- some refactoring and minor additions

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reduction / lpx_lleq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
index fd95e52cd4e52645d810586aacf0846606fcbe05..08918429bd96721ceda58d1bb11d066615bde276 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
@@ -12,16 +12,22 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/lleq_lleq.ma".
+include "basic_2/multiple/lleq_leq.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
+include "basic_2/reduction/cpx_leq.ma".
 include "basic_2/reduction/lpx_ldrop.ma".
 
 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
 
 (* Properties on lazy equivalence for local environments ********************)
 
-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 ⋕[0, T1] L1 →
-                          ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ⋕[0, T2] L2.
+axiom lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
+                      ∀L1,T,d. L1 ≡[T, d] L2 →
+                      ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & K1 ≡[T, d] K2.
+
+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
@@ -37,7 +43,7 @@ lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2,
 | #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 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
-  elim (ldrop_O1_le (e+1) K1)
+  elim (ldrop_O1_le (Ⓕ) (e+1) K1)
   [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
     #H2KL elim (lpx_ldrop_trans_O1 … H1KL1 … HL1) -L1
     #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
@@ -48,9 +54,9 @@ lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2,
 ]
 qed-.
 
-lemma lpx_lleq_fquq_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83⸮ ⦃G2, L2, T2⦄ →
-                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[0, T1] L1 →
-                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\83⸮ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[0, T2] L2.
+lemma lpx_lleq_fquq_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90⸮ ⦃G2, L2, T2⦄ →
+                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90⸮ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[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
@@ -59,9 +65,9 @@ elim (fquq_inv_gen … H) -H
 ]
 qed-.
 
-lemma lpx_lleq_fqup_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83+ ⦃G2, L2, T2⦄ →
-                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[0, T1] L1 →
-                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\83+ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[0, T2] L2.
+lemma lpx_lleq_fqup_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+ ⦃G2, L2, T2⦄ →
+                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90+ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[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/
@@ -71,9 +77,9 @@ lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2
 ]
 qed-.
 
-lemma lpx_lleq_fqus_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃G2, L2, T2⦄ →
-                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[0, T1] L1 →
-                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\83* â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[0, T2] L2.
+lemma lpx_lleq_fqus_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃G2, L2, T2⦄ →
+                           â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+                           â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90* â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[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
@@ -81,3 +87,33 @@ elim (fqus_inv_gen … H) -H
 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
 ]
 qed-.
+
+fact leq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
+                             ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                             ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                                  (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L).
+#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
+[ #d #e #_ #L2 #H >(lpx_inv_atom1 … H) -H
+  /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
+| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+  elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  lapply (ysucc_inv_Y_dx … He) -He #He
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpx_pair, leq_cpx_trans, leq_pair/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #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, leq_succ/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
+]
+qed-.
+
+lemma leq_lpx_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
+                          ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                          ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                               (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L).
+/2 width=1 by leq_lpx_trans_lleq_aux/ qed-.