milestone in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / rdsx_lpxs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma
index 06649215c638c3237c828a5e906decd5a2ac7490..0368929ea76d87cdddb62f4a0832e98733353ccc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma
@@ -21,35 +21,36 @@ include "basic_2/rt_computation/rdsx_rdsx.ma".
 (* Properties with unbound rt-computation for full local environments *******)
 
 (* Basic_2A1: uses: lsx_intro_alt *)
-lemma rdsx_intro_lpxs (h) (o) (G):
-                      ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
-                      G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
+lemma rdsx_intro_lpxs (h) (G):
+                      ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) →
+                      G ⊢ ⬈*[h, T] 𝐒⦃L1⦄.
 /4 width=1 by lpx_lpxs, rdsx_intro/ qed-.
 
 (* Basic_2A1: uses: lsx_lpxs_trans *)
-lemma rdsx_lpxs_trans (h) (o) (G): ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
-                                   ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
-#h #o #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
+lemma rdsx_lpxs_trans (h) (G):
+      ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
+      ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄.
+#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
 /2 width=3 by rdsx_lpx_trans/
 qed-.
 
 (* Eliminators with unbound rt-computation for full local environments ******)
 
-lemma rdsx_ind_lpxs_rdeq (h) (o) (G):
+lemma rdsx_ind_lpxs_rdeq (h) (G):
                          ∀T. ∀Q:predicate lenv.
-                         (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
-                               (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
+                         (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
+                               (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
                                Q L1
                          ) →
-                         ∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄  →
-                         ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[h, o, T] L2 → Q L2.
-#h #o #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
+                         ∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄  →
+                         ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+#h #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
 #L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
 @IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
 lapply (rdeq_rdneq_trans … HL02 … HnLK2) -HnLK2 #H
 elim (rdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
 lapply (rdneq_rdeq_canc_dx … H … HK02) -H #HnLK0
-elim (rdeq_dec h o L1 L0 T) #H
+elim (rdeq_dec L1 L0 T) #H
 [ lapply (rdeq_rdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
   lapply (lpxs_trans … HL10 … HLK0) -L0 #H10
   elim (lpxs_rdneq_inv_step_sn … H10 …  Hn10) -H10 -Hn10
@@ -61,31 +62,31 @@ elim (rdeq_dec h o L1 L0 T) #H
 qed-.
 
 (* Basic_2A1: uses: lsx_ind_alt *)
-lemma rdsx_ind_lpxs (h) (o) (G):
+lemma rdsx_ind_lpxs (h) (G):
                     ∀T. ∀Q:predicate lenv.
-                    (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
-                          (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
+                    (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ →
+                          (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
                           Q L1
                     ) →
-                    ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄  → Q L.
-#h #o #G #T #Q #IH #L #HL
+                    ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄  → Q L.
+#h #G #T #Q #IH #L #HL
 @(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
 qed-.
 
 (* Advanced properties ******************************************************)
 
-fact rdsx_bind_lpxs_aux (h) (o) (G):
-                        ∀p,I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
-                        ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
+fact rdsx_bind_lpxs_aux (h) (G):
+                        ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
+                        ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
                         ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
-                        G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
-#h #o #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
+                        G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
 #L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
 #Y #HY #IHY #L2 #H #HL12 destruct
 @rdsx_intro_lpxs #L0 #HL20
 lapply (lpxs_trans … HL12 … HL20) #HL10 #H
 elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
+[ #HnV elim (rdeq_dec L1 L2 V)
   [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
     /3 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
   | -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
@@ -95,23 +96,23 @@ elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
 qed-.
 
 (* Basic_2A1: uses: lsx_bind *)
-lemma rdsx_bind (h) (o) (G):
-                ∀p,I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
-                ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓑ{I}V⦄ →
-                G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+lemma rdsx_bind (h) (G):
+                ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
+                ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓑ{I}V⦄ →
+                G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
 /2 width=3 by rdsx_bind_lpxs_aux/ qed.
 
 (* Basic_2A1: uses: lsx_flat_lpxs *)
-lemma rdsx_flat_lpxs (h) (o) (G):
-                     ∀I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
-                     ∀L2,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
-                     G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L2⦄.
-#h #o #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
+lemma rdsx_flat_lpxs (h) (G):
+                     ∀I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
+                     ∀L2,T. G ⊢ ⬈*[h, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
+                     G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L2⦄.
+#h #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
 #L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2
 #L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs
 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
 #HL10 #H elim (rdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
+[ #HnV elim (rdeq_dec L1 L2 V)
   [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
     /3 width=5 by rdsx_lpxs_trans, rdeq_canc_sn/ (**) (* full auto too slow: 47s *)
   | -HnV -HL10 /3 width=4 by rdsx_lpxs_trans/
@@ -121,23 +122,23 @@ lemma rdsx_flat_lpxs (h) (o) (G):
 qed-.
 
 (* Basic_2A1: uses: lsx_flat *)
-lemma rdsx_flat (h) (o) (G):
-                ∀I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
-                ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄.
+lemma rdsx_flat (h) (G):
+                ∀I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
+                ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄.
 /2 width=3 by rdsx_flat_lpxs/ qed.
 
-fact rdsx_bind_lpxs_void_aux (h) (o) (G):
-                             ∀p,I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
-                             ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
+fact rdsx_bind_lpxs_void_aux (h) (G):
+                             ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ →
+                             ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ →
                              ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
-                             G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
-#h #o #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
+                             G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
 #L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
 #Y #HY #IHY #L2 #H #HL12 destruct
 @rdsx_intro_lpxs #L0 #HL20
 lapply (lpxs_trans … HL12 … HL20) #HL10 #H
 elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
+[ #HnV elim (rdeq_dec L1 L2 V)
   [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
     /3 width=6 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
   | -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
@@ -146,8 +147,8 @@ elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
 ]
 qed-.
 
-lemma rdsx_bind_void (h) (o) (G):
-                     ∀p,I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
-                     ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄ →
-                     G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄.
+lemma rdsx_bind_void (h) (G):
+                     ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ →
+                     ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄ →
+                     G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄.
 /2 width=3 by rdsx_bind_lpxs_void_aux/ qed.