+++ /dev/null
-lemma fqu_lpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
-[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
- #K2 #W2 #HLK2 #HVW2 #H destruct
- /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12
- elim (drop_lpx_trans … HLK1 … HK12) -HK12
- /3 width=7 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h, o] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predsn_5.ma".
+include "basic_2/static/lfxs.ma".
+include "basic_2/rt_transition/cpr_ext.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+definition lfpr: sh → genv → relation3 term lenv lenv ≝
+ λh,G. lfxs (λL. cpm h G L 0).
+
+interpretation
+ "parallel r-transition on referred entries (local environment)"
+ 'PRedSn h T G L1 L2 = (lfpr h G T L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma lfpr_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] ⋆.
+/2 width=1 by lfxs_atom/ qed.
+
+lemma lfpr_sort: ∀h,I,G,L1,L2,V1,V2,s.
+ ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_sort/ qed.
+
+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_pair/ qed.
+
+lemma lfpr_lref: ∀h,I1,I2,G,L1,L2,i.
+ ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ➡[h, #↑i] L2.ⓘ{I2}.
+/2 width=1 by lfxs_lref/ qed.
+
+lemma lfpr_gref: ∀h,I1,I2,G,L1,L2,l.
+ ⦃G, L1⦄ ⊢ ➡[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ➡[h, §l] L2.ⓘ{I2}.
+/2 width=1 by lfxs_gref/ qed.
+
+lemma lfpr_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 ***************************************************)
+
+(* Basic_2A1: uses: lpr_inv_atom1 *)
+lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆.
+/2 width=3 by lfxs_inv_atom_sn/ qed-.
+
+(* Basic_2A1: uses: lpr_inv_atom2 *)
+lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
+/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 = ⋆
+ | ∃∃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 = ⋆
+ | ∃∃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 = ⋆
+ | ∃∃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 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[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 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h, V] L2 & ⦃G, L1⦄ ⊢ ➡[h, T] L2.
+/2 width=2 by lfxs_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+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_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 &
+ Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
+
+lemma lfpr_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃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_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_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_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_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 lfpr_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-.
+
+lemma lfpr_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 lfpr_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 5:
+ lpr_inv_pair1 lpr_inv_pair2
+ cpr_conf_lpr lpr_cpr_conf_dx lpr_cpr_conf_sn
+*)
+(* Basic_1: removed theorems 7:
+ wcpr0_gen_sort wcpr0_gen_head
+ wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+ wcpr0_drop wcpr0_drop_back
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lfpx_aaa.ma".
+include "basic_2/rt_transition/lfpr_lfpx.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Properties with atomic arity assignment for terms ************************)
+
+lemma cpr_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (cpm h G L 0).
+/3 width=5 by cpx_aaa_conf, cpm_fwd_cpx/ qed-.
+
+(* Basic_2A1: uses: lpr_aaa_conf *)
+lemma lfpr_aaa_conf: ∀h,G,T. Conf3 … (λL. aaa G L T) (lfpr h G T).
+/3 width=4 by lfpx_aaa_conf, lfpr_fwd_lfpx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfxs_drops.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/lfpr.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: uses: drop_lpr_trans *)
+lemma drops_lfpr_trans: ∀h,G. dedropable_sn (λL. cpm h G L 0).
+/3 width=6 by lfxs_liftable_dedropable_sn, cpm_lifts_sn/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: uses: lpr_drop_conf *)
+lemma lfpr_drops_conf: ∀h,G. dropable_sn (λL. cpm h G L 0).
+/2 width=5 by lfxs_dropable_sn/ qed-.
+
+(* Basic_2A1: uses: lpr_drop_trans_O1 *)
+lemma lfpr_drops_trans: ∀h,G. dropable_dx (λL. cpm h G L 0).
+/2 width=5 by lfxs_dropable_dx/ qed-.
+
+lemma lfpr_inv_lref_pair_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
+
+lemma lfpr_inv_lref_pair_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfxs_fqup.ma".
+include "basic_2/rt_transition/lfpr.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: lemma 250 *)
+(* Basic_2A1: uses: lpr_refl *)
+lemma lfpr_refl: ∀h,G,T. reflexive … (lfpr h G T).
+/2 width=1 by lfxs_refl/ qed.
+
+(* Basic_2A1: uses: lpr_pair *)
+lemma lfpr_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡[h, T] L.ⓑ{I}V2.
+/2 width=1 by lfxs_pair_refl/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfpr_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ.
+/2 width=3 by lfxs_inv_bind_void/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lfpr_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ.
+/2 width=4 by lfxs_fwd_bind_dx_void/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/s_transition/fquq.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/cpm_lsubr.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_transition/lfpr_fqup.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Properties with supclosure ***********************************************)
+
+lemma fqu_cpr_trans_dx: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
+[ /5 width=5 by lsubr_cpm_trans, cpm_bind, lsubr_unit, fqu_clear, ex3_2_intro/
+| #I #G #L #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
+ /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: uses: fqu_lpr_trans *)
+lemma fqu_cpr_trans_sn: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
+[ /5 width=5 by lsubr_cpm_trans, cpm_bind, lsubr_unit, fqu_clear, ex3_2_intro/
+| #I #G #L #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
+ /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with optional supclosure **************************************)
+
+lemma fquq_cpr_trans_dx: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
+[ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: uses: fquq_lpr_trans *)
+lemma fquq_cpr_trans_sn: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
+[ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfxs_length.ma".
+include "basic_2/rt_transition/lfpr.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: uses: lpr_fwd_length *)
+lemma lfpr_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → |L1| = |L2|.
+/2 width=3 by lfxs_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_lsubr.ma".
+include "basic_2/rt_transition/cpm_fsle.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_transition/cpr_drops.ma".
+include "basic_2/rt_transition/lfpr_drops.ma".
+include "basic_2/rt_transition/lfpr_fqup.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Main properties with context-sensitive parallel r-transition for terms ***)
+
+fact cpr_conf_lfpr_atom_atom:
+ ∀h,I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T.
+/2 width=3 by ex2_intro/ qed-.
+
+fact cpr_conf_lfpr_atom_delta:
+ ∀h,G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_lref_pair_sn … HL01 … HLK0) -HL01 #K1 #V1 #HLK1 #HK01 #HV01
+elim (lfpr_inv_lref_pair_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_
+lapply (drops_isuni_fwd_drop2 … HLK2) // -W2 #HLK2
+lapply (fqup_lref (Ⓣ) … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -K2 -V2
+/3 width=6 by cpm_delta_drops, ex2_intro/
+qed-.
+
+(* Note: we don't use cpm_lifts_bi to preserve visual symmetry *)
+fact cpr_conf_lfpr_delta_delta:
+ ∀h,G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
+ ∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (drops_mono … H … HLK0) -H #H destruct
+elim (lfpr_inv_lref_pair_sn … HL01 … HLK0) -HL01 #K1 #W1 #HLK1 #HK01 #_
+lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1
+elim (lfpr_inv_lref_pair_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (cpm_lifts_sn … HV1 … HLK1 … HVT1) -K1 -V1 #T #HVT #HT1
+elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -K2 -V2 #X #HX #HT2
+lapply (lifts_mono … HX … HVT) #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_bind_bind:
+ ∀h,p,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
+#h #p #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
+elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by lfpr_bind_repl_dx, cpm_bind, ext2_pair, ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_bind_zeta:
+ ∀h,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
+elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpm_inv_lifts_sn … HT2 … L2 … HXT2) -T2 /3 width=3 by drops_refl, drops_drop, cpm_zeta, ex2_intro/
+qed-.
+
+(* Note: we don't use cpm_inv_lifts_bi to preserve visual symmetry *)
+fact cpr_conf_lfpr_zeta_zeta:
+ ∀h,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → ∀X1. ⬆*[1] X1 ≘ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
+elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -T0 #T #HT1 #HT2
+elim (cpm_inv_lifts_sn … HT1 … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #T1 #HT1 #HXT1
+elim (cpm_inv_lifts_sn … HT2 … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #T2 #HT2 #HXT2
+lapply (lifts_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_flat_flat:
+ ∀h,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
+#h #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_flat_epsilon:
+ ∀h,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpm_eps, ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_epsilon_epsilon:
+ ∀h,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_flat_beta:
+ ∀h,p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
+/4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/
+qed-.
+
+fact cpr_conf_lfpr_flat_theta:
+ ∀h,p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (cpm_lifts_sn … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU2
+elim (cpm_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0
+ /4 width=7 by cpm_bind, cpr_flat, cpm_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpm_inv_lifts_sn … HT1 … L1 … HXT1) -HXT1
+ /4 width=9 by cpr_flat, cpm_zeta, drops_refl, drops_drop, lifts_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_lfpr_beta_beta:
+ ∀h,p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
+/4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Note: we don't use cpm_lifts_bi to preserve visual symmetry *)
+fact cpr_conf_lfpr_theta_theta:
+ ∀h,p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
+elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0
+elim (cpm_lifts_sn … HV1 … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU
+elim (cpm_lifts_sn … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #X #HX
+lapply (lifts_mono … HX … HVU) -HX #H destruct
+/4 width=7 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_lfpr: ∀h,G. R_confluent2_lfxs (λL. cpm h G L 0) (λL. cpm h G L 0) (λL. cpm h G L 0) (λL. cpm h G L 0).
+#h #G #L0 #T0 @(fqup_wf_ind_eq (Ⓣ) … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1_drops … H1) -H1
+ elim (cpr_inv_atom1_drops … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_lfpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_lfpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_lfpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_lfpr_delta_delta/
+ ]
+| #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpm_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpm_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_lfpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_lfpr_bind_zeta/
+ | /3 width=11 by cpr_conf_lfpr_bind_zeta/
+ | /3 width=12 by cpr_conf_lfpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+ | #p1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #p2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_lfpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_lfpr_flat_epsilon/
+ | /4 width=12 by ex2_commute, cpr_conf_lfpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_lfpr_flat_theta/
+ | /3 width=8 by cpr_conf_lfpr_flat_epsilon/
+ | /3 width=8 by cpr_conf_lfpr_epsilon_epsilon/
+ | /3 width=12 by cpr_conf_lfpr_flat_beta/
+ | /3 width=13 by cpr_conf_lfpr_beta_beta/
+ | /3 width=14 by cpr_conf_lfpr_flat_theta/
+ | /3 width=17 by cpr_conf_lfpr_theta_theta/
+ ]
+]
+qed-.
+
+theorem cpr_conf: ∀h,G,L. confluent … (cpm h G L 0).
+/2 width=6 by cpr_conf_lfpr/ qed-.
+
+(* Properties with context-sensitive parallel r-transition for terms ********)
+
+lemma lfpr_cpr_conf_dx: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lfpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+lemma lfpr_cpr_conf_sn: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lfpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lfpr_conf: ∀h,G,T. confluent … (lfpr h G T).
+/3 width=6 by cpr_conf_lfpr, lfpr_fsge_comp, lfxs_conf/ qed-.
