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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/predtysn_5.ma".
16 include "basic_2/static/lfxs.ma".
17 include "basic_2/rt_transition/cpx.ma".
19 (* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
21 definition lfpx: sh → genv → relation3 term lenv lenv ≝
25 "uncounted parallel rt-transition on referred entries (local environment)"
26 'PRedTySn h T G L1 L2 = (lfpx h G T L1 L2).
28 (* Basic properties ***********************************************************)
30 lemma lfpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
31 /2 width=1 by lfxs_atom/ qed.
33 lemma lfpx_sort: ∀h,I,G,L1,L2,V1,V2,s.
34 ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, ⋆s] L2.ⓑ{I}V2.
35 /2 width=1 by lfxs_sort/ qed.
37 lemma lfpx_zero: ∀h,I,G,L1,L2,V.
38 ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V.
39 /2 width=1 by lfxs_zero/ qed.
41 lemma lfpx_lref: ∀h,I,G,L1,L2,V1,V2,i.
42 ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #⫯i] L2.ⓑ{I}V2.
43 /2 width=1 by lfxs_lref/ qed.
45 lemma lfpx_gref: ∀h,I,G,L1,L2,V1,V2,l.
46 ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, §l] L2.ⓑ{I}V2.
47 /2 width=1 by lfxs_gref/ qed.
49 lemma lfpx_pair_repl_dx: ∀h,I,G,L1,L2,T,V,V1.
50 ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V1 →
51 ∀V2. ⦃G, L1⦄ ⊢ V ⬈[h] V2 →
52 ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2.
53 /2 width=2 by lfxs_pair_repl_dx/ qed-.
55 (* Basic inversion lemmas ***************************************************)
57 lemma lfpx_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] Y2 → Y2 = ⋆.
58 /2 width=3 by lfxs_inv_atom_sn/ qed-.
60 lemma lfpx_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ⬈[h, ⓪{I}] ⋆ → Y1 = ⋆.
61 /2 width=3 by lfxs_inv_atom_dx/ qed-.
63 lemma lfpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
65 ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
66 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
67 /2 width=1 by lfxs_inv_sort/ qed-.
69 lemma lfpx_inv_zero: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
71 ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
72 ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
73 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
74 /2 width=1 by lfxs_inv_zero/ qed-.
76 lemma lfpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #⫯i] Y2 →
78 ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
79 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
80 /2 width=1 by lfxs_inv_lref/ qed-.
82 lemma lfpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
84 ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
85 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
86 /2 width=1 by lfxs_inv_gref/ qed-.
88 lemma lfpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
89 ⦃G, L1⦄ ⊢ ⬈[h, V] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
90 /2 width=2 by lfxs_inv_bind/ qed-.
92 lemma lfpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
93 ⦃G, L1⦄ ⊢ ⬈[h, V] L2 ∧ ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
94 /2 width=2 by lfxs_inv_flat/ qed-.
96 (* Advanced inversion lemmas ************************************************)
98 lemma lfpx_inv_sort_pair_sn: ∀h,I,G,Y2,L1,V1,s. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, ⋆s] Y2 →
99 ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
100 /2 width=2 by lfxs_inv_sort_pair_sn/ qed-.
102 lemma lfpx_inv_sort_pair_dx: ∀h,I,G,Y1,L2,V2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓑ{I}V2 →
103 ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
104 /2 width=2 by lfxs_inv_sort_pair_dx/ qed-.
106 lemma lfpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
107 ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
109 /2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
111 lemma lfpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
112 ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
114 /2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
116 lemma lfpx_inv_lref_pair_sn: ∀h,I,G,Y2,L1,V1,i. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #⫯i] Y2 →
117 ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓑ{I}V2.
118 /2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
120 lemma lfpx_inv_lref_pair_dx: ∀h,I,G,Y1,L2,V2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #⫯i] L2.ⓑ{I}V2 →
121 ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓑ{I}V1.
122 /2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
124 lemma lfpx_inv_gref_pair_sn: ∀h,I,G,Y2,L1,V1,l. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, §l] Y2 →
125 ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓑ{I}V2.
126 /2 width=2 by lfxs_inv_gref_pair_sn/ qed-.
128 lemma lfpx_inv_gref_pair_dx: ∀h,I,G,Y1,L2,V2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓑ{I}V2 →
129 ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓑ{I}V1.
130 /2 width=2 by lfxs_inv_gref_pair_dx/ qed-.
132 (* Basic forward lemmas *****************************************************)
134 lemma lfpx_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
135 ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
136 /2 width=4 by lfxs_fwd_bind_sn/ qed-.
138 lemma lfpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
139 ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
140 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
142 lemma lfpx_fwd_flat_sn: ∀h,I,G,L1,L2,V,T.
143 ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
144 /2 width=3 by lfxs_fwd_flat_sn/ qed-.
146 lemma lfpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
147 ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
148 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
150 lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
151 ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
152 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
154 (* Basic_2A1: removed theorems 14:
155 lpx_refl lpx_pair lpx_fwd_length
156 lpx_inv_atom1 lpx_inv_pair1 lpx_inv_atom2 lpx_inv_pair2 lpx_inv_pair
157 lpx_drop_conf drop_lpx_trans lpx_drop_trans_O1
158 lpx_cpx_frees_trans cpx_frees_trans lpx_frees_trans