-
Notifications
You must be signed in to change notification settings - Fork 3
/
ErasedNat.v
165 lines (147 loc) · 4.05 KB
/
ErasedNat.v
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
From Equations Require Import Equations.
Require Import Coq.Lists.List.
Require Import Coq.Arith.PeanoNat.
Require Export SystemFR.ErasedArrow.
Opaque reducible_values.
Opaque makeFresh.
Lemma reducible_value_zero:
forall ρ, [ ρ ⊨ zero : T_nat ]v.
Proof.
repeat step || simp_red.
Qed.
Lemma reducible_zero:
forall ρ, [ ρ ⊨ zero : T_nat ].
Proof.
repeat step || simp_red || unfold reduces_to || eexists || constructor.
Qed.
Lemma open_reducible_zero:
forall Θ Γ,
[ Θ; Γ ⊨ zero : T_nat ].
Proof.
unfold open_reducible; steps;
auto using reducible_zero.
Qed.
Lemma reducible_values_succ:
forall ρ v,
[ ρ ⊨ v : T_nat ]v ->
[ ρ ⊨ succ v : T_nat ]v.
Proof.
repeat step || simp_red; eauto with is_nat_value.
Qed.
Lemma reducible_succ:
forall ρ t,
valid_interpretation ρ ->
[ ρ ⊨ t : T_nat ] ->
[ ρ ⊨ succ t : T_nat ].
Proof.
unfold reduces_to; steps.
exists (succ v); repeat step || simp_red; eauto with cbvlemmas;
eauto with is_nat_value.
Qed.
Lemma reducible_nat_value:
forall ρ v,
is_nat_value v ->
valid_interpretation ρ ->
[ ρ ⊨ v : T_nat ]v.
Proof.
induction 1; repeat step;
eauto using reducible_value_zero;
eauto using reducible_values_succ.
Qed.
Lemma reducible_nat:
forall ρ v,
is_nat_value v ->
valid_interpretation ρ ->
[ ρ ⊨ v : T_nat ].
Proof.
induction 1; repeat step;
eauto using reducible_zero;
eauto using reducible_succ.
Qed.
Lemma open_reducible_succ:
forall Θ Γ t,
[ Θ; Γ ⊨ t : T_nat ] ->
[ Θ; Γ ⊨ succ t : T_nat ].
Proof.
unfold open_reducible in *; steps;
eauto using reducible_succ.
Qed.
Lemma reducible_match:
forall ρ tn t0 ts T,
fv ts = nil ->
fv t0 = nil ->
wf t0 0 ->
wf ts 1 ->
is_erased_term t0 ->
is_erased_term ts ->
valid_interpretation ρ ->
[ ρ ⊨ tn : T_nat ] ->
([ tn ≡ zero ] -> [ ρ ⊨ t0 : T ] ) ->
(forall n,
[ tn ≡ succ n ] ->
[ ρ ⊨ n : T_nat ]v ->
[ ρ ⊨ open 0 ts n : T ]) ->
[ ρ ⊨ tmatch tn t0 ts : T ].
Proof.
steps.
unfold reduces_to in H6; steps.
eapply star_backstep_reducible with (tmatch v t0 ts);
repeat step || list_utils || simp_red; t_closer;
eauto with cbvlemmas.
t_invert_nat_value; steps.
- (* zero *)
eapply backstep_reducible; eauto with smallstep;
repeat step || list_utils || apply_any; eauto with fv;
try solve [ eapply equivalent_star; steps; t_closer ].
- (* succ v *)
apply backstep_reducible with (open 0 ts v0);
repeat step || list_utils || apply reducible_nat_value ||
match goal with
| H: forall n, _ -> _ -> [ _ ⊨ _ : _ ] |- _ => apply H
end;
eauto 4 with smallstep values;
auto 2 with fv;
eauto 2 with wf;
eauto with erased;
try solve [ eapply equivalent_star; steps; t_closer ].
Qed.
Lemma open_reducible_match:
forall Θ tn t0 ts Γ T n p,
wf ts 1 ->
wf t0 0 ->
subset (fv ts) (support Γ) ->
subset (fv t0) (support Γ) ->
~(p ∈ fv tn) ->
~(p ∈ fv T) ->
~(p ∈ fv_context Γ) ->
~(n ∈ fv tn) ->
~(n ∈ fv ts) ->
~(n ∈ fv T) ->
~(n ∈ fv_context Γ) ->
~(p = n) ->
is_erased_term t0 ->
is_erased_term ts ->
[ Θ; Γ ⊨ tn : T_nat ] ->
[ Θ; (p, T_equiv tn zero) :: Γ ⊨ t0 : T ] ->
[ Θ;
(p, T_equiv tn (succ (fvar n term_var))) ::
(n, T_nat) ::
Γ ⊨
open 0 ts (fvar n term_var) : T ] ->
[ Θ; Γ ⊨ tmatch tn t0 ts : T ].
Proof.
unfold open_reducible; repeat step || t_instantiate_sat3.
apply reducible_match; repeat step || t_termlist;
eauto with wf;
eauto using subset_same with fv;
eauto with erased.
- (* zero *)
unshelve epose proof (H14 ρ ((p, uu) :: lterms) _ _ _);
repeat step || apply SatCons || simp_red || t_substitutions;
t_closer.
- (* successor *)
unshelve epose proof (H15 ρ ((p, uu) :: (n,n0) :: lterms) _ _ _);
repeat step || apply SatCons || simp_red || t_substitutions;
t_closer;
eauto with twf.
Qed.