- exclusion binder in local environments:

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / lfpx_frees.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma
index 935109f903b0756bc42e613679da1b3df17201dc..15004fe7f9064a8298c3ed7a956e43012360d803 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma
@@ -13,85 +13,81 @@
 (**************************************************************************)
 
 include "basic_2/relocation/drops_lexs.ma".
-include "basic_2/s_computation/fqup_weight.ma".
 include "basic_2/static/frees_drops.ma".
+include "basic_2/static/lsubf_frees.ma".
+include "basic_2/static/lfxs.ma".
 include "basic_2/rt_transition/cpx_drops.ma".
+include "basic_2/rt_transition/cpx_ext.ma".
 
 (* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
 
 (* Properties with context-sensitive free variables *************************)
 
-axiom pippo: ∀RN,RP,L1,i. ⬇*[Ⓕ, 𝐔❴i❵] L1 ≡ ⋆ → 
-             ∀f,L2. L1 ⦻*[RN, RP, f] L2 →
-             ⬇*[Ⓕ, 𝐔❴i❵] L2 ≡ ⋆.
-(*
-#RN #RP #L1 #i #H1 #f #L2 #H2
-lapply (lexs_co_dropable_sn … H1 … H2) // -HL1 -H2
-*)
-
-
-axiom frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f →
-                          ∀K. ⬇*[b, 𝐔❴1❵] L ≡ K → ∀T. ⬆*[1] T ≡ U →
-                          K ⊢ 𝐅*⦃T⦄ ≡ ⫱f.
-
-axiom frees_pair_flat: ∀L,T,f1,I1,V1. L.ⓑ{I1}V1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
-                       ∀f2,I2,V2. L.ⓑ{I2}V2 ⊢ 𝐅*⦃T⦄ ≡ f2 →
-                       ∀f0. f1 ⋓ f2 ≡ f0 →
-                       ∀I0,I. L.ⓑ{I0}ⓕ{I}V1.V2 ⊢ 𝐅*⦃T⦄ ≡ f0.
-
-(* Basic_2A1: was: lpx_cpx_frees_trans *)
-lemma cpx_frees_trans_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
-                            ∀L2. L1 ⦻*[cpx h G, cfull, f1] L2 →
-                            ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
-                            ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
-#h #G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1
+(* Basic_2A1: uses: lpx_cpx_frees_trans *)
+(* check sle_refl *)
+lemma cpx_frees_conf_lfpx: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
+                           ∀L2. L1 ⪤*[cpx_ext h G, cfull, f1] L2 →
+                           ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
+                           ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
+#h #G #L1 #T1 @(fqup_wf_ind_eq (Ⓣ) … G L1 T1) -G -L1 -T1
 #G0 #L0 #U0 #IH #G #L1 * *
 [ -IH #s #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
   lapply (frees_inv_sort … H1) -H1 #Hg1
   elim (cpx_inv_sort1 … H0) -H0 #H destruct
-  /3 width=3 by frees_sort_gen, sle_refl, ex2_intro/
+  /3 width=3 by frees_sort, sle_refl, ex2_intro/
 | #i #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
   elim (frees_inv_lref_drops … H1) -H1 *
-  [ -IH #HL1 #Hg1
+  [ -IH #f1 #HL1 #Hf1 #H destruct
     elim (cpx_inv_lref1_drops … H0) -H0
-    [ #H destruct lapply (pippo … HL1 … H2) -HL1 -H2
-      /3 width=3 by frees_lref_atom, sle_refl, ex2_intro/
-    | * -H2 -Hg1 #I #K1 #V1 #V2 #HLK1
+    [ #H destruct
+      /4 width=9 by frees_atom_drops, drops_atom2_lexs_conf, sle_refl, ex2_intro/
+    | * -H2 -Hf1 #I #K1 #V1 #V2 #HLK1
       lapply (drops_TF … HLK1) -HLK1 #HLK1
       lapply (drops_mono … HLK1 … HL1) -L1 #H destruct
     ]
   | #f1 #I #K1 #V1 #Hf1 #HLK1 #H destruct
     elim (cpx_inv_lref1_drops … H0) -H0
     [ #H destruct
-      elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #K2 #V2 #HLK2 #HK12 #HV12
+      elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #Z #K2 #HLK2 #HK12 #H
