X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Flfpx_frees.ma;h=1ec892cbf054a980d323bc394eee386e5eff667d;hb=fed8c1a61273b0eb4a719fda70e2b5dd31933c8a;hp=47d35f2e56453600da7de42fadc44a36627fe43a;hpb=a041f17ec3bee5a321e6feee9937229e93c39a72;p=helm.git

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 47d35f2e5..1ec892cbf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma
@@ -11,49 +11,22 @@
 (*        v         GNU General Public License Version 2                  *)
 (*                                                                        *)
 (**************************************************************************)
-(*
-include "basic_2/multiple/frees_lreq.ma".
-include "basic_2/multiple/frees_lift.ma".
-*)
+
 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".
 
 (* 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
-*)
-
-lemma coafter_uni_sn: ∀i,f. 𝐔❴i❵ ~⊚ f ≡ ↑*[i] f.
-#i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
-qed.
-
-lemma sle_pushs: ∀f1,f2. f1 ⊆ f2 → ∀i. ↑*[i] f1 ⊆ ↑*[i] f2.
-#f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_push/
-qed.
-
-axiom monotonic_sle_sor: ∀f1,g1. f1 ⊆ g1 → ∀f2,g2. f2 ⊆ g2 →
-                         ∀f. f1 ⋓ f2 ≡ f → ∀g. g1 ⋓ g2 ≡ g → f ⊆ g.
-
-axiom sle_tl: ∀f1,f2. f1 ⊆ f2 → ⫱f1 ⊆ ⫱f2.
-
-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.
-
-(* 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.
+(* Basic_2A1: uses: lpx_cpx_frees_trans *)
+lemma cpx_frees_conf_lfpx: ∀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
 #G0 #L0 #U0 #IH #G #L1 * *
 [ -IH #s #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
@@ -64,8 +37,8 @@ lemma cpx_frees_trans_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
   elim (frees_inv_lref_drops … H1) -H1 *
   [ -IH #HL1 #Hg1
     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/
+    [ #H destruct
+      /5 width=9 by frees_lref_atom, drops_atom2_lexs_conf, coafter_isuni_isid, sle_refl, ex2_intro/
     | * -H2 -Hg1 #I #K1 #V1 #V2 #HLK1
       lapply (drops_TF … HLK1) -HLK1 #HLK1
       lapply (drops_mono … HLK1 … HL1) -L1 #H destruct
@@ -102,12 +75,82 @@ lemma cpx_frees_trans_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
   | #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
+    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/
+  ]
+| #I #V1 #T0 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
+  elim (frees_inv_flat … H1) -H1 #gV1 #gT0 #HgV1 #HgT0 #Hg1
+  elim (cpx_inv_flat1 … 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 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+    elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
+    elim (IH … HgT0 … HL12T … HT12) // -IH -HgT0 -HL12T -HT12 #gT2 #HgT2 #HgT21
+    elim (sor_isfin_ex gV2 gT2) /2 width=3 by frees_fwd_isfin/
+    /3 width=10 by frees_flat, monotonic_sle_sor, ex2_intro/
+  | #HU2 #H destruct -HgV1
+    lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+    elim (IH … HgT0 … HL12T … HU2) // -L1 -T0 -V1
+    /4 width=6 by sor_inv_sle_dx, sle_trans, ex2_intro/
+  | #HU2 #H destruct -HgT0
+    lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ -H2 #HL12V
+    elim (IH … HgV1 … HL12V … HU2) // -L1 -T0 -V1
+    /4 width=6 by sor_inv_sle_sn, sle_trans, ex2_intro/
+  | #p #V2 #W1 #W2 #T1 #T2 #HV12 #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 … Abst … HL12T … HW12 ?) // -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_sym … 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 gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
+    elim (lsubf_frees_trans … HgT2 (⫯gVT2) … (L2.ⓓⓝW2.V2))
+    [2: /4 width=4 by lsubf_refl, lsubf_beta, sor_inv_sle_dx, sor_inv_sle_sn, sle_inv_tl_sn/ ]
+    -HgT2 #gT0 #HgT0 #HgT20
+    elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #gV0 #HgV0 #H
+    elim (sor_isfin_ex gV0 (⫱gT0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
+    @(ex2_intro … g2)
+    [ @(frees_bind … Hg2) /2 width=5 by frees_flat/
+    | -L2 @(sor_inv_sle … Hg2) -Hg2
+      [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
+      | @sle_xn_tl [2:|*: // ] @(sle_trans … HgT20) -HgT20
+        /4 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl, sle_next/
+      ] (**) (* 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
+    elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
+    lapply (sor_trans2 … 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
-    lapply (frees_inv_lifts_SO (Ⓣ) … HgT2 … L2 … HUT2) [ /3 width=1 by drops_refl, drops_drop/ ]
+    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: lapply (sor_sym … 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).
+/3 width=7 by cpx_frees_conf_lfpx, lexs_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 h G) cfull.
+/2 width=7 by cpx_frees_conf_lfpx/ qed-.