blob: 4a253f28b77d4a6113577f348736e7d82ca65770 (
plain)
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
|
Require Import Crypto.Reflection.Syntax.
Require Import Crypto.Reflection.ExprInversion.
Require Import Crypto.Util.Sigma.
Require Import Crypto.Util.Option.
Require Import Crypto.Util.Equality.
Require Import Crypto.Util.Tactics.
Require Import Crypto.Util.Notations.
Section language.
Context {base_type_code : Type}
{op : flat_type base_type_code -> flat_type base_type_code -> Type}.
Local Notation flat_type := (flat_type base_type_code).
Local Notation type := (type base_type_code).
Local Notation exprf := (@exprf base_type_code op).
Local Notation expr := (@expr base_type_code op).
Local Notation Expr := (@Expr base_type_code op).
Local Notation wff := (@wff base_type_code op).
Local Notation wf := (@wf base_type_code op).
Local Notation Wf := (@Wf base_type_code op).
Section with_var.
Context {var1 var2 : base_type_code -> Type}.
Local Notation eP := (fun t => var1 t * var2 t)%type (only parsing).
Local Notation "x == y" := (existT eP _ (x, y)).
Definition wff_code (G : list (sigT eP)) {t} (e1 : @exprf var1 t) : forall (e2 : @exprf var2 t), Prop
:= match e1 in Syntax.exprf _ _ t return exprf t -> Prop with
| TT
=> fun e2
=> TT = e2
| Var t v1
=> fun e2
=> match invert_Var e2 with
| Some v2 => List.In (v1 == v2) G
| None => False
end
| Op t1 tR opc1 args1
=> fun e2
=> match invert_Op e2 with
| Some (existT t2 (opc2, args2))
=> { pf : t1 = t2
| eq_rect _ (fun t => op t tR) opc1 _ pf = opc2
/\ wff G (eq_rect _ exprf args1 _ pf) args2 }
| None => False
end
| LetIn tx1 ex1 tC1 eC1
=> fun e2
=> match invert_LetIn e2 with
| Some (existT tx2 (ex2, eC2))
=> { pf : tx1 = tx2
| wff G (eq_rect _ exprf ex1 _ pf) ex2
/\ (forall x1 x2,
wff (flatten_binding_list base_type_code x1 x2 ++ G)%list
(eC1 x1) (eC2 (eq_rect _ _ x2 _ pf))) }
| None => False
end
| Pair tx1 ex1 ty1 ey1
=> fun e2
=> match invert_Pair e2 with
| Some (ex2, ey2) => wff G ex1 ex2 /\ wff G ey1 ey2
| None => False
end
end.
Definition wff_encode {G t e1 e2} (v : @wff var1 var2 G t e1 e2) : @wff_code G t e1 e2.
Proof.
destruct v;
repeat first [ reflexivity
| assumption
| progress simpl
| exists eq_refl
| split ].
Defined.
Definition wff_decode {G t e1 e2} (v : @wff_code G t e1 e2) : @wff var1 var2 G t e1 e2.
Proof.
destruct e1;
repeat match goal with
| _ => progress simpl in *
| _ => progress subst
| _ => progress inversion_option
| [ H : Some _ = _ |- _ ] => symmetry in H
| [ H : invert_Var _ = _ |- _ ] => apply invert_Var_Some in H
| [ H : invert_Op _ = _ |- _ ] => apply invert_Op_Some in H
| [ H : invert_LetIn _ = _ |- _ ] => apply invert_LetIn_Some in H
| [ H : invert_Pair _ = _ |- _ ] => apply invert_Pair_Some in H
| _ => constructor
| _ => assumption
| _ => progress destruct_head False
| _ => progress destruct_head and
| _ => progress destruct_head sig
| _ => progress break_match_hyps
end.
Defined.
Definition wff_endecode {G t e1 e2} v : @wff_decode G t e1 e2 (@wff_encode G t e1 e2 v) = v.
Proof.
destruct v; reflexivity.
Qed.
Local Ltac t' :=
repeat first [ progress simpl in *
| progress subst
| reflexivity
| progress destruct_head and
| progress inversion_option
| intro
| progress break_match
| tauto ].
Definition wff_deencode {G t e1 e2} v : @wff_encode G t e1 e2 (@wff_decode G t e1 e2 v) = v.
Proof.
destruct e1; simpl in *;
move e2 at top;
lazymatch type of e2 with
| exprf Unit
=> subst; reflexivity
| exprf (Tbase ?t)
=> revert dependent t;
intros ? e2
| exprf (Prod ?A ?B)
=> revert dependent A;
intros ? e2;
move e2 at top;
revert dependent B;
intros ? e2
| exprf ?t
=> revert dependent t;
intros ? e2
end;
refine match e2 with
| TT => _
| _ => _
end;
t'.
Qed.
End with_var.
End language.
Ltac inversion_wff_step :=
match goal with
| [ H : wff _ ?x ?y |- _ ]
=> first [ is_var x; is_var y; fail 1
| idtac ];
apply wff_encode in H; unfold wff_code in H; simpl in H
end.
Ltac inversion_wff := repeat inversion_wff_step.
|