+      elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
       elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -L1 -K1 -V1 #f2 #Hf2 #Hf21
-      /4 width=7 by frees_lref_pushs, frees_lref_pair, drops_refl, sle_next, ex2_intro, sle_pushs/
-    | * #J #Y #X #V2 #H #HV12 #HVU2
+      /4 width=7 by frees_lref_pushs, frees_pair_drops, drops_refl, sle_pushs, sle_next, ex2_intro/
+    | * #Z #Y #X #V2 #H #HV12 #HVU2
       lapply (drops_mono … H … HLK1) -H #H destruct
-      elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #K2 #V0 #HLK2 #HK12 #_
+      elim (lexs_drops_conf_next … H2 … HLK1) -H2 [ |*: // ] #I2 #K2 #HLK2 #HK12 #H
+      elim (ext2_inv_pair_sn … H) -H #V0 #_ #H destruct
       lapply (drops_isuni_fwd_drop2 … HLK2) // -V0 #HLK2
       elim (IH … Hf1 … HK12 … HV12) /2 width=2 by fqup_lref/ -I -L1 -K1 -V1 #f2 #Hf2 #Hf21
       lapply (frees_lifts … Hf2 … HLK2 … HVU2 ??) /4 width=7 by sle_weak, ex2_intro, sle_pushs/
     ]
+  | #f1 #I #K1 #HLK1 #Hf1 #H destruct
+    elim (cpx_inv_lref1_drops … H0) -H0
+    [ -IH #H destruct
+      elim (lexs_drops_conf_next … H2 … HLK1) -H2 -HLK1 [ |*: // ] #Z #K2 #HLK2 #_ #H
+      lapply (ext2_inv_unit_sn … H) -H #H destruct
+      /3 width=3 by frees_unit_drops, sle_refl, ex2_intro/
+    | * -H2 -Hf1 #Z #Y1 #X1 #X2 #HLY1
+      lapply (drops_mono … HLK1 … HLY1) -L1 #H destruct
+    ]
   ]
 | -IH #l #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
   lapply (frees_inv_gref … H1) -H1 #Hg1
   lapply (cpx_inv_gref1 … H0) -H0 #H destruct
-  /3 width=3 by frees_gref_gen, sle_refl, ex2_intro/
+  /3 width=3 by frees_gref, sle_refl, ex2_intro/
 | #p #I #V1 #T1 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
   elim (frees_inv_bind … H1) -H1 #gV1 #gT1 #HgV1 #HgT1 #Hg1
   elim (cpx_inv_bind1 … H0) -H0 *
   [ #V2 #T2 #HV12 #HT12 #H destruct
     lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
     lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
-    lapply (lexs_inv_tl … I … HL12T … HV12 ?) // -HL12T #HL12T
+    lapply (lexs_inv_tl … (BPair I V1) (BPair I V2) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
     elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
     elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
     elim (sor_isfin_ex gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/
     /4 width=10 by frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
   | #T2 #HT12 #HUT2 #H0 #H1 destruct -HgV1
     lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
-    lapply (lexs_inv_tl … Abbr … V1 V1 HL12T ??) // -HL12T #HL12T
+    lapply (lexs_inv_tl … (BPair Abbr V1) (BPair Abbr V1) … HL12T ??) /2 width=1 by ext2_pair/ -HL12T #HL12T
     elim (IH … HgT1 … HL12T … HT12) // -L1 -T1 #gT2 #HgT2 #HgT21
     lapply (frees_inv_lifts_SO (Ⓣ) … HgT2 … L2 … HUT2) [ /3 width=1 by drops_refl, drops_drop/ ] -V1 -T2
     /5 width=6 by sor_inv_sle_dx, sle_trans, sle_tl, ex2_intro/
@@ -120,39 +116,52 @@ lemma cpx_frees_trans_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
     lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
     lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
     lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
-    lapply (lexs_inv_tl … Abst … HL12T … HW12 ?) // -HL12T #HL12T
+    lapply (lexs_inv_tl … (BPair Abst W1) (BPair Abst W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
+    elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
+    lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+    lapply (sor_comm … Hg0) -Hg0 #Hg0
     elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
     elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
     elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
+    elim (sor_isfin_ex (⫱gT2) gV2) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
+    elim (lsubf_beta_tl_dx … HgV2 … HgVT2 … W2) [ |*: /1 width=3 by lsubf_refl/ ] #gT0 #HL2 #HgT02
+    lapply (lsubf_frees_trans … HgT2 … HL2) -HgT2 -HL2 #HgT0
+    lapply (sor_comm … HgVT2) -HgVT2 #HgVT2 (**) (* this should be removed *)
     elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #gV0 #HgV0 #H
-    elim (sor_isfin_ex gV0 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
-    @(ex2_intro … g2)
-    [ @(frees_bind … Hg2) /2 width=5 by frees_flat/ ]
+    elim (sor_isfin_ex gV0 (⫱gT0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
+    @(ex2_intro … g2) /3 width=5 by frees_flat, frees_bind/ -h -p -G -L1 -L2 -V1 -V2 -W1 -W2 -T1 -T2
+    @(sor_inv_sle … Hg2) -Hg2
+    [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
+    | @(sle_trans … HgT02) -HgT02
+      /3 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl/
+    ] (**) (* full auto too slow *)
   | #p #V2 #V #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #H0 #H1 #H destruct
     elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
     lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
     lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
     lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
     lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
-    lapply (lexs_inv_tl … Abbr … HL12T … HW12 ?) // -HL12T #HL12T
+    lapply (lexs_inv_tl … (BPair Abbr W1) (BPair Abbr W2) … HL12T ??) /2 width=1 by ext2_pair, I/ -HL12T #HL12T
     elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
-    lapply (sor_trans2 … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+    lapply (sor_assoc_sn … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
     elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
     elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
     elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
     elim (sor_isfin_ex (↑gV2) gT2) /3 width=3 by frees_fwd_isfin, isfin_push/ #gV0 #HgV0 #H
     elim (sor_isfin_ex gW2 (⫱gV0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
     elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
-    lapply (sor_trans2 … Hg2 … (⫱gT2) … Hg) /2 width=1 by sor_tl/ #Hg2
-    lapply (frees_lifts (Ⓣ) … HgV2 … (L2.ⓓW2) … HV2 ??) [4: |*: /3 width=3 by drops_refl, drops_drop/ ] -V2 #HgV
-    lapply (sor_sym … Hg) -Hg #Hg
+    lapply (sor_assoc_sn … Hg2 … (⫱gT2) … Hg) /2 width=1 by sor_tl/ #Hg2
+    lapply (frees_lifts (Ⓣ) … HgV2 … (L2.ⓓW2) … HV2 ??)
+    [4: lapply (sor_comm … Hg) |*: /3 width=3 by drops_refl, drops_drop/ ] -V2 (**) (* full auto too slow *)
     /4 width=10 by frees_flat, frees_bind, monotonic_sle_sor, sle_tl, ex2_intro/
   ]
 ]
+qed-.
 
-lemma cpx_frees_trans: ∀h,o,G. frees_trans (cpx h o G).
-/2 width=8 by lpx_cpx_frees_trans/ qed-.
+(* Basic_2A1: uses: cpx_frees_trans *)
+lemma cpx_frees_conf: ∀h,G. R_frees_confluent (cpx h G).
+/4 width=7 by cpx_frees_conf_lfpx, lexs_refl, ext2_refl/ qed-.
 
-lemma lpx_frees_trans: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 →
-                       ∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄.
-/2 width=8 by lpx_cpx_frees_trans/ qed-.
+(* Basic_2A1: uses: lpx_frees_trans *)
+lemma lfpx_frees_conf: ∀h,G. lexs_frees_confluent (cpx_ext h G) cfull.
+/2 width=7 by cpx_frees_conf_lfpx/ qed-.