+
+theorem lfpr_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
+ ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, T] L2.ⓑ{I}V2 →
+ ∀p. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V1.T] L2.
+/2 width=2 by lfxs_bind/ qed.
+
+theorem lfpr_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ➡[h, V] L2 →
+ ∀I,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2.
+/2 width=1 by lfxs_flat/ qed.
+
+theorem lfpr_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ➡[h, V] L2 →
+ ∀T. ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ →
+ ∀p,I. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2.
+/2 width=1 by lfxs_bind_void/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lfpx.ma".
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/lfpr.ma".
+
+(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
+
+(* Fwd. lemmas with unc. rt-transition for local env.s on referred entries **)
+
+(* Basic_2A1: uses: lpr_lpx *)
+lemma lfpr_fwd_lfpx: ∀h,T,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/3 width=3 by cpm_fwd_cpx, lfxs_co/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_drops.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/rt_transition/lfpx_fqup.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+
+(* Properties with atomic arity assignment for terms ************************)
+
+lemma cpx_aaa_conf_lfpx: ∀h,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #s #X2 #HX
+ elim (cpx_inv_sort1 … HX) -HX //
+| #I #G #L1 #V1 #B #_ #IH #X2 #HX #Y #HY
+ elim (lfpx_inv_zero_pair_sn … HY) -HY #L2 #V2 #HL12 #HV12 #H destruct
+ elim (cpx_inv_zero1_pair … HX) -HX
+ [ #H destruct /3 width=1 by aaa_zero/
+ | -HV12 * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
+ ]
+| #I #G #L1 #B #i #_ #IH #X2 #HX #Y #HY
+ elim (lfpx_inv_lref_bind_sn … HY) -HY #I2 #L2 #HL12 #H destruct
+ elim (cpx_inv_lref1_bind … HX) -HX
+ [ #H destruct /3 width=1 by aaa_lref/
+ | * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
+ ]
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
+ elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
+ elim (cpx_inv_abbr1 … HX) -HX *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=2 by lfpx_bind_repl_dx, aaa_abbr, ext2_pair/
+ | #T2 #HT12 #HXT2 #H destruct -IHV
+ /4 width=7 by aaa_inv_lifts, drops_drop, drops_refl/
+ ]
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
+ elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
+ elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=2 by lfpx_bind_repl_dx, aaa_abst, ext2_pair/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
+ elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
+ elim (cpx_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
+ | #q #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+ lapply (IHV … HV12 … HV) -IHV -HV12 -HV #HV2
+ lapply (IHT (ⓛ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+ /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
+ | #q #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+ lapply (aaa_lifts G L2 … B … (L2.ⓓW2) … HV2) -HV2 /3 width=2 by drops_drop, drops_refl/ #HV2
+ lapply (IHT (ⓓ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
+ ]
+| #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
+ elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
+ elim (cpx_inv_cast1 … HX) -HX
+ [ * #V2 #T2 #HV12 #HT12 #H destruct ] /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma lfpx_aaa_conf: ∀h,G,T. Conf3 … (λL. aaa G L T) (lfpx h G T).
+/2 width=6 by cpx_aaa_conf_lfpx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfxs_drops.ma".
+include "basic_2/rt_transition/cpx_drops.ma".
+include "basic_2/rt_transition/lfpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: uses: drop_lpx_trans *)
+lemma drops_lfpx_trans: ∀h,G. dedropable_sn (cpx h G).
+/3 width=6 by lfxs_liftable_dedropable_sn, cpx_lifts_sn/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: uses: lpx_drop_conf *)
+lemma lfpx_drops_conf: ∀h,G. dropable_sn (cpx h G).
+/2 width=5 by lfxs_dropable_sn/ qed-.
+
+(* Basic_2A1: uses: lpx_drop_trans_O1 *)
+lemma lfpx_drops_trans: ∀h,G. dropable_dx (cpx h G).
+/2 width=5 by lfxs_dropable_dx/ qed-.
+
+lemma lfpx_inv_lref_pair_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
+
+lemma lfpx_inv_lref_pair_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lfpx_drops.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
+include "basic_2/s_transition/fquq.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+
+(* Properties with extended structural successor for closures ***************)
+
+lemma lfpx_fqu_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #K #V #K1 #H
+ elim (lfpx_inv_zero_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
+ elim (lifts_total V (𝐔❴1❵)) #T #HVT
+ /3 width=7 by lfpx_cpx_conf, cpx_delta, fqu_drop, ex3_2_intro/
+| /3 width=7 by lfpx_fwd_pair_sn, cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=6 by lfpx_fwd_bind_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| /3 width=8 by lfpx_fwd_bind_dx_void, cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=7 by lfpx_fwd_flat_dx, cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #T #U #HTU #K1 #H
+ elim (lfpx_drops_trans … H (Ⓣ) … HTU) -H
+ [|*: /3 width=2 by drops_refl, drops_drop/ ] -I #K0 #HK10 #HK0
+ elim (drops_inv_succ … HK10) -HK10 #I #Y #HY #H destruct
+ lapply (drops_fwd_isid … HY ?) -HY // #H destruct
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with extended optional structural successor for closures ******)
+
+lemma lfpx_fquq_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
+[ #H12 elim (lfpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ [ break term 46 h, break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSn $h $T $G $L1 $L2 }.
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-definition tc_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤**[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤**[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤**[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤**[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-definition tc_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤**[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤**[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤**[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤**[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
-definition tc_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤**[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤**[R, T] K2.
+definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⪤**[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤**[R, T] K2.
(* Properties with generic slicing for local environments *******************)
-lemma dedropable_sn_CTC: ∀R. dedropable_sn R → tc_dedropable_sn R.
+lemma dedropable_sn_CTC: ∀R. f_dedropable_sn R → tc_f_dedropable_sn R.
#R #HR #b #f #L1 #K1 #HLK1 #K2 #T #H elim H -K2
[ #K2 #HK12 #U #HTU elim (HR … HLK1 … HK12 … HTU) -K1 -T -HR
/3 width=4 by ex3_intro, inj/
(* Inversion lemmas with generic slicing for local environments *************)
-lemma dropable_sn_CTC: ∀R. dropable_sn R → tc_dropable_sn R.
+lemma dropable_sn_CTC: ∀R. f_dropable_sn R → tc_f_dropable_sn R.
#R #HR #b #f #L1 #K1 #HLK1 #Hf #L2 #U #H elim H -L2
[ #L2 #HL12 #T #HTU elim (HR … HLK1 … HL12 … HTU) -L1 -U -HR
/3 width=3 by inj, ex2_intro/
]
qed-.
-lemma dropable_dx_CTC: ∀R. dropable_dx R → tc_dropable_dx R.
+lemma dropable_dx_CTC: ∀R. f_dropable_dx R → tc_f_dropable_dx R.
#R #HR #L1 #L2 #U #H elim H -L2
[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #T #HTU
elim (HR … HL12 … HLK2 … HTU) -L2 -U -HR
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡[ break term 46 h ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSn $h $G $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ [ break term 46 h, break term 46 T ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSn $h $T $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex.ma".
+include "basic_2/relocation/drops_cext2.ma".
+include "basic_2/relocation/drops_lexs.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition dedropable_sn: predicate … ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
+ ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+definition dropable_sn: predicate … ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition dropable_dx: predicate … ≝
+ λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
+
+(* Properties with generic extension ****************************************)
+
+(* Basic_2A1: was: lpx_sn_liftable_dedropable *)
+lemma lex_liftable_dedropable_sn (R): c_reflexive … R →
+ d_liftable2_sn … lifts R → dedropable_sn R.
+#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 * #f1 #Hf1 #HK12
+elim (lexs_liftable_co_dedropable_sn … HLK1 … HK12) -K1
+/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, coafter_isid_dx, ex3_intro, ex2_intro/
+qed-.
+
+(* Inversion lemmas with generic extension **********************************)
+
+(* Basic_2A1: was: lpx_sn_deliftable_dropable *)
+lemma lex_dropable_sn (R): dropable_sn R.
+#R #b #f #L1 #K1 #HLK1 #H1f #L2 * #f2 #Hf2 #HL12
+elim (lexs_co_dropable_sn … HLK1 … HL12) -L1
+/3 width=3 by coafter_isid_dx, ex2_intro/
+qed-.
+
+(* Basic_2A1: was: lpx_sn_dropable *)
+lemma lex_dropable_dx (R): dropable_dx R.
+#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #Hf
+elim (lexs_co_dropable_dx … HL12 … HLK2) -L2
+/3 width=5 by coafter_isid_dx, ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lpx_sn_drop_conf *)
+lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
+ ∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
+ ∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
+#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf
+elim (lexs_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
+[ #Z2 #K2 #HLK2 #HK12 #H
+ elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
+ /3 width=5 by ex3_2_intro, ex2_intro/
+| /3 width=3 by coafter_isid_dx, isid_push/
+]
+qed-.
+
+(* Basic_2A1: includes: lpx_sn_drop_trans *)
+lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
+ ∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
+ ∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
+#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf
+elim (lexs_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
+[ #Z1 #K1 #HLK1 #HK12 #H
+ elim (ext2_inv_pair_dx … H) -H #V1 #HV12 #H destruct
+ /3 width=5 by ex3_2_intro, ex2_intro/
+| /3 width=3 by coafter_isid_dx, isid_push/
+]
+qed-.
(* Properties with entrywise extension of context-sensitive relations *******)
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *) (**) (* changed after commit 13218 *)
+(**) (* changed after commit 13218 *)
lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
]
qed-.
-(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
co_dedropable_sn (lexs RN RP).
]
qed-.
-(* Basic_2A1: includes: lpx_sn_dropable *)
lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP).
/2 width=5 by lexs_dropable_dx_aux/ qed-.
-(* Basic_2A1: includes: lpx_sn_drop_conf *) (**)
lemma lexs_drops_conf_next: ∀RN,RP.
∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-(* Basic_2A1: includes: lpx_sn_drop_trans *)
lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
(* *)
(**************************************************************************)
+include "ground_2/pull/pull_2.ma".
+include "ground_2/pull/pull_4.ma".
include "ground_2/relocation/rtmap_uni.ma".
include "basic_2/notation/relations/relation_3.ma".
include "basic_2/syntax/cext2.ma".
(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
-(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
-definition lex: (lenv → relation term) → relation lenv ≝
- λR,L1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤*[cfull, cext2 R, f] L2.
+definition lex (R): relation lenv ≝
+ λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤*[cfull, cext2 R, f] L2.
interpretation "generic extension (local environment)"
'Relation R L1 L2 = (lex R L1 L2).
+definition lex_confluent: relation (relation3 …) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. L0 ⪤[R1] L1 → ∀L2. L0 ⪤[R2] L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lex_transitive: relation (relation3 …) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⪤[R1] L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
(* Basic properties *********************************************************)
-lemma lex_bind: ∀R,I1,I2,K1,K2. K1 ⪤[R] K2 → cext2 R K1 I1 I2 →
- K1.ⓘ{I1} ⪤[R] K2.ⓘ{I2}.
