- improved definition of lsx allows more invariants

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
index 1f1ebc834316952f6c2de1dad1e31a14b0b080eb..2130d8b5144420af9d328f606907983fa0ff7476 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
@@ -20,35 +20,89 @@ include "basic_2/computation/lsx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_lleq_trans: ∀h,g,T,G,L1. G ⊢ ⋕⬊*[h, g, T] L1 →
-                      ∀L2. L1 ⋕[T, 0] L2 → G ⊢ ⋕⬊*[h, g, T] L2.
-#h #g #T #G #L1 #H @(lsx_ind … H) -L1
+lemma lsx_leqy_conf: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+                     ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → G ⊢ ⋕⬊*[h, g, T, d] L2.
+#h #g #G #L1 #T #d #H @(lsx_ind … H) -L1
+#L1 #_ #IHL1 #L2 #H1L12 #H2L12 @lsx_intro
+#L3 #H1L23 #HnL23 lapply (lpxs_fwd_length … H1L23)
+#H2L23 elim (lsuby_lpxs_trans_lleq … H1L12 … H1L23) -H1L12 -H1L23
+#L0 #H1L03 #H1L10 #H lapply (lpxs_fwd_length … H1L10)
+#H2L10 elim (H T) -H //
+#_ #H @(IHL1 … H1L10) -IHL1 -H1L10 /3 width=1 by/
+qed-.
+
+lemma lsx_lleq_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+                      ∀L2. L1 ⋕[T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
+#h #g #T #G #L1 #d #H @(lsx_ind … H) -L1
 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
-#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HL12 … HLK2) -HLK2
+#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
 /5 width=4 by lleq_canc_sn, lleq_trans/
 qed-.
 
-lemma lsx_lpxs_trans: ∀h,g,T,G,L1. G ⊢ ⋕⬊*[h, g, T] L1 →
-                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⋕⬊*[h, g, T] L2.
-#h #g #T #G #L1 #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 0) /3 width=4 by lsx_lleq_trans/
+lemma lsx_lpxs_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
+                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
+#h #g #T #G #L1 #d #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 d) /3 width=4 by lsx_lleq_trans/
+qed-.
+
+fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
+                        ∀Y,T. G ⊢ ⋕⬊*[h, g, T, ⫯d] Y →
+                        ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                        G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L2.
+#h #g #a #I #G #L1 #V #d #H @(lsx_ind … H) -L1
+#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 d)
+  [ #HV @(IHL1 … L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
+  | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
+  ]
+| /3 width=4 by lpxs_pair/
+]
 qed-.
 
-lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V. G ⊢ ⋕⬊*[h, g, V] L1 →
-                     ∀L2,T. G ⊢ ⋕⬊*[h, g, T] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                     G ⊢ ⋕⬊*[h, g, ⓕ{I}T.V] L2.
-#h #g #I #G #L1 #V #H @(lsx_ind … H) -L1
+lemma lsx_bind: ∀h,g,a,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
+                ∀T. G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V →
+                G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L.
+/2 width=3 by lsx_bind_lpxs_aux/ qed.
+
+lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
+                     ∀L2,T. G ⊢ ⋕⬊*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                     G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L2.
+#h #g #I #G #L1 #V #d #H @(lsx_ind … H) -L1
 #L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind … H) -L2
 #L2 #HL2 #IHL2 #HL12 @lsx_intro
 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL2 -IHL1 | -HL1 -IHL2 ]
-[ /3 width=1 by/
-| #HnV elim (lleq_dec V L1 L2 0)
+#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 d)
   [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
   | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
+  ]
+| /3 width=1 by/
 ]
 qed-.
 
-lemma lsx_flat: ∀h,g,I,G,L,V. G ⊢ ⋕⬊*[h, g, V] L →
-                ∀T. G ⊢ ⋕⬊*[h, g, T] L → G ⊢ ⋕⬊*[h, g, ⓕ{I}T.V] L.
+lemma lsx_flat: ∀h,g,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
+                ∀T. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L.
 /2 width=3 by lsx_flat_lpxs/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L →
+                       G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
+#h #g #a #I #G #L #V1 #T #d #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#Y #H #HT elim (lpxs_inv_pair1 … H) -H
+#L2 #V2 #HL12 #_ #H destruct
+@(lsx_leqy_conf … (L2.ⓑ{I}V1)) /2 width=1 by lsuby_succ/
+@IHL1 // #H @HT -IHL1 -HL12 -HT
+@(lleq_lsuby_trans … (L2.ⓑ{I}V1))
+/2 width=2 by lleq_fwd_bind_dx, lsuby_succ/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a, I}V.T, d] L →
+                    G ⊢ ⋕⬊*[h, g, V, d] L ∧ G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
+/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.