-
Notifications
You must be signed in to change notification settings - Fork 1
/
13-fixedpoints.mm1
73 lines (59 loc) · 3.12 KB
/
13-fixedpoints.mm1
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
import "01-propositional.mm1";
import "12-proof-system-p.mm1";
--- Helpers
-----------
theorem unfold_r (psi phi: Pattern X)
(p: $_Positive X psi$)
(h: $phi -> s[ mu X psi / X ] psi$):
$phi -> mu X psi$ =
'(syl (pre_fixpoint p) h );
theorem propag_floor (phi psi ctx: Pattern box):
$app[ phi /\ |_ psi _| / box ] ctx
-> app[ phi / box ] ctx /\ |_ psi _| $
= '(iand (framing anl) @ rsyl (framing anr) (rsyl lemma_56 (bi1 ceil_floor_floor)) );
--- Appendix G: Properties of Proof System H_mu
-----------------------------------------------
--- Appendix I: Notations and Proofs about Recursive Symbols
------------------------------------------------------------
theorem ctximp_in_ctx_forward {box: SVar} (ctx psi: Pattern box)
: $app[ (ctximp_app box ctx psi) / box ] ctx -> psi$
= (named '(rsyl (propag_exists eFresh_disjoint) @ exists_generalization eFresh_disjoint @ rsyl propag_floor @ impcom @ rsyl corollary_57_floor id));
theorem wrap_lemma {x: EVar} (phi psi: Pattern x)
(h: $ (x in phi) -> psi $):
$ (x in phi) -> |_ psi _| $ =
'(syl (con3 @ framing_def @ con3 h) @ con2 @ rsyl (framing_def membership_not_reverse)
(rsyl (norm (norm_imp
(norm_trans appCtxR_disjoint @ norm_app norm_refl @ norm_trans appCtxR_disjoint @ norm_app norm_refl appCtxVar)
norm_refl) (! lemma_56 box)) (con2 @ dne singletonDef)));
theorem unwrap_lemma {x y: EVar} {box: SVar} (ctx phi: Pattern box):
$ ((y in eVar x) /\ y in |_ app[ eVar x / box ] ctx -> phi _|) -> |_ (app[ eVar y / box ] ctx) -> phi _| $ =
(named '(curry @ rsyl membership_var_forward @ syl (imim1i mem_floor_forward) @
con3d @ norm (norm_imp_r @ norm_imp defNorm defNorm) (lemma_14 @ con3d @ rsyl eq_to_framing imim1)));
--- lemma 91 (Specialized to Application contexts)
theorem unwrap {box: SVar} (ctx phi psi: Pattern box)
(h: $ phi -> (ctximp_app box ctx psi) $):
$ (app[ phi / box ] ctx) -> psi $ =
(named '( imim1 (syl (rsyl propag_exists_disjoint @ exists_framing lemma_60_forward)
@ framing lemma_62_b_reverse)
@ exists_generalization_disjoint @ impcom (imim2 corollary_57_floor (
rsyl (membership_imp_forward @ con1 exists_intro_same_var @ membership_intro h)
@ rsyl membership_exists_forward
@ rsyl (exists_framing membership_and_forward)
@ exists_generalization_disjoint unwrap_lemma
))));
theorem wrap {box: SVar} (ctx psi: Pattern box) (phi: Pattern box)
(h: $ (app[ phi / box ] ctx) -> psi $):
$ phi -> (ctximp_app box ctx psi) $ =
(named '(rsyl lemma_62_b_reverse @ exists_framing @ anim id
(wrap_lemma @ expcom @ rsyl exists_intro_same_var (imim1i (rsyl (exists_framing lemma_60_reverse) prop_43_exists) (rsyl (framing (rsyl (exists_framing ancom) lemma_62_forward)) h)))
));
theorem wrap_subst {box: SVar} (ctx phi psi rho: Pattern box)
(subst: $ Norm (app[ phi / box ] ctx) rho $)
(h: $ rho -> psi $):
$ phi -> (ctximp_app box ctx psi) $ =
'(wrap @ norm (norm_sym @ norm_imp_l subst) h);
theorem unwrap_subst {box: SVar} (ctx phi psi rho: Pattern box)
(subst: $ Norm (app[ phi / box ] ctx) rho $)
(h: $ phi -> (ctximp_app box ctx psi) $):
$ rho -> psi $ =
'(norm (norm_imp_l subst) @ unwrap h);