+(* Basic_2A1: was: lpx_sn_atom *)
+lemma lex_atom (R): ⋆ ⪤[R] ⋆.
+/2 width=3 by lexs_atom, ex2_intro/ qed.
+
+lemma lex_bind (R): ∀I1,I2,K1,K2. K1 ⪤[R] K2 → cext2 R K1 I1 I2 →
+ K1.ⓘ{I1} ⪤[R] K2.ⓘ{I2}.
#R #I1 #I2 #K1 #K2 * #f #Hf #HK12 #HI12
/3 width=3 by lexs_push, isid_push, ex2_intro/
qed.
(* Basic_2A1: was: lpx_sn_refl *)
-lemma lex_refl: ∀R. c_reflexive … R → reflexive … (lex R).
+lemma lex_refl (R): c_reflexive … R → reflexive … (lex R).
/4 width=3 by lexs_refl, ext2_refl, ex2_intro/ qed.
+lemma lex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2. L1 ⪤[R1] L2 → L1 ⪤[R2] L2.
+#R1 #R2 #HR #L1 #L2 * /5 width=7 by lexs_co, cext2_co, ex2_intro/
+qed-.
+
(* Advanced properties ******************************************************)
-lemma lex_bind_refl_dx: ∀R. c_reflexive … R →
- ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓘ{I} ⪤[R] K2.ⓘ{I}.
+lemma lex_bind_refl_dx (R): c_reflexive … R →
+ ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓘ{I} ⪤[R] K2.ⓘ{I}.
/3 width=3 by ext2_refl, lex_bind/ qed.
+lemma lex_unit (R): ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓤ{I} ⪤[R] K2.ⓤ{I}.
+/3 width=1 by lex_bind, ext2_unit/ qed.
+
+(* Basic_2A1: was: lpx_sn_pair *)
+lemma lex_pair (R): ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → R K1 V1 V2 →
+ K1.ⓑ{I}V1 ⪤[R] K2.ⓑ{I}V2.
+/3 width=1 by lex_bind, ext2_pair/ qed.
+
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpx_sn_inv_atom1: *)
-lemma lex_inv_atom_sn: ∀R,L2. ⋆ ⪤[R] L2 → L2 = ⋆.
+lemma lex_inv_atom_sn (R): ∀L2. ⋆ ⪤[R] L2 → L2 = ⋆.
#R #L2 * #f #Hf #H >(lexs_inv_atom1 … H) -L2 //
qed-.
-lemma lex_inv_bind_sn: ∀R,I1,L2,K1. K1.ⓘ{I1} ⪤[R] L2 →
- ∃∃I2,K2. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L2 = K2.ⓘ{I2}.
+lemma lex_inv_bind_sn (R): ∀I1,L2,K1. K1.ⓘ{I1} ⪤[R] L2 →
+ ∃∃I2,K2. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L2 = K2.ⓘ{I2}.
#R #I1 #L2 #K1 * #f #Hf #H
lapply (lexs_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
elim (lexs_inv_push1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct
qed-.
(* Basic_2A1: was: lpx_sn_inv_atom2 *)
-lemma lex_inv_atom_dx: ∀R,L1. L1 ⪤[R] ⋆ → L1 = ⋆.
+lemma lex_inv_atom_dx (R): ∀L1. L1 ⪤[R] ⋆ → L1 = ⋆.
#R #L1 * #f #Hf #H >(lexs_inv_atom2 … H) -L1 //
qed-.
-lemma lex_inv_bind_dx: ∀R,I2,L1,K2. L1 ⪤[R] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L1 = K1.ⓘ{I1}.
+lemma lex_inv_bind_dx (R): ∀I2,L1,K2. L1 ⪤[R] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L1 = K1.ⓘ{I1}.
#R #I2 #L1 #K2 * #f #Hf #H
lapply (lexs_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
elim (lexs_inv_push2 … H) -H #I1 #K1 #HK12 #HI12 #H destruct
(* Advanced inversion lemmas ************************************************)
+lemma lex_inv_unit_sn (R): ∀I,L2,K1. K1.ⓤ{I} ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & L2 = K2.ⓤ{I}.
+#R #I #L2 #K1 #H
+elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
+elim (ext2_inv_unit_sn … HZ2) -HZ2
+/2 width=3 by ex2_intro/
+qed-.
+
(* Basic_2A1: was: lpx_sn_inv_pair1 *)
-lemma lex_inv_pair_sn: ∀R,I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R] L2 →
- ∃∃K2,V2. K1 ⪤[R] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+lemma lex_inv_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R] L2 →
+ ∃∃K2,V2. K1 ⪤[R] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
#R #I #L2 #K1 #V1 #H
elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
elim (ext2_inv_pair_sn … HZ2) -HZ2 #V2 #HV12 #H destruct
/2 width=5 by ex3_2_intro/
qed-.
+lemma lex_inv_unit_dx (R): ∀I,L1,K2. L1 ⪤[R] K2.ⓤ{I} →
+ ∃∃K1. K1 ⪤[R] K2 & L1 = K1.ⓤ{I}.
+#R #I #L1 #K2 #H
+elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
+elim (ext2_inv_unit_dx … HZ1) -HZ1
+/2 width=3 by ex2_intro/
+qed-.
+
(* Basic_2A1: was: lpx_sn_inv_pair2 *)
-lemma lex_inv_pair_dx: ∀R,I,L1,K2,V2. L1 ⪤[R] K2.ⓑ{I}V2 →
- ∃∃K1,V1. K1 ⪤[R] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+lemma lex_inv_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
#R #I #L1 #K2 #V2 #H
elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
elim (ext2_inv_pair_dx … HZ1) -HZ1 #V1 #HV12 #H destruct
qed-.
(* Basic_2A1: was: lpx_sn_inv_pair *)
-lemma lex_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
- L1.ⓑ{I1}V1 ⪤[R] L2.ⓑ{I2}V2 →
- ∧∧ L1 ⪤[R] L2 & R L1 V1 V2 & I1 = I2.
+lemma lex_inv_pair (R): ∀I1,I2,L1,L2,V1,V2.
+ L1.ⓑ{I1}V1 ⪤[R] L2.ⓑ{I2}V2 →
+ ∧∧ L1 ⪤[R] L2 & R L1 V1 V2 & I1 = I2.
#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lex_inv_pair_sn … H) -H
#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma lex_ind (R) (Q:relation2 …):
+ Q (⋆) (⋆) →
+ (
+ ∀I,K1,K2. K1 ⪤[R] K2 → Q K1 K2 → Q (K1.ⓤ{I}) (K2.ⓤ{I})
+ ) → (
+ ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → Q K1 K2 → R K1 V1 V2 →Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L1,L2. L1 ⪤[R] L2 → Q L1 L2.
+#R #Q #IH1 #IH2 #IH3 #L1 #L2 * #f @pull_2 #H
+elim H -f -L1 -L2 // #f #I1 #I2 #K1 #K2 @pull_4 #H
+[ elim (isid_inv_next … H)
+| lapply (isid_inv_push … H ??)
+] -H [5:|*: // ] #Hf @pull_2 #H
+elim H -H /3 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: was: lpx_sn_trans *)
+theorem lex_trans (R): lex_transitive R R → Transitive … (lex R).
+#R #HR #L1 #L #H @(lex_ind … H) -L1 -L //
+[ #I #K1 #K #HK1 #IH #Y #H
+ elim (lex_inv_unit_sn … H) -H #K2 #K2 #H destruct
+ /3 width=1 by lex_unit/
+| #I #K1 #L #V1 #V #HK1 #IH #HV1 #X #H
+ elim (lex_inv_pair_sn … H) -H #K2 #V2 #HK2 #HV2 #H destruct
+ /3 width=5 by lex_pair/
+]
+qed-.
+
+(* Basic_2A1: was: lpx_sn_conf *)
+theorem lex_conf (R1) (R2): lex_confluent R1 R2 → confluent2 … (lex R1) (lex R2).
+#R1 #R2 #HR12 #L0 elim L0 -L0 [| #K0 * ]
+[ #Y1 #H1 #Y2 #H2
+ >(lex_inv_atom_sn … H1) -Y1
+ >(lex_inv_atom_sn … H2) -Y2
+ /2 width=3 by lex_atom, ex2_intro/
+| #I #IH #Y1 #H1 #Y2 #H2
+ elim (lex_inv_unit_sn … H1) -H1 #K1 #HK01 #H destruct
+ elim (lex_inv_unit_sn … H2) -H2 #K2 #HK02 #H destruct
+ elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
+ /3 width=3 by lex_unit, ex2_intro/
+| #I #V0 #IH #Y1 #H1 #Y2 #H2
+ elim (lex_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+ elim (lex_inv_pair_sn … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (HR12 … HV01 … HV02 … HK01 … HK02) -V0 #V #HV1 #HV2
+ elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
+ /3 width=5 by lex_pair, ex2_intro/
+]
+qed-.
]
qed-.
-(* Basic_2A1: includes: lpx_sn_trans *)
theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) →
(∀g,I,K. lexs_transitive RP RP RP RN RP g K I) →
Transitive … (lexs RN RP f).
/3 width=1 by lexs_push/
qed-.
-(* Basic_2A1: includes: lpx_sn_conf *)
theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
∀L,f.
(∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/lpx_aaa.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with atomic arity assignment for terms ************************)
+
+(* Basic_2A1: includes: cpr_aaa_conf *)
+lemma cpm_aaa_conf (n) (h): ∀G,L. Conf3 … (aaa G L) (cpm h G L n).
+/3 width=5 by cpx_aaa_conf, cpm_fwd_cpx/ qed-.
(* Properties with restricted refinement for local environments *************)
(* Basic_2A1: includes: lsubr_cpr_trans *)
-lemma lsubr_cpm_trans: ∀n,h,G. lsub_trans … (λL. cpm h G L n) lsubr.
+lemma lsubr_cpm_trans (n) (h) (G): lsub_trans … (λL. cpm h G L n) lsubr.
#n #h #G #L1 #T1 #T2 * /3 width=5 by lsubr_cpg_trans, ex2_intro/
qed-.
+
+lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
+ ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ➡[n, h] T2 →
+ ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+/4 width=4 by lsubr_cpm_trans, cpm_bind, lsubr_unit/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lpr_lpr.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf (h) (G) (L): confluent … (cpm h G L 0).
+/2 width=6 by cpr_conf_lpr/ qed-.
(* Properties with restricted refinement for local environments *************)
-lemma lsubr_cpx_trans: ∀h,G. lsub_trans … (cpx h G) lsubr.
+lemma lsubr_cpx_trans (h) (G): lsub_trans … (cpx h G) lsubr.
#h #G #L1 #T1 #T2 * /3 width=4 by lsubr_cpg_trans, ex_intro/
qed-.
