- some renaming according to the written version of basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / lpxs_lleq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
index a3ca7a1acad43e7ec715e1fd324ae9dffb539ba9..6f6045dabd917744886cff100e01f28e8d292795 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
@@ -14,7 +14,7 @@
 
 include "basic_2/multiple/lleq_lleq.ma".
 include "basic_2/reduction/lpx_lleq.ma".
-include "basic_2/computation/cpxs_leq.ma".
+include "basic_2/computation/cpxs_lreq.ma".
 include "basic_2/computation/lpxs_drop.ma".
 include "basic_2/computation/lpxs_cpxs.ma".
 
@@ -23,19 +23,19 @@ include "basic_2/computation/lpxs_cpxs.ma".
 (* Properties on lazy equivalence for local environments ********************)
 
 lemma lleq_lpxs_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.
+                       ∀L1,T,l. L1 ≡[T, l] L2 →
+                       ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[T, l] K2.
 #h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
-#K #K2 #_ #HK2 #IH #L1 #T #d #HT elim (IH … HT) -L2
+#K #K2 #_ #HK2 #IH #L1 #T #l #HT elim (IH … HT) -L2
 #L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
 /3 width=3 by lpxs_strap1, ex2_intro/
 qed-.
 
-lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, d] L2 → ⊥) →
-                              ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, g] L & L1 ≡[T, d] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L0 & L0 ≡[T, d] L2.
-#h #g #G #L1 #L2 #T #d #H @(lpxs_ind_dx … H) -L1
+lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) →
+                              ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, g] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L0 & L0 ≡[T, l] L2.
+#h #g #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
 [ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L d) #H
+| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
   [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
     #L0 #L3 #H1 #H2 #H3 #H4 lapply (lleq_nlleq_trans … H … H2) -H2
     #H2 elim (lleq_lpx_trans … H1 … H) -L
@@ -64,8 +64,8 @@ lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2,
   /3 width=4 by lpxs_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 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
-  elim (drop_O1_le (Ⓕ) (e+1) K1)
+| #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 (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
     #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
@@ -110,32 +110,32 @@ elim (fqus_inv_gen … H) -H
 ]
 qed-.
 
-fact leq_lpxs_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 >(lpxs_inv_atom1 … H) -H
+fact lreq_lpxs_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 >(lpxs_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
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
-  lapply (ysucc_inv_Y_dx … He) -He #He
+  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 lpxs_pair, leq_cpxs_trans, leq_pair/
+  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_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
+  @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 (lpxs_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 lpxs_pair, leq_succ/
+  @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
   #T elim (IH T) #HL0dx #HL0sn
-  @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
 ]
 qed-.
 
-lemma leq_lpxs_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_lpxs_trans_lleq_aux/ qed-.
+lemma lreq_lpxs_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_lpxs_trans_lleq_aux/ qed-.