milestone in basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma
index 0f46012343a86093eb52c43438100bff443becb6..d976e52d1c62b410405edae0f54083f41c8c00dc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma
@@ -12,45 +12,45 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/lsubeqx_6.ma".
+include "basic_2/notation/relations/lsubeqx_5.ma".
 include "basic_2/rt_computation/rdsx.ma".
 
 (* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
 
 (* Note: this should be an instance of a more general sex *)
 (* Basic_2A1: uses: lcosx *)
-inductive lsubsx (h) (o) (G): rtmap → relation lenv ≝
-| lsubsx_atom: ∀f. lsubsx h o G f (⋆) (⋆)
-| lsubsx_push: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
-               lsubsx h o G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
-| lsubsx_unit: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
-               lsubsx h o G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ →
-               lsubsx h o G f K1 K2 → lsubsx h o G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
+inductive lsubsx (h) (G): rtmap → relation lenv ≝
+| lsubsx_atom: ∀f. lsubsx h G f (⋆) (⋆)
+| lsubsx_push: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
+               lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
+| lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
+               lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
+| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ →
+               lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
 .
 
 interpretation
   "local environment refinement (clear)"
-  'LSubEqX h o f G L1 L2 = (lsubsx h o G f L1 L2).
+  'LSubEqX h f G L1 L2 = (lsubsx h G f L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsubsx_inv_atom_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
+fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
                              L1 = ⋆ → L2 = ⋆.
-#h #o #g #G #L1 #L2 * -g -L1 -L2 //
+#h #g #G #L1 #L2 * -g -L1 -L2 //
 [ #f #I #K1 #K2 #_ #H destruct
 | #f #I #K1 #K2 #_ #H destruct
 | #f #I #K1 #K2 #V #_ #_ #H destruct
 ]
 qed-.
 
-lemma lsubsx_inv_atom_sn: ∀h,o,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, o, g] L2 → L2 = ⋆.
+lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, g] L2 → L2 = ⋆.
 /2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
 
-fact lsubsx_inv_push_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
+fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
                              ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
-                             ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
+                             ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
+#h #g #G #L1 #L2 * -g -L1 -L2
 [ #f #g #J #L1 #_ #H destruct
 | #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
   <(injective_push … H1) -g /2 width=3 by ex2_intro/
@@ -61,14 +61,14 @@ fact lsubsx_inv_push_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
 ]
 qed-.
 
-lemma lsubsx_inv_push_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, o, ⫯f] L2 →
-                          ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
+lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, ⫯f] L2 →
+                          ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
 /2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
 
-fact lsubsx_inv_unit_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
+fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
                              ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
-                             ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
+                             ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+#h #g #G #L1 #L2 * -g -L1 -L2
 [ #f #g #J #L1 #_ #H destruct
 | #f #I #K1 #K2 #_ #g #J #L1 #H
   elim (discr_push_next … H)
@@ -78,15 +78,15 @@ fact lsubsx_inv_unit_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
 ]
 qed-.
 
-lemma lsubsx_inv_unit_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, o, ↑f] L2 →
-                          ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, ↑f] L2 →
+                          ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
 /2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
 
-fact lsubsx_inv_pair_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
+fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
                              ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
-                             ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
-                                   G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
+                             ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
+                                   G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+#h #g #G #L1 #L2 * -g -L1 -L2
 [ #f #g #J #L1 #W #_ #H destruct
 | #f #I #K1 #K2 #_ #g #J #L1 #W #H
   elim (discr_push_next … H)
@@ -97,18 +97,18 @@ fact lsubsx_inv_pair_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
 qed-.
 
 (* Basic_2A1: uses: lcosx_inv_pair *)
-lemma lsubsx_inv_pair_sn: ∀h,o,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, ↑f] L2 →
-                          ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
-                                G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, ↑f] L2 →
+                          ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
+                                G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
 /2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lsubsx_inv_pair_sn_gen: ∀h,o,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, g] L2 →
-                              ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
-                               | ∃∃f,K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
-                                         G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ↑f & L2 = K2.ⓧ.
-#h #o #g #I #G #K1 #L2 #V #H
+lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, g] L2 →
+                              ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
+                               | ∃∃f,K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
+                                         G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ↑f & L2 = K2.ⓧ.
+#h #g #I #G #K1 #L2 #V #H
 elim (pn_split g) * #f #Hf destruct
 [ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
 | elim (lsubsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
@@ -117,9 +117,9 @@ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma lsubsx_fwd_bind_sn: ∀h,o,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, o, g] L2 →
-                          ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, o, ⫱g] K2 & L2 = K2.ⓘ{I2}.
-#h #o #g #I1 #G #K1 #L2
+lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, g] L2 →
+                          ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, ⫱g] K2 & L2 = K2.ⓘ{I2}.
+#h #g #I1 #G #K1 #L2
 elim (pn_split g) * #f #Hf destruct
 [ #H elim (lsubsx_inv_push_sn … H) -H
 | cases I1 -I1 #I1
@@ -132,8 +132,8 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsubsx_eq_repl_back: ∀h,o,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
-#h #o #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
+lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
 [ #f #I #L1 #L2 #_ #IH #x #H
   elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
 | #f #I #L1 #L2 #_ #IH #x #H
@@ -143,15 +143,15 @@ lemma lsubsx_eq_repl_back: ∀h,o,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆
 ]
 qed-.
 
-lemma lsubsx_eq_repl_fwd: ∀h,o,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
-#h #o #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
+lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+#h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
 qed-.
 
 (* Advanced properties ******************************************************)
 
 (* Basic_2A1: uses: lcosx_O *)
-lemma lsubsx_refl: ∀h,o,f,G. 𝐈⦃f⦄ → reflexive … (lsubsx h o G f).
-#h #o #f #G #Hf #L elim L -L
+lemma lsubsx_refl: ∀h,f,G. 𝐈⦃f⦄ → reflexive … (lsubsx h G f).
+#h #f #G #Hf #L elim L -L
 /3 width=3 by lsubsx_eq_repl_back, lsubsx_push, eq_push_inv_isid/
 qed.