-lemma cpx_bind_unit: ∀h,L,G,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
- ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
+ ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
/4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit/ qed.
include "basic_2/notation/relations/predsubtyproper_8.ma".
include "basic_2/s_transition/fqu.ma".
include "basic_2/static/lfdeq.ma".
-include "basic_2/rt_transition/lfpr_lfpx.ma".
+include "basic_2/rt_transition/lpr_lpx.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
inductive fpb (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fqu : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h o G1 L1 T1 G2 L2 T2
-| fpb_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2
-| fpb_lfpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h, T1] L2 → (L1 ≛[h, o, T1] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1
+| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h o G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T1] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1
.
interpretation
(* Basic properties *********************************************************)
(* Basic_2A1: includes: cpr_fpb *)
-lemma cpm_fpb: ∀n,h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≛[h, o] T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄.
+lemma cpm_fpb (n) (h) (o) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≛[h, o] T2 → ⊥) →
+ ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄.
/3 width=2 by fpb_cpx, cpm_fwd_cpx/ qed.
-(* Basic_2A1: includes: lpr_fpb *)
-lemma lfpr_fpb: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
- ⦃G, L1, T⦄ ≻[h, o] ⦃G, L2, T⦄.
-/3 width=1 by fpb_lfpx, lfpr_fwd_lfpx/ qed.
+lemma lpr_fpb (h) (o) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
+ ⦃G, L1, T⦄ ≻[h, o] ⦃G, L2, T⦄.
+/3 width=1 by fpb_lpx, lpr_fwd_lpx/ qed.
include "basic_2/static/lfdeq_fqus.ma".
include "basic_2/rt_transition/cpx_lfdeq.ma".
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_transition/lpx_lfdeq.ma".
include "basic_2/rt_transition/fpb.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
| #T1 #HUT1 #HnUT1
elim (tdeq_cpx_trans … HU21 … HUT1) -HUT1
/6 width=5 by fpb_cpx, tdeq_canc_sn, tdeq_trans, ex3_2_intro/
-| /6 width=5 by fpb_lfpx, lfpx_tdeq_div, tdeq_lfdeq_conf, ex3_2_intro/
+| /6 width=5 by fpb_lpx, lfpx_tdeq_div, tdeq_lfdeq_conf, ex3_2_intro/
]
qed-.
/3 width=5 by fpb_fqu, ex3_2_intro/
| #U #HTU #HnTU elim (lfdeq_cpx_trans … HT … HTU) -HTU
/5 width=10 by fpb_cpx, cpx_lfdeq_conf_sn, tdeq_trans, tdeq_lfdeq_conf, ex3_2_intro/
-| #L2 #HKL2 #HnKL2 elim (lfdeq_lfpx_trans … HKL2 … HT) -HKL2
- /6 width=5 by fpb_lfpx, (* 2x *) lfdeq_canc_sn, ex3_2_intro/
+| #L2 #HKL2 #HnKL2 elim (lfdeq_lpx_trans … HKL2 … HT) -HKL2
+ /6 width=5 by fpb_lpx, (* 2x *) lfdeq_canc_sn, ex3_2_intro/
]
qed-.
include "basic_2/notation/relations/predsubty_8.ma".
include "basic_2/static/ffdeq.ma".
include "basic_2/s_transition/fquq.ma".
-include "basic_2/rt_transition/lfpr_lfpx.ma".
+include "basic_2/rt_transition/lpr_lpx.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
| fpbq_fquq : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
-| fpbq_lfpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h, T1] L2 → fpbq h o G1 L1 T1 G1 L2 T1
+| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h o G1 L1 T1 G1 L2 T1
| fpbq_ffdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
.
(* Basic properties *********************************************************)
-lemma fpbq_refl: ∀h,o. tri_reflexive … (fpbq h o).
+lemma fpbq_refl (h) (o): tri_reflexive … (fpbq h o).
/2 width=1 by fpbq_cpx/ qed.
(* Basic_2A1: includes: cpr_fpbq *)
-lemma cpm_fpbq: ∀n,h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
+lemma cpm_fpbq (n) (h) (o) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
/3 width=2 by fpbq_cpx, cpm_fwd_cpx/ qed.
-lemma lfpr_fpbq: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1, T⦄ ≽[h, o] ⦃G, L2, T⦄.
-/3 width=1 by fpbq_lfpx, lfpr_fwd_lfpx/ qed.
+lemma lpr_fpbq (h) (o) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1, T⦄ ≽[h, o] ⦃G, L2, T⦄.
+/3 width=1 by fpbq_lpx, lpr_fwd_lpx/ qed.
(* Basic_2A1: removed theorems 2:
fpbq_fpbqa fpbqa_inv_fpbq
include "basic_2/static/aaa_fqus.ma".
include "basic_2/static/aaa_ffdeq.ma".
-include "basic_2/rt_transition/lfpx_aaa.ma".
+include "basic_2/rt_transition/lpx_aaa.ma".
include "basic_2/rt_transition/fpbq.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
lemma fpbq_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=8 by lfpx_aaa_conf, cpx_aaa_conf, aaa_ffdeq_conf, aaa_fquq_conf, ex_intro/
+/3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_ffdeq_conf, aaa_fquq_conf, ex_intro/
qed-.
lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lfpx, fqu_fquq/
+/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
qed.
(* Basic_2A1: fpb_fpbq_alt *)
| #T2 #H elim (tdeq_dec h o T1 T2)
/4 width=1 by fpb_cpx, ffdeq_intro_sn, or_intror, or_introl/
| #L2 elim (lfdeq_dec h o L1 L2 T1)
- /4 width=1 by fpb_lfpx, ffdeq_intro_sn, or_intror, or_introl/
+ /4 width=1 by fpb_lpx, ffdeq_intro_sn, or_intror, or_introl/
| /2 width=1 by or_introl/
]
qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/predsn_5.ma".
-include "basic_2/static/lfxs.ma".
-include "basic_2/rt_transition/cpr_ext.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-definition lfpr: sh → genv → relation3 term lenv lenv ≝
- λh,G. lfxs (λL. cpm h G L 0).
-
-interpretation
- "parallel r-transition on referred entries (local environment)"
- 'PRedSn h T G L1 L2 = (lfpr h G T L1 L2).
-
-(* Basic properties ***********************************************************)
-
-lemma lfpr_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] ⋆.
-/2 width=1 by lfxs_atom/ qed.
-
-lemma lfpr_sort: ∀h,I,G,L1,L2,V1,V2,s.
- ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_sort/ qed.
-
-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_pair/ qed.
-
-lemma lfpr_lref: ∀h,I1,I2,G,L1,L2,i.
- ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ➡[h, #↑i] L2.ⓘ{I2}.
-/2 width=1 by lfxs_lref/ qed.
-
-lemma lfpr_gref: ∀h,I1,I2,G,L1,L2,l.
- ⦃G, L1⦄ ⊢ ➡[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ➡[h, §l] L2.ⓘ{I2}.
-/2 width=1 by lfxs_gref/ qed.
-
-lemma lfpr_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 ***************************************************)
-
-(* Basic_2A1: uses: lpr_inv_atom1 *)
-lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆.
-/2 width=3 by lfxs_inv_atom_sn/ qed-.
-
-(* Basic_2A1: uses: lpr_inv_atom2 *)
-lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
-/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 = ⋆
- | ∃∃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 = ⋆
- | ∃∃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 = ⋆
- | ∃∃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 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[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 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h, V] L2 & ⦃G, L1⦄ ⊢ ➡[h, T] L2.
-/2 width=2 by lfxs_inv_flat/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-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_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 &
- Y2 = L2.ⓑ{I}V2.
-/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
-
-lemma lfpr_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃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_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_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_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_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 lfpr_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-.
-
-lemma lfpr_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 lfpr_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 5:
- lpr_inv_pair1 lpr_inv_pair2
- cpr_conf_lpr lpr_cpr_conf_dx lpr_cpr_conf_sn
-*)
-(* Basic_1: removed theorems 7:
- wcpr0_gen_sort wcpr0_gen_head
- wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
- wcpr0_drop wcpr0_drop_back
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/lfpx_aaa.ma".
-include "basic_2/rt_transition/lfpr_lfpx.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Properties with atomic arity assignment for terms ************************)
-
-lemma cpr_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (cpm h G L 0).
-/3 width=5 by cpx_aaa_conf, cpm_fwd_cpx/ qed-.
-
-(* Basic_2A1: uses: lpr_aaa_conf *)
-lemma lfpr_aaa_conf: ∀h,G,T. Conf3 … (λL. aaa G L T) (lfpr h G T).
-/3 width=4 by lfpx_aaa_conf, lfpr_fwd_lfpx/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lfxs_drops.ma".
-include "basic_2/rt_transition/cpm_drops.ma".
-include "basic_2/rt_transition/lfpr.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Properties with generic slicing for local environments *******************)
-
-(* Basic_2A1: uses: drop_lpr_trans *)
-lemma drops_lfpr_trans: ∀h,G. dedropable_sn (λL. cpm h G L 0).
-/3 width=6 by lfxs_liftable_dedropable_sn, cpm_lifts_sn/ qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-(* Basic_2A1: uses: lpr_drop_conf *)
-lemma lfpr_drops_conf: ∀h,G. dropable_sn (λL. cpm h G L 0).
-/2 width=5 by lfxs_dropable_sn/ qed-.
-
-(* Basic_2A1: uses: lpr_drop_trans_O1 *)
-lemma lfpr_drops_trans: ∀h,G. dropable_dx (λL. cpm h G L 0).
-/2 width=5 by lfxs_dropable_dx/ qed-.
-
-lemma lfpr_inv_lref_pair_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
-/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
-
-lemma lfpr_inv_lref_pair_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
-/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lfxs_fqup.ma".
-include "basic_2/rt_transition/lfpr.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Advanced properties ******************************************************)
-
-(* Note: lemma 250 *)
-(* Basic_2A1: uses: lpr_refl *)
-lemma lfpr_refl: ∀h,G,T. reflexive … (lfpr h G T).
-/2 width=1 by lfxs_refl/ qed.
-
-(* Basic_2A1: uses: lpr_pair *)
-lemma lfpr_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡[h, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair_refl/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfpr_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ.
-/2 width=3 by lfxs_inv_bind_void/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lfpr_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ.
-/2 width=4 by lfxs_fwd_bind_dx_void/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/s_transition/fquq.ma".
-include "basic_2/rt_transition/cpm_drops.ma".
-include "basic_2/rt_transition/cpm_lsubr.ma".
-include "basic_2/rt_transition/cpr.ma".
-include "basic_2/rt_transition/lfpr_fqup.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Properties with supclosure ***********************************************)
-
-lemma fqu_cpr_trans_dx: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
-[ /5 width=5 by lsubr_cpm_trans, cpm_bind, lsubr_unit, fqu_clear, ex3_2_intro/
-| #I #G #L #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
- /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: fqu_lpr_trans *)
-lemma fqu_cpr_trans_sn: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
-[ /5 width=5 by lsubr_cpm_trans, cpm_bind, lsubr_unit, fqu_clear, ex3_2_intro/
-| #I #G #L #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
- /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
-]
-qed-.
