]> matita.cs.unibo.it Git - helm.git/blob - matita/matita/contribs/lambdadelta/basic_2/etc/fpr/fpcs_cpcs.etc
- extended multiple substitutions now uses bounds in ynat (ie. they
[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / fpr / fpcs_cpcs.etc
1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "basic_2/computation/fprs_cprs.ma".
16 include "basic_2/equivalence/cpcs_cpcs.ma".
17 include "basic_2/equivalence/fpcs_fpcs.ma".
18
19 (* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
20
21 (* Advanced properties ******************************************************)
22
23 lemma fpcs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
24                         ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⬌* ⦃L2, ⓕ{I}V2.T2⦄.
25 #L1 #L2 #T1 #T2 #HT12
26 elim (fpcs_inv_fprs … HT12) -HT12
27 /3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *)
28 qed.
29
30 lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
31                   ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄.
32 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12
33 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
34 elim (fprs_bind2_minus … H1) -H1 #W1 #U1 #HTU1 #H destruct
35 elim (fprs_bind2_minus … H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
36 qed.
37
38 (* Advanced inversion lemmas ************************************************)
39
40 lemma fpcs_inv_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
41                       ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
42 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12
43 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
44 elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
45 elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
46 qed-.
47
48 (* Advanced forward lemmas **************************************************)
49
50 lemma fpcs_fwd_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
51                            ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
52 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
53 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
54 elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
55 elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
56 qed-.
57
58 lemma fpcs_fwd_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
59                       ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
60 /3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-.
61
62 lemma fpcs_fwd_abst24: ∀a,L1,L2,V1,V2,T1,T2. ⦃L1, ⓛ{a}V1.T1⦄ ⬌* ⦃L2, ⓛ{a}V2.T2⦄ →
63                        ∀b,I,W. ⦃L1, ⓑ{b,I}W.T1⦄ ⬌* ⦃L2, ⓑ{b,I}W.T2⦄.
64 #a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W
65 elim (fpcs_inv_fprs … H12) -H12 #L #U #H1 #H2
66 elim (fprs_fwd_abst2 … H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct
67 elim (fprs_fwd_abst2 … H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
68 qed-.
69
70 lemma fpcs_fwd_abst13: ∀L1,L2,V1,V2,T1,T2. ⦃L1.ⓛV1, T1⦄ ⬌* ⦃L2.ⓛV2, T2⦄ →
71                        ∀I,W. ⦃L1.ⓑ{I}W, T1⦄ ⬌* ⦃L2.ⓑ{I}W, T2⦄.
72 /4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-.
73
74 (* Properties on context-sensitive parallel equivalence for terms ***********)
75
76 lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
77 #L #T1 #T2 #H
78 elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
79 qed.
80
81 (* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
82
83 lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
84 #L #T1 #T2 #H
85 elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/
86 qed-.