include "basic_2/static/lfxs.ma".
include "basic_2/rt_transition/cpx_ext.ma".
-(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-definition lfpx: sh → genv → relation3 term lenv lenv ≝
- λh,G. lfxs (cpx h G).
+definition lfpx (h) (G): relation3 term lenv lenv ≝
+ lfxs (cpx h G).
interpretation
- "uncounted parallel rt-transition on referred entries (local environment)"
+ "unbound parallel rt-transition on referred entries (local environment)"
'PRedTySn h T G L1 L2 = (lfpx h G T L1 L2).
(* Basic properties ***********************************************************)
/2 width=1 by lfxs_pair/ qed.
lemma lfpx_lref: ∀h,I1,I2,G,L1,L2,i.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, #i] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, #⫯i] L2.ⓘ{I2}.
+ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, #i] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, #â\86\91i] L2.ⓘ{I2}.
/2 width=1 by lfxs_lref/ qed.
lemma lfpx_gref: ∀h,I1,I2,G,L1,L2,l.
(* Basic inversion lemmas ***************************************************)
-(* Basic_2A1: uses: lpx_inv_atom1 *)
lemma lfpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
-(* Basic_2A1: uses: lpx_inv_atom2 *)
lemma lfpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
| ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by lfxs_inv_sort/ qed-.
-(*
-lemma lfpx_inv_zero: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
- ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-/2 width=1 by lfxs_inv_zero/ qed-.
-*)
-lemma lfpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #⫯i] Y2 →
+
+lemma lfpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
| ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
Y1 = L1.ⓑ{I}V1.
/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-lemma lfpx_inv_lref_bind_sn: â\88\80h,I1,G,Y2,L1,i. â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, #⫯i] Y2 →
+lemma lfpx_inv_lref_bind_sn: â\88\80h,I1,G,Y2,L1,i. â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, #â\86\91i] Y2 →
∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-lemma lfpx_inv_lref_bind_dx: â\88\80h,I2,G,Y1,L2,i. â¦\83G, Y1â¦\84 â\8a¢ â¬\88[h, #⫯i] L2.ⓘ{I2} →
+lemma lfpx_inv_lref_bind_dx: â\88\80h,I2,G,Y1,L2,i. â¦\83G, Y1â¦\84 â\8a¢ â¬\88[h, #â\86\91i] L2.ⓘ{I2} →
∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
lemma lfpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-
-(* Basic_2A1: removed theorems 3:
- lpx_inv_pair1 lpx_inv_pair2 lpx_inv_pair
-*)