-
-(* Properties with optional supclosure **************************************)
-
-lemma fquq_cpr_trans_dx: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
-[ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: fquq_lpr_trans *)
-lemma fquq_cpr_trans_sn: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
-[ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lfxs_length.ma".
-include "basic_2/rt_transition/lfpr.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: uses: lpr_fwd_length *)
-lemma lfpr_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → |L1| = |L2|.
-/2 width=3 by lfxs_fwd_length/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/cpm_lsubr.ma".
-include "basic_2/rt_transition/cpm_fsle.ma".
-include "basic_2/rt_transition/cpr.ma".
-include "basic_2/rt_transition/cpr_drops.ma".
-include "basic_2/rt_transition/lfpr_drops.ma".
-include "basic_2/rt_transition/lfpr_fqup.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Main properties with context-sensitive parallel r-transition for terms ***)
-
-fact cpr_conf_lfpr_atom_atom:
- ∀h,I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T.
-/2 width=3 by ex2_intro/ qed-.
-
-fact cpr_conf_lfpr_atom_delta:
- ∀h,G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V2. ⦃G, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ #i ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
-#h #G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_lref_pair_sn … HL01 … HLK0) -HL01 #K1 #V1 #HLK1 #HK01 #HV01
-elim (lfpr_inv_lref_pair_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_
-lapply (drops_isuni_fwd_drop2 … HLK2) // -W2 #HLK2
-lapply (fqup_lref (Ⓣ) … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -K2 -V2
-/3 width=6 by cpm_delta_drops, ex2_intro/
-qed-.
-
-(* Note: we don't use cpm_lifts_bi to preserve visual symmetry *)
-fact cpr_conf_lfpr_delta_delta:
- ∀h,G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V1. ⦃G, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
- ∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
- ∀V2. ⦃G, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
-#h #G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (drops_mono … H … HLK0) -H #H destruct
-elim (lfpr_inv_lref_pair_sn … HL01 … HLK0) -HL01 #K1 #W1 #HLK1 #HK01 #_
-lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1
-elim (lfpr_inv_lref_pair_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_
-lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
-lapply (fqup_lref (Ⓣ) … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (cpm_lifts_sn … HV1 … HLK1 … HVT1) -K1 -V1 #T #HVT #HT1
-elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -K2 -V2 #X #HX #HT2
-lapply (lifts_mono … HX … HVT) #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_bind_bind:
- ∀h,p,I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
-#h #p #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
-elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
-/3 width=5 by lfpr_bind_repl_dx, cpm_bind, ext2_pair, ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_bind_zeta:
- ∀h,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
-#h #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
-elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpm_inv_lifts_sn … HT2 … L2 … HXT2) -T2 /3 width=3 by drops_refl, drops_drop, cpm_zeta, ex2_intro/
-qed-.
-
-(* Note: we don't use cpm_inv_lifts_bi to preserve visual symmetry *)
-fact cpr_conf_lfpr_zeta_zeta:
- ∀h,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → ∀X1. ⬆*[1] X1 ≘ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
-#h #G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0
-elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -T0 #T #HT1 #HT2
-elim (cpm_inv_lifts_sn … HT1 … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #T1 #HT1 #HXT1
-elim (cpm_inv_lifts_sn … HT2 … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #T2 #HT2 #HXT2
-lapply (lifts_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_flat_flat:
- ∀h,I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
-#h #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_flat_epsilon:
- ∀h,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
-#h #G #L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpm_eps, ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_epsilon_epsilon:
- ∀h,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
-#h #G #L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_flat_beta:
- ∀h,p,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
-#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
-/4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/
-qed-.
-
-fact cpr_conf_lfpr_flat_theta:
- ∀h,p,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
-#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (cpm_lifts_sn … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU2
-elim (cpm_inv_abbr1 … H) -H *
-[ #W1 #T1 #HW01 #HT01 #H destruct
- elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
- elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0
- /4 width=7 by cpm_bind, cpr_flat, cpm_theta, ex2_intro/
-| #T1 #HT01 #HXT1 #H destruct
- elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
- elim (cpm_inv_lifts_sn … HT1 … L1 … HXT1) -HXT1
- /4 width=9 by cpr_flat, cpm_zeta, drops_refl, drops_drop, lifts_flat, ex2_intro/
-]
-qed-.
-
-fact cpr_conf_lfpr_beta_beta:
- ∀h,p,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
-#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
-lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
-/4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-(* Note: we don't use cpm_lifts_bi to preserve visual symmetry *)
-fact cpr_conf_lfpr_theta_theta:
- ∀h,p,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
- ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
-#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0
-elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /3 width=4 by lfpr_bind_repl_dx, ext2_pair/ -L0 -V0 -W0 -T0
-elim (cpm_lifts_sn … HV1 … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU
-elim (cpm_lifts_sn … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #X #HX
-lapply (lifts_mono … HX … HVU) -HX #H destruct
-/4 width=7 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-theorem cpr_conf_lfpr: ∀h,G. R_confluent2_lfxs (λL. cpm h G L 0) (λL. cpm h G L 0) (λL. cpm h G L 0) (λL. cpm h G L 0).
-#h #G #L0 #T0 @(fqup_wf_ind_eq (Ⓣ) … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
-[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_atom1_drops … H1) -H1
- elim (cpr_inv_atom1_drops … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpr_conf_lfpr_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpr_conf_lfpr_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpr_conf_lfpr_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpr_conf_lfpr_delta_delta/
- ]
-| #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpm_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpm_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpr_conf_lfpr_bind_bind/
- | /4 width=11 by ex2_commute, cpr_conf_lfpr_bind_zeta/
- | /3 width=11 by cpr_conf_lfpr_bind_zeta/
- | /3 width=12 by cpr_conf_lfpr_zeta_zeta/
- ]
-| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
- | #p1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
- ]
- elim (cpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #HX2 #H2
- |3,7,11,15: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
- |4,8,12,16: #p2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
- ] destruct
- [ /3 width=10 by cpr_conf_lfpr_flat_flat/
- | /4 width=8 by ex2_commute, cpr_conf_lfpr_flat_epsilon/
- | /4 width=12 by ex2_commute, cpr_conf_lfpr_flat_beta/
- | /4 width=14 by ex2_commute, cpr_conf_lfpr_flat_theta/
- | /3 width=8 by cpr_conf_lfpr_flat_epsilon/
- | /3 width=8 by cpr_conf_lfpr_epsilon_epsilon/
- | /3 width=12 by cpr_conf_lfpr_flat_beta/
- | /3 width=13 by cpr_conf_lfpr_beta_beta/
- | /3 width=14 by cpr_conf_lfpr_flat_theta/
- | /3 width=17 by cpr_conf_lfpr_theta_theta/
- ]
-]
-qed-.
-
-(* Basic_1: includes: pr0_confluence pr2_confluence *)
-theorem cpr_conf: ∀h,G,L. confluent … (cpm h G L 0).
-/2 width=6 by cpr_conf_lfpr/ qed-.
-
-(* Properties with context-sensitive parallel r-transition for terms ********)
-
-lemma lfpr_cpr_conf_dx: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T.
-#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lfpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
-qed-.
-
-lemma lfpr_cpr_conf_sn: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T.
-#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lfpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lfpr_conf: ∀h,G,T. confluent … (lfpr h G T).
-/3 width=6 by cpr_conf_lfpr, lfpr_fsge_comp, lfxs_conf/ qed-.
-
-theorem lfpr_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
- ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, T] L2.ⓑ{I}V2 →
- ∀p. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V1.T] L2.
-/2 width=2 by lfxs_bind/ qed.
-
-theorem lfpr_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ➡[h, V] L2 →
- ∀I,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2.
-/2 width=1 by lfxs_flat/ qed.
-
-theorem lfpr_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ➡[h, V] L2 →
- ∀T. ⦃G, L1.ⓧ⦄ ⊢ ➡[h, T] L2.ⓧ →
- ∀p,I. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2.
-/2 width=1 by lfxs_bind_void/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/lfpx.ma".
-include "basic_2/rt_transition/cpm_cpx.ma".
-include "basic_2/rt_transition/lfpr.ma".
-
-(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
-
-(* Fwd. lemmas with unc. rt-transition for local env.s on referred entries **)
-
-(* Basic_2A1: uses: lpr_lpx *)
-lemma lfpr_fwd_lfpx: ∀h,T,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
-/3 width=3 by cpm_fwd_cpx, lfxs_co/ qed-.
include "basic_2/static/lfxs.ma".
include "basic_2/rt_transition/cpx_ext.ma".
-(* UNBOUND 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
"unbound parallel rt-transition on referred entries (local environment)"
(* 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-.
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
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_drops.ma".
-include "basic_2/static/lsuba_aaa.ma".
-include "basic_2/rt_transition/lfpx_fqup.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
-
-(* Properties with atomic arity assignment for terms ************************)
-
-(* Note: lemma 500 *)
-(* Basic_2A1: uses: cpx_lpx_aaa_conf *)
-lemma cpx_aaa_conf_lfpx: ∀h,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #s #X2 #HX
- elim (cpx_inv_sort1 … HX) -HX //
-| #I #G #L1 #V1 #B #_ #IH #X2 #HX #Y #HY
- elim (lfpx_inv_zero_pair_sn … HY) -HY #L2 #V2 #HL12 #HV12 #H destruct
- elim (cpx_inv_zero1_pair … HX) -HX
- [ #H destruct /3 width=1 by aaa_zero/
- | -HV12 * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
- ]
-| #I #G #L1 #B #i #_ #IH #X2 #HX #Y #HY
- elim (lfpx_inv_lref_bind_sn … HY) -HY #I2 #L2 #HL12 #H destruct
- elim (cpx_inv_lref1_bind … HX) -HX
- [ #H destruct /3 width=1 by aaa_lref/
- | * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
- ]
-| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
- elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
- elim (cpx_inv_abbr1 … HX) -HX *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- /5 width=2 by lfpx_bind_repl_dx, aaa_abbr, ext2_pair/
- | #T2 #HT12 #HXT2 #H destruct -IHV
- /4 width=7 by aaa_inv_lifts, drops_drop, drops_refl/
- ]
-| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
- elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
- elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /5 width=2 by lfpx_bind_repl_dx, aaa_abst, ext2_pair/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
- elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
- elim (cpx_inv_appl1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
- | #q #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
- lapply (IHV … HV12 … HV) -IHV -HV12 -HV #HV2
- lapply (IHT (ⓛ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
- elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
- /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
- | #q #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
- lapply (aaa_lifts G L2 … B … (L2.ⓓW2) … HV2) -HV2 /3 width=2 by drops_drop, drops_refl/ #HV2
- lapply (IHT (ⓓ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
- elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
- ]
-| #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
- elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
- elim (cpx_inv_cast1 … HX) -HX
- [ * #V2 #T2 #HV12 #HT12 #H destruct ] /3 width=1 by aaa_cast/
-]
-qed-.
