- new component "s_transition" for the restored fqu and fquq

[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 9e731c5a9e50b3e9dabeb2df93e3cec1e112740f..8f2f7497b190ca60051e5037abf546d1558d9946 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
@@ -26,10 +26,10 @@ include "basic_2/reduction/lpx_frees.ma".
 (* 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 →
+lemma lleq_lpx_trans: ∀h,o,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, o] 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
+                      ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, o] K1 & K1 ≡[T, l] K2.
+#h #o #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
@@ -42,10 +42,10 @@ lapply (llpx_sn_llor_dx_sym … H … HLK1)
 ]
 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
+lemma lpx_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+                          ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #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 //
@@ -59,22 +59,22 @@ lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2,
   /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)
+| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+  elim (drop_O1_le (Ⓕ) (k+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 (drop_fwd_length_le2 … HL1) -L -T1 -o
     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
+lemma lpx_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #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/
@@ -82,10 +82,10 @@ elim (fquq_inv_gen … H) -H
 ]
 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
+lemma lpx_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #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
@@ -94,10 +94,10 @@ lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2
 ]
 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
+lemma lpx_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #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/
@@ -105,22 +105,22 @@ elim (fqus_inv_gen … H) -H
 ]
 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 &
+fact lreq_lpx_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
+                              ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
+                              ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] 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
+#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
+[ #l #k #_ #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
+| #I #L1 #L0 #V1 #k #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
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #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/
@@ -129,8 +129,8 @@ fact lreq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ 
 ]
 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 &
+lemma lreq_lpx_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+                           ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
+                           ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] L &
                                 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
 /2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.