X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Flfpx_lfdeq.ma;h=668246dd1c56a09f551083e30d85e662336485ca;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=d2314455748accf17e35029c43bd6f718d605d5d;hpb=b11e1907f99bea1de50db890d849ba5469d2e0e7;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_lfdeq.ma
index d23144557..668246dd1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_lfdeq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_lfdeq.ma
@@ -12,15 +12,37 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/lifts_tdeq.ma".
 include "basic_2/static/lfdeq_fqup.ma".
-include "basic_2/rt_transition/lfpx.ma".
+include "basic_2/static/lfdeq_lfdeq.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
 
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
 
-(* Properties with degree-based equivalence for terms ***********************)
+(* Properties with degree-based equivalence for local environments **********)
+
+lemma lfpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
+                          ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
+/3 width=5 by lfpx_fsge_comp, lfxs_pair_sn_split/ qed-.
+
+lemma lfpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
+                          ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
+/3 width=5 by lfpx_fsge_comp, lfxs_flat_dx_split/ qed-.
+
+lemma lfpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
+                          ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
+/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split/ qed-.
+
+lemma lfpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
+                               ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
+/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split_void/ qed-.
+
+lemma lfpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (lfpx h G).
+/2 width=5 by tdeq_lfxs_conf/ qed-.
+
+lemma lfpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+                     ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
+/2 width=5 by tdeq_lfxs_div/ qed-.
 
-(* Basic_2A1: was just: cpx_lleq_conf *)
 lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
 #h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
 [ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
@@ -31,18 +53,18 @@ lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (
   elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
   elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
   elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
-  elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
+  elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
   >(tdeq_inv_lref1 … H0) -H0
-  elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
-  elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
-  elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
-  elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
+  elim (lfpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
+  elim (lfdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
+  elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
+  elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
 | #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
   elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
   elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
   elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
-  lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
+  lapply (lfdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
   elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
   elim (IHT … HT02 … H1 … H2) -L0 -T0
   /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
@@ -57,9 +79,9 @@ lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (
   elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
   elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
   elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
-  lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
+  lapply (lfdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
   elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
-  elim (tdeq_inv_lifts … HT1 … HUT1) -T1
+  elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
   /3 width=5 by cpx_zeta, ex2_intro/
 | #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
   elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
@@ -80,7 +102,7 @@ lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (
   elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
   elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
   elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
-  lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
+  lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
   elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
   elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
   elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
@@ -92,39 +114,55 @@ lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (
   elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
   elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
   elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
-  lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
+  lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
   elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
   elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
   elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
-  elim (tdeq_lifts … HV1 … HVU1) -V1
+  elim (tdeq_lifts_sn … HV1 … HVU1) -V1
   /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
 ]
 qed-.
 
-lemma cpx_tdeq_conf: ∀h,o,G,L,T0,T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
-                     ∀T2. T0 ≡[h, o] T2 →
-                     ∃∃T. T1 ≡[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
+lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+                     ∀T2. T0 ≛[h, o] T2 →
+                     ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
 #h #o #G #L #T0 #T1 #HT01 #T2 #HT02
 elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
 /2 width=3 by lfxs_refl, ex2_intro/
 qed-.
 
-lemma tdeq_cpx_trans: ∀h,o,G,L,T2,T0. T2 ≡[h, o] T0 →
+lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
                       ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 → 
-                      ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≡[h, o] T1.
+                      ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
 #h #o #G #L #T2 #T0 #HT20 #T1 #HT01
 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
 qed-.
 
-(*
-lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
-                     ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2.
-/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
+(* Basic_2A1: uses: cpx_lleq_conf *)
+lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+                      ∀L2. L0 ≛[h, o, T0] L2 →
+                      ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
+#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
+elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
+/2 width=3 by lfxs_refl, ex2_intro/
+qed-.
 
-lemma cpx_lleq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h o G) (lleq 0).
-/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
+(* Basic_2A1: uses: lleq_cpx_trans *)
+lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
+                       ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+                       ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
+#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
+elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
+/3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
+qed-.
 
-lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
-                        ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
-/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.
-*)
\ No newline at end of file
+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-.
+
+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.
+#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
+elim (lfpx_lfdeq_conf … o … HLK2 L1)
+/3 width=3 by lfdeq_sym, ex2_intro/
+qed-.