain ⟨hπ₁, hπ₂⟩ := PermSet.fix_union 𝔸 hπ
exact Prod.mk_inj.mpr ⟨NominalSet.supp_is_support 𝔸 hπ₁, NominalSet.supp_is_support 𝔸 hπ₂⟩
instance : NominalSet 𝔸 (X × Y) where
fin_supp := by
intro ⟨x, y⟩; use NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y
exact ⟨supp' 𝔸, Set.Finite.union (NominalSet.supp_finite 𝔸) (NominalSet.supp_finite 𝔸)⟩
@[simp] lemma supp {x : X} {y : Y} : NominalSet.supp 𝔸 (x, y) = NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp' 𝔸
· exact Set.Finite.union (NominalSet.supp_finite 𝔸) (NominalSet.supp_finite 𝔸)
· intro hb; cases hb with
| inl hx =>
intro S hS
suffices PermSet.HasSupport 𝔸 x S by apply hx; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (x, y) by
simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
exact this.left
exact (hS.out.left hπ).out
| inr hy =>
intro S hS
suffices PermSet.HasSupport 𝔸 y S by apply hy; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (x, y) by
simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
exact this.right
exact (hS.out.left hπ).out
@[simp] lemma fresh {a : 𝔸} {x : X} {y : Y} : a # (x, y) ↔ a # x ∧ a # y := by
simp [NominalSet.IsFresh]
lemma supp_comm {x : X} {y : Y} : NominalSet.supp 𝔸 (x, y) = NominalSet.supp 𝔸 (y, x) := by
simp; apply Set.union_comm
lemma fst_equivariant : Equivariant 𝔸 (Prod.fst : X × Y → X) := by
simp [Equivariant]
lemma snd_equivariant : Equivariant 𝔸 (Prod.snd : X × Y → Y) := by
simp [Equivariant]
end Prod
end Products
/-!
## Coproducts
Sums of two nominal sets also form a nominal set, where supp (inl x) = supp x and supp (inr y) = supp y.
-/
section Coproducts
namespace Sum
instance {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] : PermSet 𝔸 (X ⊕ Y) where
variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
private lemma supp_inl' {x : X} : PermSet.HasSupport 𝔸 (.inl x : X ⊕ Y) (NominalSet.supp 𝔸 x) := by
intro π hπ
simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq]
exact NominalSet.supp_is_support 𝔸 hπ
private lemma supp_inr' {y : Y} : PermSet.HasSupport 𝔸 (.inr y : X ⊕ Y) (NominalSet.supp 𝔸 y) := by
intro π hπ
simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq]
exact NominalSet.supp_is_support 𝔸 hπ
instance : NominalSet 𝔸 (X ⊕ Y) where
fin_supp := by
intro xy
cases xy with
| inl x => use NominalSet.supp 𝔸 x; exact ⟨supp_inl' 𝔸, NominalSet.supp_finite 𝔸⟩
| inr y => use NominalSet.supp 𝔸 y; exact ⟨supp_inr' 𝔸, NominalSet.supp_finite 𝔸⟩
@[simp] lemma supp_inl {x : X} : NominalSet.supp 𝔸 (.inl x : X ⊕ Y) = NominalSet.supp 𝔸 x := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp_inl' 𝔸
· exact NominalSet.supp_finite 𝔸
· intro hb S hS
suffices PermSet.HasSupport 𝔸 x S by apply hb; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (.inl x : X ⊕ Y) by
simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq] at this
exact this
exact (hS.out.left hπ).out
@[simp] lemma supp_inr {y : Y} : NominalSet.supp 𝔸 (.inr y : X ⊕ Y) = NominalSet.supp 𝔸 y := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp_inr' 𝔸
· exact NominalSet.supp_finite 𝔸
· intro hb S hS
suffices PermSet.HasSupport 𝔸 y S by apply hb; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (.inr y : X ⊕ Y) by
simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq] at this
exact this
exact (hS.out.left hπ).out
@[simp] lemma fresh_inl {a : 𝔸} {x : X} : a # (.inl x : X ⊕ Y) ↔ a # x := by
simp [NominalSet.IsFresh]
@[simp] lemma fresh_inr {a : 𝔸} {y : Y} : a # (.inr y : X ⊕ Y) ↔ a # y := by
simp [NominalSet.IsFresh]
lemma inl_equivariant : Equivariant 𝔸 (Sum.inl : X → X ⊕ Y) := by
simp [Equivariant]
lemma inr_equivariant : Equivariant 𝔸 (Sum.inr : Y → X ⊕ Y) := by
simp [Equivariant]
end Sum
end Coproducts
|