-
-lemma cpx_aaa_conf: ∀h,G,L. Conf3 … (aaa G L) (cpx h G L).
-/2 width=6 by cpx_aaa_conf_lfpx/ qed-.
-
-(* Basic_2A1: uses: lpx_aaa_conf *)
-lemma lfpx_aaa_conf: ∀h,G,T. Conf3 … (λL. aaa G L T) (lfpx h G T).
-/2 width=6 by cpx_aaa_conf_lfpx/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lfxs_drops.ma".
-include "basic_2/rt_transition/cpx_drops.ma".
-include "basic_2/rt_transition/lfpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
-
-(* Properties with generic slicing for local environments *******************)
-
-(* Basic_2A1: uses: drop_lpx_trans *)
-lemma drops_lfpx_trans: ∀h,G. dedropable_sn (cpx h G).
-/3 width=6 by lfxs_liftable_dedropable_sn, cpx_lifts_sn/ qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-(* Basic_2A1: uses: lpx_drop_conf *)
-lemma lfpx_drops_conf: ∀h,G. dropable_sn (cpx h G).
-/2 width=5 by lfxs_dropable_sn/ qed-.
-
-(* Basic_2A1: uses: lpx_drop_trans_O1 *)
-lemma lfpx_drops_trans: ∀h,G. dropable_dx (cpx h G).
-/2 width=5 by lfxs_dropable_dx/ qed-.
-
-lemma lfpx_inv_lref_pair_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2.
-/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
-
-lemma lfpx_inv_lref_pair_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2.
-/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.
include "basic_2/static/lfxs_fqup.ma".
include "basic_2/rt_transition/lfpx.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* Advanced properties ******************************************************)
-(* Basic_2A1: uses: lpx_refl *)
lemma lfpx_refl: ∀h,G,T. reflexive … (lfpx h G T).
/2 width=1 by lfxs_refl/ qed.
-(* Basic_2A1: uses: lpx_pair *)
lemma lfpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
/2 width=1 by lfxs_pair_refl/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/lfpx_drops.ma".
-include "basic_2/rt_transition/lfpx_fsle.ma".
-include "basic_2/s_transition/fquq.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
-
-(* Properties with extended structural successor for closures ***************)
-
-(* Basic_2A1: uses: lpx_fqu_trans *)
-lemma lfpx_fqu_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
-#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #K #V #K1 #H
- elim (lfpx_inv_zero_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
- elim (lifts_total V (𝐔❴1❵)) #T #HVT
- /3 width=7 by lfpx_cpx_conf, cpx_delta, fqu_drop, ex3_2_intro/
-| /3 width=7 by lfpx_fwd_pair_sn, cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
-| /3 width=6 by lfpx_fwd_bind_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
-| /3 width=8 by lfpx_fwd_bind_dx_void, cpx_pair_sn, fqu_clear, ex3_2_intro/
-| /3 width=7 by lfpx_fwd_flat_dx, cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
-| #I #G #K #T #U #HTU #K1 #H
- elim (lfpx_drops_trans … H (Ⓣ) … HTU) -H
- [|*: /3 width=2 by drops_refl, drops_drop/ ] -I #K0 #HK10 #HK0
- elim (drops_inv_succ … HK10) -HK10 #I #Y #HY #H destruct
- lapply (drops_fwd_isid … HY ?) -HY // #H destruct
- /3 width=5 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-(* Properties with extended optional structural successor for closures ******)
-
-(* Basic_2A1: uses: lpx_fquq_trans *)
-lemma lfpx_fquq_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
-[ #H12 elim (lfpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
include "basic_2/rt_transition/lfpx_length.ma".
include "basic_2/rt_transition/lfpx_fqup.ma".
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* Forward lemmas with free variables inclusion for restricted closures *****)
(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
(* Note: This invalidates lfpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (lfpxs h G)" *)
-(* Basic_2A1: uses: lpx_cpx_frees_trans *)
-lemma lfpx_cpx_conf_fsge: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+lemma lfpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
#G #L #T #IH #G0 #L0 * *
[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
]
]
]
-qed.
+qed-.
-(* Basic_2A1: uses: cpx_frees_trans *)
-lemma cpx_fsge_comp: ∀h,G. R_fsge_compatible (cpx h G).
-/2 width=4 by lfpx_cpx_conf_fsge/ qed-.
-
-(* Basic_2A1: uses: lpx_frees_trans *)
-lemma lfpx_fsge_comp: ∀h,G. lfxs_fsge_compatible (cpx h G).
+lemma lfpx_fsge_comp (h) (G): lfxs_fsge_compatible (cpx h G).
/2 width=4 by lfpx_cpx_conf_fsge/ qed-.
+(**) (* this section concerns cpx *)
(* Properties with generic extension on referred entries ********************)
+(* Basic_2A1: uses: cpx_frees_trans *)
+lemma cpx_fsge_comp (h) (G): R_fsge_compatible (cpx h G).
+/2 width=4 by lfpx_cpx_conf_fsge/ qed-.
+
(* Note: lemma 1000 *)
(* Basic_2A1: uses: cpx_llpx_sn_conf *)
-lemma cpx_lfxs_conf: ∀R,h,G. s_r_confluent1 … (cpx h G) (lfxs R).
+lemma cpx_lfxs_conf (R) (h) (G): s_r_confluent1 … (cpx h G) (lfxs R).
/3 width=3 by fsge_lfxs_trans, cpx_fsge_comp/ qed-.
(* Advanced properties ******************************************************)
-lemma lfpx_cpx_conf: ∀h,G. s_r_confluent1 … (cpx h G) (lfpx h G).
+lemma lfpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (lfpx h G).
/2 width=5 by cpx_lfxs_conf/ qed-.
-lemma lfpx_cpx_conf_fsge_dx: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
+lemma lfpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
/3 width=5 by lfpx_cpx_conf, lfpx_fsge_comp/ qed-.
include "basic_2/static/lfxs_length.ma".
include "basic_2/rt_transition/lfpx.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* Forward lemmas with length for local environments ************************)
-(* Basic_2A1: uses: lpx_fwd_length *)
lemma lfpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
/2 width=3 by lfxs_fwd_length/ qed-.
include "basic_2/static/lfdeq_lfdeq.ma".
include "basic_2/rt_transition/lfpx_fsle.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* Properties with degree-based equivalence for local environments **********)
lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
/3 width=6 by lfpx_fsge_comp, lfdeq_fsge_comp, cpx_tdeq_conf_lexs, lfxs_conf/ qed-.
-(* Basic_2A1: uses: lleq_lpx_trans *)
lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
∀L1. L1 ≛[h, o, T] L2 →
∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
include "basic_2/static/lfxs_lfxs.ma".
include "basic_2/rt_transition/lfpx.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* Main properties **********************************************************)
include "basic_2/rt_transition/lfpx_fsle.ma".
include "basic_2/rt_transition/lpx.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-(* Properties with unbound parallel rt-transition for local environments ****)
+(* Properties with syntactic equivalence for referred local environments ****)
-lemma lfpx_lpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma fleq_lfpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/2 width=1 by lfeq_fwd_lfxs/ qed.
+
+(* Properties with unbound parallel rt-transition for full local envs *******)
+
+lemma lpx_lfpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
/2 width=1 by lfxs_lex/ qed.
-(* Inversion lemmas with unbound parallel rt-transition for local envs ******)
+(* Inversion lemmas with unbound parallel rt-transition for full local envs *)
lemma lfpx_inv_lpx_lfeq: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predsn_4.ma".
+include "basic_2/relocation/lex.ma".
+include "basic_2/rt_transition/cpr_ext.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+definition lpr (h) (G): relation lenv ≝
+ lex (λL. cpm h G L 0).
+
+interpretation
+ "parallel rt-transition (full local environment)"
+ 'PRedSn h G L1 L2 = (lpr h G L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lpr_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
+ ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ➡[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
+/2 width=1 by lex_bind/ qed.
+
+(* Note: lemma 250 *)
+lemma lpr_refl (h) (G): reflexive … (lpr h G).
+/2 width=1 by lex_refl/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
+ ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
+/2 width=1 by lex_bind_refl_dx/ qed.
+
+lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h] V2 →
+ ∀I. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
+/2 width=1 by lex_pair/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: was: lpr_inv_atom1 *)
+(* Basic_1: includes: wcpr0_gen_sort *)
+lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
+/2 width=2 by lex_inv_atom_sn/ qed-.
+
+lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
+ ∃∃I2,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+ L2 = K2.ⓘ{I2}.
+/2 width=1 by lex_inv_bind_sn/ qed-.
+
+(* Basic_2A1: was: lpr_inv_atom2 *)
+lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
+
+lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+ L1 = K1.ⓘ{I1}.
+/2 width=1 by lex_inv_bind_dx/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
+ ∃∃K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
+/2 width=1 by lex_inv_unit_sn/ qed-.
+
+(* Basic_2A1: was: lpr_inv_pair1 *)
+(* Basic_1: includes: wcpr0_gen_head *)
+lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+ L2 = K2.ⓑ{I}V2.
+/2 width=1 by lex_inv_pair_sn/ qed-.
+
+lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
+/2 width=1 by lex_inv_unit_dx/ qed-.
+
+(* Basic_2A1: was: lpr_inv_pair2 *)
+lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+ L1 = K1.ⓑ{I}V1.
+/2 width=1 by lex_inv_pair_dx/ qed-.
+
+lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
+/2 width=1 by lex_inv_pair/ qed-.
+
+(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lpr_lpx.ma".
+include "basic_2/rt_transition/lpx_aaa.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Properties with atomic arity assignment for terms ************************)
+
+lemma lpr_aaa_conf (h): ∀G,T. Conf3 … (λL. aaa G L T) (lpr h G).
+/3 width=5 by lpx_aaa_conf, lpr_fwd_lpx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/drops_lex.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/lpr.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: was: drop_lpr_trans *)
+lemma drops_lpr_trans (h) (G): dedropable_sn (λL. cpm h G L 0).
+/3 width=6 by lex_liftable_dedropable_sn, cpm_lifts_sn/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_1: includes: wcpr0_drop *)
+(* Basic_2A1: was: lpr_drop_conf *)
+lemma lpr_drops_conf (h) (G): dropable_sn (λL. cpm h G L 0).
+/2 width=3 by lex_dropable_sn/ qed-.
+
+(* Basic_1: includes: wcpr0_drop_back *)
+(* Basic_2A1: was: lpr_drop_trans_O1 *)
+lemma lpr_drops_trans (h) (G): dropable_dx (λL. cpm h G L 0).
