X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Freduction%2Flpx_lleq.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Freduction%2Flpx_lleq.ma;h=0563baecc7c783d18e9f616673f217726e6dea12;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0000000000000000000000000000000000000000;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_lleq.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_2A/multiple/llor_drop.ma".
+include "basic_2A/multiple/llpx_sn_llor.ma".
+include "basic_2A/multiple/llpx_sn_lpx_sn.ma".
+include "basic_2A/multiple/lleq_lreq.ma".
+include "basic_2A/multiple/lleq_llor.ma".
+include "basic_2A/reduction/cpx_lreq.ma".
+include "basic_2A/reduction/cpx_lleq.ma".
+include "basic_2A/reduction/lpx_frees.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+(* Note: contains a proof of llpx_cpx_conf *)
+lemma lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
+                      ∀L1,T,l. L1 ≡[T, l] L2 →
+                      ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & K1 ≡[T, l] K2.
+#h #g #G #L2 #K2 #HLK2 #L1 #T #l #HL12
+lapply (lpx_fwd_length … HLK2) #H1
+lapply (lleq_fwd_length … HL12) #H2
+lapply (lpx_sn_llpx_sn … T … l HLK2) // -HLK2 #H
+lapply (lleq_llpx_sn_trans … HL12 … H) /2 width=3 by lleq_cpx_trans/ -HL12 -H #H
+elim (llor_total L1 K2 T l) // -H1 -H2 #K1 #HLK1
+lapply (llpx_sn_llor_dx_sym … H … HLK1)
+[ /2 width=6 by cpx_frees_trans/
+| /3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/
+| /3 width=5 by llpx_sn_llor_fwd_sn, ex2_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                          ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
+  #K0 #V0 #H1KL1 #_ #H destruct
+  elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
+  #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
+  /2 width=4 by fqu_lref_O, ex3_intro/
+| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
+  [ elim (lleq_inv_bind … H)
+  | elim (lleq_inv_flat … H)
+  ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
+| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
+| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
+  /2 width=4 by fqu_flat_dx, ex3_intro/
+| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+  elim (drop_O1_le (Ⓕ) (m+1) K1)
+  [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
+    #H2KL elim (lpx_drop_trans_O1 … H1KL1 … HL1) -L1
+    #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
+    /3 width=4 by fqu_drop, ex3_intro/
+  | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
+    lapply (lleq_fwd_length … H2KL1) //
+  ]
+]
+qed-.
+
+lemma lpx_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fquq_inv_gen … H) -H
+[ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
+  #K #HT1 #H1KL #H2KL elim (lpx_lleq_fqu_trans … HT2 … H1KL H2KL) -L
+  /3 width=5 by fqup_strap1, ex3_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fqus_inv_gen … H) -H
+[ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+fact lreq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
+                              ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                              ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                                   (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
+[ #l #m #_ #L2 #H >(lpx_inv_atom1 … H) -H
+  /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpx_pair, lreq_cpx_trans, lreq_pair/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpx_pair, lreq_succ/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
+]
+qed-.
+
+lemma lreq_lpx_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+                           ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                           ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                                (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+/2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.