include "basic_2/notation/relations/predtysn_5.ma".
include "basic_2/static/lfxs.ma".
-include "basic_2/rt_transition/cpx.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 ***********************************************************)
lemma lfpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.
-lemma lfpx_sort: ∀h,I,G,L1,L2,V1,V2,s.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\8b\86s] L2 â\86\92 â¦\83G, L1.â\93\91{I}V1â¦\84 â\8a¢ â¬\88[h, â\8b\86s] L2.â\93\91{I}V2.
+lemma lfpx_sort: ∀h,I1,I2,G,L1,L2,s.
+ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\8b\86s] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, â\8b\86s] L2.â\93\98{I2}.
/2 width=1 by lfxs_sort/ qed.
-lemma lfpx_zero: ∀h,I,G,L1,L2,V.
- ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V.
-/2 width=1 by lfxs_zero/ qed.
+lemma lfpx_pair: ∀h,I,G,L1,L2,V1,V2.
+ ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_pair/ qed.
-lemma lfpx_lref: ∀h,I,G,L1,L2,V1,V2,i.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, #i] L2 â\86\92 â¦\83G, L1.â\93\91{I}V1â¦\84 â\8a¢ â¬\88[h, #⫯i] L2.â\93\91{I}V2.
+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, #â\86\91i] L2.â\93\98{I2}.
/2 width=1 by lfxs_lref/ qed.
-lemma lfpx_gref: ∀h,I,G,L1,L2,V1,V2,l.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, §l] L2 â\86\92 â¦\83G, L1.â\93\91{I}V1â¦\84 â\8a¢ â¬\88[h, §l] L2.â\93\91{I}V2.
+lemma lfpx_gref: ∀h,I1,I2,G,L1,L2,l.
+ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, §l] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â¬\88[h, §l] L2.â\93\98{I2}.
/2 width=1 by lfxs_gref/ qed.
+lemma lfpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
+ ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
+ ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
+ ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
+/2 width=2 by lfxs_bind_repl_dx/ qed-.
+
(* Basic inversion lemmas ***************************************************)
-lemma lfpx_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] Y2 → Y2 = ⋆.
+lemma lfpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
-lemma lfpx_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ⬈[h, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma lfpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ 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 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+lemma lfpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃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_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}.
/2 width=1 by lfxs_inv_lref/ qed-.
+lemma lfpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+/2 width=1 by lfxs_inv_gref/ qed-.
+
lemma lfpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, V] L2 â\88§ ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+ â\88§â\88§ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
lemma lfpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, V] L2 â\88§ ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+ â\88§â\88§ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
+lemma lfpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
+ ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by lfxs_inv_sort_bind_sn/ qed-.
+
+lemma lfpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by lfxs_inv_sort_bind_dx/ qed-.
+
lemma lfpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
Y2 = L2.ⓑ{I}V2.
Y1 = L1.ⓑ{I}V1.
/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-lemma lfpx_inv_lref_pair_sn: ∀h,I,G,Y2,L1,V1,i. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #⫯i] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓑ{I}V2.
-/2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
+lemma lfpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] 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: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
-lemma lfpx_inv_lref_pair_dx: ∀h,I,G,Y1,L2,V2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #⫯i] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓑ{I}V1.
-/2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
+lemma lfpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
+ ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by lfxs_inv_gref_bind_sn/ qed-.
+
+lemma lfpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by lfxs_inv_gref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lfpx_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\93\91{p,I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
+ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\91¡{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
lemma lfpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-lemma lfpx_fwd_flat_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=3 by lfxs_fwd_flat_sn/ 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-.
-
-lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-(* Basic_2A1: removed theorems 14:
- lpx_refl lpx_pair lpx_fwd_length
- lpx_inv_atom1 lpx_inv_pair1 lpx_inv_atom2 lpx_inv_pair2 lpx_inv_pair
- lpx_drop_conf drop_lpx_trans lpx_drop_trans_O1
- lpx_cpx_frees_trans cpx_frees_trans lpx_frees_trans
-*)
projects / helm.git / blobdiff