+/2 width=3 by lex_dropable_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/s_transition/fquq.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/cpm_lsubr.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_transition/lpr.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Properties with extended structural successor for closures ***************)
+
+lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=5 by cpm_bind, fqu_bind_dx, ex3_2_intro/
+| /3 width=5 by cpm_bind_unit, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpr_flat, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #U #T #HUT #U2 #HU2
+ elim (cpm_lifts_sn … HU2 (Ⓣ) … (K.ⓘ{I}) … HUT) -U
+ /3 width=5 by lpr_bind_refl_dx, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=5 by cpm_bind, fqu_bind_dx, ex3_2_intro/
+| /3 width=5 by cpm_bind_unit, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpr_flat, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #U #T #HUT #U2 #HU2
+ elim (cpm_lifts_sn … HU2 (Ⓣ) … (K.ⓘ{I}) … HUT) -U
+ /3 width=5 by lpr_bind_refl_dx, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpr_bind_refl_dx, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #X #H
+ elim (lpr_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
+ /3 width=5 by cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
+ elim (lpr_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
+ /3 width=5 by cpr_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpr_pair_sn, fqu_flat_dx, ex3_2_intro/
+| /3 width=5 by lpr_bind_refl_dx, fqu_drop, ex3_2_intro/
+]
+qed-.
+
+(* Note: does not hold in Basic_2A1 because it requires cpm *)
+(* Note: L1 = K0.ⓛV0 and T1 = #0 require n = 1 *)
+lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ * #G #K #V #K1 #H
+ elim (lpr_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
+ elim (lifts_total V (𝐔❴1❵)) #T #HVT
+ /3 width=7 by cpm_ell, cpm_delta, fqu_drop, ex4_3_intro/
+| /3 width=7 by cpr_pair_sn, fqu_pair_sn, ex4_3_intro/
+| /3 width=7 by lpr_bind_refl_dx, cpr_pair_sn, fqu_bind_dx, ex4_3_intro/
+| /3 width=7 by lpr_bind_refl_dx, cpr_pair_sn, fqu_clear, ex4_3_intro/
+| /3 width=7 by cpr_pair_sn, fqu_flat_dx, ex4_3_intro/
+| #I #G #K #T #U #HTU #K1 #H
+ elim (lpr_inv_bind_dx … H) -H #I0 #K0 #HK0 #HI0 #H destruct
+ /3 width=7 by fqu_drop, ex4_3_intro/
+]
+qed-.
+
+(* Properties with extended optional structural successor for closures ******)
+
+lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
+[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
+[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
+[ #H12 elim (fqu_lpr_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
+[ #H12 elim (lpr_fqu_trans … H12 … HKL1) -L1 /3 width=7 by fqu_fquq, ex4_3_intro/
+| * #H1 #H2 #H3 destruct /2 width=7 by ex4_3_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex_length.ma".
+include "basic_2/rt_transition/lpr.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|.
+/2 width=2 by lex_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex_lex.ma".
+include "basic_2/rt_transition/cpm_lsubr.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_transition/cpr_drops.ma".
+include "basic_2/rt_transition/lpr_drops.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Main properties with context-sensitive parallel reduction for terms ******)
+
+fact cpr_conf_lpr_atom_atom (h):
+ ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T.
+/2 width=3 by cpr_refl, ex2_intro/ qed-.
+
+fact cpr_conf_lpr_atom_delta (h):
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -V2 -HLK2 #T #HVT #HT2
+/3 width=6 by cpm_delta_drops, ex2_intro/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_lpr_delta_delta (h):
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
+ ∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (drops_mono … H … HLK0) -H #H destruct
+elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1
+elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (cpm_lifts_sn … HV1 … HLK1 … HVT1) -V1 -HLK1 #T #HVT #HT1
+/3 width=11 by cpm_lifts_bi, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_bind (h):
+ ∀p,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
+#h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by lpr_pair, cpm_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_zeta (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpm_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2
+/3 width=3 by cpm_zeta, drops_refl, drops_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_zeta_zeta (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → ∀X1. ⬆*[1] X1 ≘ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G0 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2
+elim (cpm_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #T1 #HT1 #HXT1
+elim (cpm_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #T2 #HT2 #HXT2
+lapply (lifts_inj … HT2 … HT1) -T #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_flat (h):
+ ∀I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
+#h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) //
+/3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_eps (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
+/3 width=3 by cpm_eps, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_eps_eps (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T.
+#h #G0 #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_beta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
+/4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/
+qed-.
+
+(* Basic-1: includes:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_lpr_flat_theta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU2
+elim (cpm_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+ /4 width=7 by cpm_bind, cpm_appl, cpm_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpm_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -HXT1 /3 width=2 by drops_refl, drops_drop/
+ /4 width=9 by cpm_appl, cpm_zeta, lifts_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_lpr_beta_beta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
+/4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact cpr_conf_lpr_theta_theta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU1
+lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -HVU2 /3 width=2 by drops_refl, drops_drop/
+/4 width=7 by cpm_bind, cpm_appl, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_lpr (h): ∀G. lex_confluent (λL.cpm h G L 0) (λL.cpm h G L 0).
+#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓣ) … G0 L0 T0) -G0 -L0 -T0
+#G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1_drops … H1) -H1
+ elim (cpr_inv_atom1_drops … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_lpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_lpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_lpr_delta_delta/
+ ]
+| #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpm_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpm_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
+ | /3 width=11 by cpr_conf_lpr_bind_zeta/
+ | /3 width=12 by cpr_conf_lpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+ | #p1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #p2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_eps/
+ | /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
+ | /3 width=8 by cpr_conf_lpr_flat_eps/
+ | /3 width=7 by cpr_conf_lpr_eps_eps/
+ | /3 width=12 by cpr_conf_lpr_flat_beta/
+ | /3 width=13 by cpr_conf_lpr_beta_beta/
+ | /3 width=14 by cpr_conf_lpr_flat_theta/
+ | /3 width=17 by cpr_conf_lpr_theta_theta/
+ ]
+]
+qed-.
+
+(* Properties with context-sensitive parallel reduction for terms ***********)
+
+lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpr_conf (h) (G): confluent … (lpr h G).
+/3 width=6 by lex_conf, cpr_conf_lpr/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lpx.ma".
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/lpr.ma".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* Forward lemmas with unbound parallel rt-transition for ref local envs ****)
+
+(* Basic_2A1: was: lpr_lpx *)
+lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h] L2.
+/3 width=3 by cpm_fwd_cpx, lex_co/ qed-.
include "basic_2/relocation/lex.ma".
include "basic_2/rt_transition/cpx_ext.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENVIRONMENTS ********************)
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
-definition lpx: sh → genv → relation lenv ≝
- λh,G. lex (cpx h G).
+definition lpx (h) (G): relation lenv ≝
+ lex (cpx h G).
interpretation
- "unbound parallel rt-transition (local environment)"
+ "unbound parallel rt-transition on all entries (local environment)"
'PRedTySn h G L1 L2 = (lpx h G L1 L2).
(* Basic properties *********************************************************)
-lemma lpx_bind: ∀h,G,K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
+lemma lpx_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
+ ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
/2 width=1 by lex_bind/ qed.
-lemma lpx_refl: ∀h,G. reflexive … (lpx h G).
+lemma lpx_refl (h) (G): reflexive … (lpx h G).
/2 width=1 by lex_refl/ qed.
(* Advanced properties ******************************************************)
-lemma lpx_bind_refl_dx: ∀h,G,K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
+lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
+ ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-(*
-lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
-/2 width=1 by lpx_sn_pair/ qed.
-*)
+
+lemma lpx_pair (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I.⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
+/2 width=1 by lex_pair/ qed.
+
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpx_inv_atom1 *)
-lemma lpx_inv_atom_sn: ∀h,G,L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
+lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpx_inv_bind_sn: ∀h,I1,G,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
- L2 = K2.ⓘ{I2}.
+lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
+ ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+ L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpx_inv_atom2 *)
-lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
+lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
-lemma lpx_inv_bind_dx: ∀h,I2,G,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
- L1 = K1.ⓘ{I1}.
+lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+ L1 = K1.ⓘ{I1}.
/2 width=1 by lex_inv_bind_dx/ qed-.
(* Advanced inversion lemmas ************************************************)
+lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
+/2 width=1 by lex_inv_unit_sn/ qed-.
+
(* Basic_2A1: was: lpx_inv_pair1 *)
-lemma lpx_inv_pair_sn: ∀h,I,G,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
- L2 = K2.ⓑ{I}V2.
+lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+ L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
+lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
+/2 width=1 by lex_inv_unit_dx/ qed-.
+
(* Basic_2A1: was: lpx_inv_pair2 *)
-lemma lpx_inv_pair_dx: ∀h,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
- L1 = K1.ⓑ{I}V1.
+lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+ L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
-lemma lpx_inv_pair: ∀h,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
+lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
/2 width=1 by lex_inv_pair/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/aaa_drops.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/rt_transition/lpx_drops.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with atomic arity assignment for terms ************************)
+
+(* Note: lemma 500 *)
+(* Basic_2A1: was: cpx_lpx_aaa_conf *)
+lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #s #X #H
+ elim (cpx_inv_sort1 … H) -H #H destruct //
+| #I #G #K1 #V1 #B #_ #IH #X #HX #Y #HY
+ elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK12 #HV12 #H destruct
+ elim (cpx_inv_zero1_pair … HX) -HX
+ [ #H destruct /3 width=1 by aaa_zero/
+ | * #V #HV1 #HVX -HV12
+ /4 width=7 by aaa_lifts, drops_refl, drops_drop, true/
+ ]
+| #I1 #G #K1 #A #i #_ #IH #X #HX #Y #HY
+ elim (lpx_inv_bind_sn … HY) -HY #I2 #K2 #HK12 #_ #H destruct
+ elim (cpx_inv_lref1_bind … HX) -HX
+ [ #H destruct /3 width=1 by aaa_lref/
+ | * #T #HT #HTX
+ /4 width=7 by aaa_lifts, drops_refl, drops_drop, true/
+ ]
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpx_inv_abbr1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2 by lpx_pair, aaa_abbr/
+ | #T2 #HT12 #HT2 #H destruct -IHV1
+ /4 width=8 by lpx_pair, aaa_inv_lifts, drops_refl, drops_drop/
+ ]
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=1 by lpx_pair, aaa_abst/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpx_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
+ | #q #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+ lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
+ lapply (IHT1 (ⓛ{q}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+ /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
+ | #q #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+ lapply (aaa_lifts G L2 … B … (L2.ⓓW2) … HV2) -HV2 /3 width=2 by drops_refl, drops_drop/ #HV2
+ lapply (IHT1 (ⓓ{q}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
+ ]
+| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpx_inv_cast1 … H) -H
+ [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
+ | -IHV1 /2 width=1 by/
+ | -IHT1 /2 width=1 by/
+ ]
+]
+qed-.
+
+lemma cpx_aaa_conf (h): ∀G,L. Conf3 … (aaa G L) (cpx h G L).
+/2 width=7 by cpx_aaa_conf_lpx/ qed-.
+
+lemma lpx_aaa_conf (h): ∀G,T. Conf3 … (λL. aaa G L T) (lpx h G).
+/2 width=7 by cpx_aaa_conf_lpx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/drops_lex.ma".
