include "basic_2/notation/relations/predsn_5.ma".
include "basic_2/static/lfxs.ma".
-include "basic_2/rt_transition/cpm.ma".
+include "basic_2/rt_transition/cpr_ext.ma".
(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
/2 width=1 by lfxs_sort/ qed.
-lemma lfpr_zero: ∀h,I,G,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
+lemma lfpr_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_zero/ qed.
+/2 width=1 by lfxs_pair/ qed.
-lemma lfpr_lref: ∀h,I,G,L1,L2,V1,V2,i.
- â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, #i] L2 â\86\92 â¦\83G, L1.â\93\91{I}V1â¦\84 â\8a¢ â\9e¡[h, #⫯i] L2.â\93\91{I}V2.
+lemma lfpr_lref: ∀h,I1,I2,G,L1,L2,i.
+ â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, #i] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â\9e¡[h, #⫯i] L2.â\93\98{I2}.
/2 width=1 by lfxs_lref/ qed.
-lemma lfpr_gref: ∀h,I,G,L1,L2,V1,V2,l.
- â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, §l] L2 â\86\92 â¦\83G, L1.â\93\91{I}V1â¦\84 â\8a¢ â\9e¡[h, §l] L2.â\93\91{I}V2.
+lemma lfpr_gref: ∀h,I1,I2,G,L1,L2,l.
+ â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, §l] L2 â\86\92 â¦\83G, L1.â\93\98{I1}â¦\84 â\8a¢ â\9e¡[h, §l] L2.â\93\98{I2}.
/2 width=1 by lfxs_gref/ qed.
-lemma lfpr_pair_repl_dx: ∀h,I,G,L1,L2,T,V,V1.
- â¦\83G, L1.â\93\91{I}Vâ¦\84 â\8a¢ â\9e¡[h, T] L2.â\93\91{I}V1 →
- ∀V2. ⦃G, L1⦄ ⊢ V ➡[h] V2 →
- â¦\83G, L1.â\93\91{I}Vâ¦\84 â\8a¢ â\9e¡[h, T] L2.â\93\91{I}V2.
-/2 width=2 by lfxs_pair_repl_dx/ qed-.
+lemma lfpr_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
+ â¦\83G, L1.â\93\98{I}â¦\84 â\8a¢ â\9e¡[h, T] L2.â\93\98{I1} →
+ ∀I2. ⦃G, L1⦄ ⊢ I ➡[h] I2 →
+ â¦\83G, L1.â\93\98{I}â¦\84 â\8a¢ â\9e¡[h, T] L2.â\93\98{I2}.
+/2 width=2 by lfxs_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
/2 width=3 by lfxs_inv_atom_dx/ qed-.
lemma lfpr_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+ ∨∨ 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 lfpr_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 lfpr_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.
+ ∨∨ 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 lfpr_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+ ∨∨ 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 lfpr_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\9e¡[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\9e¡[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
lemma lfpr_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 →
- â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, V] L2 â\88§ ⦃G, L1⦄ ⊢ ➡[h, T] L2.
+ â\88§â\88§ â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, V] L2 & ⦃G, L1⦄ ⊢ ➡[h, T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfpr_inv_sort_pair_sn: ∀h,I,G,Y2,L1,V1,s. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
-/2 width=2 by lfxs_inv_sort_pair_sn/ qed-.
+lemma lfpr_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 lfpr_inv_sort_pair_dx: ∀h,I,G,Y1,L2,V2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
-/2 width=2 by lfxs_inv_sort_pair_dx/ qed-.
+lemma lfpr_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 lfpr_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 &
Y1 = L1.ⓑ{I}V1.
/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-lemma lfpr_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 lfpr_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 lfpr_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 lfpr_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 lfpr_inv_gref_pair_sn: ∀h,I,G,Y2,L1,V1,l. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y2 = L2.ⓑ{I}V2.
-/2 width=2 by lfxs_inv_gref_pair_sn/ qed-.
+lemma lfpr_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 lfpr_inv_gref_pair_dx: ∀h,I,G,Y1,L2,V2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y1 = L1.ⓑ{I}V1.
-/2 width=2 by lfxs_inv_gref_pair_dx/ qed-.
+lemma lfpr_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 *****************************************************)
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