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
|
import Mathlib
import Nominal.Permutations
import Nominal.Support
import Nominal.Equivariance
import Nominal.Instances.Basic
import Nominal.Helpers.CountableString
/-!
# λ-Preterms
As a practical example for nominal sets, this file defines λ-preterms (that is, λ-terms without α-equivalence).
For any preterm t, supp t is exactly the set of variables occurring in t (either bound or free).
-/
namespace Lambda
/-!
## Basic Syntax Definition
Define the AST and a parser for it.
-/
section Syntax
/-- We will use strings as the set of atoms, representing variable names. -/
def Var := String
deriving DecidableEq, Countable, Infinite
/-- Basic λ-preterm definition.
We furthermore allow defining λ-terms as ⟫ t ⟪ with
t, u ::= x | λ x y ..., t | t u,
where x, y, ... are variables. -/
inductive Preterm
| var : Var → Preterm
| abs : Var → Preterm → Preterm
| app : Preterm → Preterm → Preterm
deriving DecidableEq
/-- The set of variables mentioned by a preterm. -/
def Preterm.vars : Preterm → Set Var
| .var x => {x}
| .abs x t => {x} ∪ vars t
| .app t u => vars t ∪ vars u
/-- The set of free variables in a preterm. -/
def Preterm.fv : Preterm → Set Var
| .var x => {x}
| .abs x t => fv t \ {x}
| .app t u => fv t ∪ fv u
open Lean Elab Meta
declare_syntax_cat lam
syntax ident : lam
syntax "(" lam ")" : lam
syntax:10 (name := stxabs) "λ " ident+ ", " lam : lam
syntax:45 lam:45 lam:46 : lam
partial def elabLam : Syntax → MetaM Expr
| `(lam | $x:ident) => mkAppM ``Preterm.var #[mkStrLit x.getId.toString]
| `(lam | ($t:lam)) => elabLam t
| `(lam | $t:lam $u:lam) => do mkAppM ``Preterm.app #[← elabLam t, ← elabLam u]
| `(lam | λ $xs:ident*, $t:lam) => do
xs.foldrM (fun x t' => mkAppM ``Preterm.abs #[mkStrLit x.getId.toString, t']) (← elabLam t)
| _ => throwUnsupportedSyntax
elab "⟫ " t:lam " ⟪" : term => elabLam t
example : ⟫ x y z ⟪ = ⟫ (x y) z ⟪ := by decide
example : ⟫ λ x, λ y, x ⟪ = ⟫ λ x y, x ⟪ := by decide
end Syntax
/-!
## Nominal Set Instance
The set of preterms forms a nominal set, where the permutation action just renames all occurrences of a variable.
-/
section NominalSet
namespace Preterm
private def rename (π : FPerm Var) : Preterm → Preterm
| .var x => .var (π • x)
| .abs x t => .abs (π • x) (rename π t)
| .app t u => .app (rename π t) (rename π u)
private lemma one_rename (t : Preterm) : rename 1 t = t := by
induction' t <;> all_goals simp [rename, *]
private lemma mul_rename (π τ : FPerm Var) (t : Preterm) : rename (π * τ) t = rename π (rename τ t) := by
induction' t <;> all_goals simp [rename, mul_smul, *]
instance : PermSet Var Preterm where
smul := rename
one_smul := one_rename
mul_smul := mul_rename
@[simp] lemma smul_var {π : FPerm Var} {x : Var} : π • Preterm.var x = Preterm.var (π • x) := rfl
@[simp] lemma smul_abs {π : FPerm Var} {x : Var} {t : Preterm} : π • Preterm.abs x t = Preterm.abs (π • x) (π • t) := rfl
@[simp] lemma smul_app {π : FPerm Var} {t u : Preterm} : π • Preterm.app t u = Preterm.app (π • t) (π • u) := rfl
instance : NominalSet Var Preterm where
fin_supp t := by
use vars t
induction' t <;> all_goals expose_names
· constructor
· simp [PermSet.HasSupport, PermSet.fix, vars, PermSet.fix']
· simp [vars]
· constructor
· intro π hπ
simp only [PermSet.fix', smul_abs, Preterm.abs.injEq, Set.mem_setOf_eq]
simp only [PermSet.fix, vars, Set.singleton_union, Set.mem_insert_iff, forall_eq_or_imp, Set.mem_setOf_eq] at hπ
exact ⟨hπ.left, a_ih.left hπ.right⟩
· simp [vars, Set.Finite.insert, a_ih]
· constructor
· intro π hπ
simp only [PermSet.fix', smul_app, Preterm.app.injEq, Set.mem_setOf_eq]
simp only [PermSet.fix, vars, Set.mem_union, Set.mem_setOf_eq] at hπ
constructor
· exact a_ih.left (fun x hx => hπ x (Or.inl hx))
· exact a_ih_1.left (fun x hx => hπ x (Or.inr hx))
· simp [vars, a_ih, a_ih_1]
/-- It's easiest to prove the least support via an equivalence between preterms and a well-known nominal type. -/
private def equiv : EquivariantEquiv Var Preterm (Var ⊕ (Var × Preterm) ⊕ (Preterm × Preterm)) :=
{ toFun
| .var x => .inl x
| .abs x t => .inr (.inl ⟨x, t⟩)
| .app t u => .inr (.inr ⟨t, u⟩)
invFun
| .inl x => .var x
| .inr (.inl x) => .abs x.fst x.snd
| .inr (.inr x) => .app x.fst x.snd
left_inv := by intro x; cases x <;> all_goals simp
right_inv := by
intro x
cases x with
| inl _ => simp
| inr x' => cases x' <;> all_goals simp
toFun_equivariant := by
intro t π
cases t <;> all_goals simp
}
@[simp] lemma equiv_var {x : Var} : equiv.toFun (Preterm.var x) = .inl x := rfl
@[simp] lemma equiv_abs {x : Var} {t : Preterm} : equiv.toFun (Preterm.abs x t) = .inr (.inl (x, t)) := rfl
@[simp] lemma equiv_app {t u : Preterm} : equiv.toFun (Preterm.app t u) = .inr (.inr (t, u)) := rfl
/-- The support of a preterm consists exactly of the variables mentioned by it. -/
lemma supp {t : Preterm} : NominalSet.supp Var t = vars t := by
fun_induction vars
all_goals rw [EquivariantEquiv.supp_eq Var equiv]; simp [*]
end Preterm
end NominalSet
/-!
## Variables
Helpers for reasoning about variables in a preterm.
-/
section Variables
namespace Preterm
/-- The free variables are a subset of all variables in a preterm. -/
lemma fv_subset_vars (t : Preterm) : t.fv ⊆ t.vars := by
induction' t <;> all_goals grind [Preterm.vars, Preterm.fv]
/-- The set of variables in a preterm is finite. -/
lemma vars_finite (t : Preterm) : t.vars.Finite := by
rw [← Preterm.supp]
exact NominalSet.supp_finite Var
/-- The set of free variables in a preterm is finite. -/
lemma fv_finite (t : Preterm) : t.fv.Finite := by
apply Set.Finite.subset (vars_finite t) (fv_subset_vars t)
/-- The vars map is equivariant. -/
lemma vars_equivariant : Function.Equivariant Var vars := by
intro t π
fun_induction vars <;> all_goals simp [vars, *, Set.smul_set_insert, Set.smul_set_union]
/-- The fv map is equivariant. -/
lemma fv_equivariant : Function.Equivariant Var fv := by
intro t π
fun_induction fv <;> all_goals simp [fv, *, Set.smul_set_sdiff, Set.smul_set_union]
end Preterm
end Variables
end Lambda
|