+include "basic_2/rt_transition/cpx_drops.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: was: drop_lpx_trans *)
+lemma drops_lpx_trans (h) (G): dedropable_sn (cpx h G).
+/3 width=6 by lex_liftable_dedropable_sn, cpx_lifts_sn/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: was: lpx_drop_conf *)
+lemma lpx_drops_conf (h) (G): dropable_sn (cpx h G).
+/2 width=3 by lex_dropable_sn/ qed-.
+
+(* Basic_2A1: was: lpx_drop_trans_O1 *)
+lemma lpx_drops_trans (h) (G): dropable_dx (cpx h G).
+/2 width=3 by lex_dropable_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/s_transition/fquq.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with extended structural successor for closures ***************)
+
+lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #K #V #K1 #H
+ elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
+ elim (lifts_total V (𝐔❴1❵)) #T #HVT
+ /3 width=5 by cpx_delta, fqu_drop, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #T #U #HTU #K1 #H
+ elim (lpx_inv_bind_dx … H) -H #I0 #K0 #HK0 #HI0 #H destruct
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #X #H
+ elim (lpx_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
+ /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
+ elim (lpx_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
+ /3 width=5 by cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| /3 width=5 by lpx_bind_refl_dx, fqu_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with extended optional structural successor for closures ******)
+
+lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
+[ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
+[ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lfpx_lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Forward lemmas with free variables inclusion for restricted closures *****)
+
+(* Basic_2A1: uses: lpx_cpx_frees_trans *)
+lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+/3 width=4 by lfpx_cpx_conf_fsge, lpx_lfpx/ qed-.
+
+(* Basic_2A1: uses: lpx_frees_trans *)
+lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄.
+/2 width=4 by lpx_cpx_conf_fsge/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex_length.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|.
+/2 width=2 by lex_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfdeq_lfeq.ma".
+include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_transition/lfpx_lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: uses: lleq_lpx_trans *)
+lemma lfdeq_lpx_trans (h) (o) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 →
+ ∀L1. ∀T:term. L1 ≛[h, o, T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[h, o, T] K2.
+#h #o #G #L2 #K2 #HLK2 #L1 #T #HL12
+lapply (lpx_lfpx … T HLK2) -HLK2 #HLK2
+elim (lfdeq_lfpx_trans … HLK2 … HL12) -L2 #K #H #HK2
+elim (lfpx_inv_lpx_lfeq … H) -H #K1 #HLK1 #HK1
+/3 width=5 by lfeq_lfdeq_trans, ex2_intro/
+qed-.
(* Properties with generic slicing for local environments *******************)
-lemma lfdeq_lifts_sn: ∀h,o. dedropable_sn (cdeq h o).
+lemma lfdeq_lifts_sn: ∀h,o. f_dedropable_sn (cdeq h o).
/3 width=5 by lfxs_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma lfdeq_inv_lifts_sn: ∀h,o. dropable_sn (cdeq h o).
+lemma lfdeq_inv_lifts_sn: ∀h,o. f_dropable_sn (cdeq h o).
/2 width=5 by lfxs_dropable_sn/ qed-.
(* Note: missing in basic_2A1 *)
-lemma lfdeq_inv_lifts_dx: ∀h,o. dropable_dx (cdeq h o).
+lemma lfdeq_inv_lifts_dx: ∀h,o. f_dropable_dx (cdeq h o).
/2 width=5 by lfxs_dropable_dx/ qed-.
(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-definition dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-definition dropable_sn: predicate (relation3 lenv term term) ≝
+definition f_dropable_sn: predicate (relation3 lenv term term) ≝
λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
-definition dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+definition f_dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
-definition lfxs_transitive_next: relation3 … ≝ λR1,R2,R3.
- ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
- lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
+definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
+ ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
+ lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
(* Properties with generic slicing for local environments *******************)
lemma lfxs_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
- d_liftable2_sn … lifts R → dedropable_sn R.
+ d_liftable2_sn … lifts R → f_dedropable_sn R.
#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
elim (frees_total L1 U) #f2 #Hf2
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
qed-.
-lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → lfxs_transitive_next R1 R2 R3.
+lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
generalize in match HLK; -HLK elim H -I1 -I
[ #I #_ #L2 #_ #I2 #H
(* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
(* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma lfxs_dropable_sn: ∀R. dropable_sn R.
+lemma lfxs_dropable_sn: ∀R. f_dropable_sn R.
#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
elim (frees_total K1 T) #f1 #Hf1
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
(* Basic_2A1: was: llpx_sn_drop_trans_O *)
(* Note: the proof might be simplified *)
-lemma lfxs_dropable_dx: ∀R. dropable_dx R.
+lemma lfxs_dropable_dx: ∀R. f_dropable_dx R.
#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
elim (drops_isuni_ex … H1f L1) #K1 #HLK1
elim (frees_total K1 T) #f1 #Hf1
qed-.
theorem lfxs_trans_fsle: ∀R1,R2,R3.
- lfxs_fsle_compatible R1 → lfxs_transitive_next R1 R2 R3 →
+ lfxs_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
∀L1,L,T. L1 ⪤*[R1, T] L →
∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
]
}
]
-(*
class "wine"
- [ { "higher order dynamic typing" * } {
- [ { "higher order native type assignment" * } {
- [ [ "" ] "ntas ( ⦃?,?⦄ ⊢ ? :* ? )" "nta_lift" * ]
+ [ { "iterated dynamic typing" * } {
+ [ { "" (* "higher order native type assignment" *) * } {
+ [ [ "" ] (* "ntas ( ⦃?,?⦄ ⊢ ? :* ? )" "nta_lift" *) * ]
}
]
}
]
-*)
class "magenta"
[ { "dynamic typing" * } {
(*
}
]
[ { "context-sensitive parallel r-transition" * } {
- [ [ "for lenvs on referred entries" ] "lfpr" + "( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fquq" + "lfpr_fqup" + "lfpr_aaa" + "lfpr_lfpx" + "lfpr_lfpr" * ]
+ [ [ "for lenvs on all entries" ] "lpr" + "( ⦃?,?⦄ ⊢ ➡[?] ? )" "lpr_length" + "lpr_drops" + "lpr_fquq" + "lpr_aaa" + "lpr_lpx" (* + "lfpr_lfpr" *) * ]
[ [ "for binders" ] "cpr_ext" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" * ]
- [ [ "for terms" ] "cpr" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" * ]
+ [ [ "for terms" ] "cpr" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" + "cpr_cpr" * ]
}
]
[ { "t-bound context-sensitive parallel rt-transition" * } {
- [ [ "for terms" ] "cpm" + "( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_drops" + "cpm_lsubr" + "cpm_fsle" + "cpm_cpx" * ]
+ [ [ "for terms" ] "cpm" + "( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_drops" + "cpm_lsubr" + "cpm_fsle" + "cpm_aaa" + "cpm_cpx" * ]
}
]
[ { "unbound context-sensitive parallel rt-transition" * } {
[ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_cnx" * ]
- [ [ "for lenvs on referred entries" ] "lfpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fquq" + "lfpx_fqup" + "lfpx_fsle" + "lfpx_lfdeq" + "lfpx_aaa" + "lfpx_lpx" + "lfpx_lfpx" * ]
- [ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" * ]
+ [ [ "for lenvs on referred entries" ] "lfpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_fqup" + "lfpx_fsle" + "lfpx_lfdeq" + "lfpx_lpx" + "lfpx_lfpx" * ]
+ [ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" "lpx_length" + "lpx_drops" + "lpx_fquq" + "lpx_fsle" + "lpx_lfdeq" + "lpx_aaa" * ]
[ [ "for binders" ] "cpx_ext" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" * ]
[ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_lfeq" + "cpx_lfdeq" + "cpx_ffdeq" * ]
}
class "orange"
[ { "relocation" * } {
[ { "generic slicing" * } {
- [ [ "for lenvs" ] "drops" + "( ⬇*[?,?] ? ≘ ? )" + "( ⬇*[?] ? ≘ ? )" "drops_ctc" + "drops_ltc" + "drops_weight" + "drops_length" + "drops_cext2" + "drops_lexs" + "drops_lreq" + "drops_drops" + "drops_vector" * ]
+ [ [ "for lenvs" ] "drops" + "( ⬇*[?,?] ? ≘ ? )" + "( ⬇*[?] ? ≘ ? )" "drops_ctc" + "drops_ltc" + "drops_weight" + "drops_length" + "drops_cext2" + "drops_lexs" + "drops_lex" + "drops_lreq" + "drops_drops" + "drops_vector" * ]
}
]
[ { "generic relocation" * } {
}
]
[ { "generic entrywise extension" * } {
- [ [ "for lenvs of one contex-sensitive relation" ] "lex" + "( ? ⦻[?] ? )" "lex_tc" + "lex_length" * ]
+ [ [ "for lenvs of one contex-sensitive relation" ] "lex" + "( ? ⦻[?] ? )" "lex_tc" + "lex_length" + "lex_lex" * ]
[ [ "for lenvs of two contex-sensitive relations" ] "lexs" + "( ? ⦻*[?,?,?] ? )" "lexs_tc" + "lexs_length" + "lexs_lexs" * ]
}
]
class "italic" { 2 }
(*
+ [ [ "for lenvs on referred entries" ]
+ "lfpr" + "( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fquq" + "lfpr_fqup" + "lfpr_aaa" + "lfpr_lfpx" + "lfpr_lfpr" * ]
[ { "evaluation for context-sensitive rt-reduction" * } {
[ [ "" ] "cpxe ( ⦃?,?⦄ ⊢ ➡*[?,?] 𝐍⦃?⦄ )" * ]
}
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basics/pts.ma".
+
+(* GENERATED LIBRARY ********************************************************)
+
+lemma pull_2 (A1:Type[0])
+ (A2:Type[0])
+ (A:A1→A2→Type[0]):
+ (∀x2,x1. A x1 x2) →
+ (∀x1,x2. A x1 x2).
+/2 width=1 by/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basics/pts.ma".
+
+(* GENERATED LIBRARY ********************************************************)
+
+lemma pull_4 (A1:Type[0])
+ (A2:A1→Type[0])
+ (A3:∀x1.A2 x1→Type[0])
+ (A4:Type[0])
+ (A:∀x1:A1.∀x2:A2 x1.A3 x1 x2 → A4 → Type[0]):
+ (∀x4,x1,x2,x3. A x1 x2 x3 x4) →
+ (∀x1,x2,x3,x4. A x1 x2 x3 x4).
+/2 width=1 by/ qed-.
]
class "orange"
[ { "generated library" * } {
- [ { "equality insertion" * } {
+ [ { "generalization with equality" * } {
[ "insert_eq" * ]
}
]
+ [ { "permutation of quantifiers" * } {
+ [ "pull" * ]
+ }
+ ]
[ { "logical decomposables" * } {
[ "xoa ( ∃∃ ) ( ∨∨ ) ( ∧∧ )" * ]
}
projects / helm.git / commitdiff