partial commit in delayed_updating

[helm.git] / matita / matita / contribs / lambdadelta / delayed_updating / syntax / bdd_term.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
index 0101b8d9361671cef4edcdb00f89c8fa34d70c05..1eb11fc030bc8c9f0676f2d82b52b9450ad371aa 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
@@ -12,38 +12,40 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_eq.ma".
+include "delayed_updating/notation/functions/class_d_tau_0.ma".
 include "ground/xoa/or_5.ma".
 include "ground/xoa/ex_3_1.ma".
 include "ground/xoa/ex_4_2.ma".
 include "ground/xoa/ex_4_3.ma".
 include "ground/xoa/ex_5_3.ma".
-include "delayed_updating/syntax/preterm_equivalence.ma".
-include "delayed_updating/syntax/preterm_constructors.ma".
-include "delayed_updating/notation/functions/class_d_phi_0.ma".
 
 (* BY-DEPTH DELAYED (BDD) TERM **********************************************)
 
-inductive bdd: 𝒫❨preterm❩ ≝
-| bdd_oref: ∀n. bdd (#n)
-| bdd_iref: ∀t,n. bdd t → bdd (𝛗n.t)
+inductive bdd: 𝒫❨prototerm❩ ≝
+| bdd_oref: ∀k. bdd (⧣k)
+| bdd_iref: ∀t,k. bdd t → bdd (𝛕k.t)
 | bdd_abst: ∀t. bdd t → bdd (𝛌.t)
-| bdd_appl: ∀u,t. bdd u → bdd t → bdd (@u.t)
+| bdd_appl: ∀u,t. bdd u → bdd t → bdd (＠u.t)
 | bdd_conv: ∀t1,t2. t1 ⇔ t2 → bdd t1 → bdd t2
 .
 
 interpretation
-  "by-depth delayed (preterm)"
-  'ClassDPhi = (bdd).
+  "by-depth delayed (prototerm)"
+  'ClassDTau = (bdd).
+
+(* COMMENT
 
 (* Basic inversions *********************************************************)
 
 lemma bdd_inv_in_comp_gen:
-      â\88\80t,p. t ϵ ð\9d\90\83ð\9d\9b\97 → p ϵ t →
+      â\88\80t,p. t ϵ ð\9d\90\83ð\9d\9b\95 → p ϵ t →
       ∨∨ ∃∃n. #n ⇔ t & 𝗱n◗𝐞 = p
-       | â\88\83â\88\83u,q,n. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & ð\9d\9b\97n.u â\87\94 t & ð\9d\97±n◗q = p
-       | â\88\83â\88\83u,q. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & 𝛌.u ⇔ t & 𝗟◗q = p
-       | â\88\83â\88\83v,u,q. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & @v.u ⇔ t & 𝗔◗q = p
-       | â\88\83â\88\83v,u,q. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ v & @v.u ⇔ t & 𝗦◗q = p
+       | â\88\83â\88\83u,q,n. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ u & ð\9d\9b\95n.u â\87\94 t & ð\9d\97±nâ\97\97ð\9d\97º◗q = p
+       | â\88\83â\88\83u,q. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ u & 𝛌.u ⇔ t & 𝗟◗q = p
+       | â\88\83â\88\83v,u,q. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ u & @v.u ⇔ t & 𝗔◗q = p
+       | â\88\83â\88\83v,u,q. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ v & @v.u ⇔ t & 𝗦◗q = p
 .
 #t #p #H elim H -H
 [ #n * /3 width=3 by or5_intro0, ex2_intro/
@@ -62,14 +64,15 @@ lemma bdd_inv_in_comp_gen:
 qed-.
 
 lemma bdd_inv_in_comp_d:
-      â\88\80t,q,n. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗱n◗q ϵ t →
+      â\88\80t,q,n. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗱n◗q ϵ t →
       ∨∨ ∧∧ #n ⇔ t & 𝐞 = q
-       | â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & ð\9d\9b\97n.u ⇔ t
+       | â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ ɱ.u & ð\9d\9b\95n.u ⇔ t
 .
 #t #q #n #Ht #Hq
 elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 [ #n0 #H1 #H2 destruct /3 width=1 by or_introl, conj/
-| #u0 #q0 #n0 #Hu0 #Hq0 #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
+| #u0 #q0 #n0 #Hu0 #Hq0 #H1 #H2 destruct
+ /4 width=4 by ex3_intro, ex2_intro, or_intror/
 | #u0 #q0 #_ #_ #_ #H0 destruct
 | #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
 | #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
@@ -77,9 +80,9 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_d:
-      â\88\80t,q,n. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗱n◗q ϵ ▵t →
+      â\88\80t,q,n. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗱n◗q ϵ ▵t →
       ∨∨ ∧∧ #n ⇔ t & 𝐞 = q
-       | â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ â\96µu & ð\9d\9b\97n.u ⇔ t
+       | â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ â\96µÉ±.u & ð\9d\9b\95n.u ⇔ t
 .
 #t #q #n #Ht * #r #Hq
 elim (bdd_inv_in_comp_d … Ht Hq) -Ht -Hq *
@@ -92,8 +95,8 @@ elim (bdd_inv_in_comp_d … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_comp_L:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗟◗q ϵ t →
-      â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & 𝛌.u ⇔ t
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗟◗q ϵ t →
+      â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ u & 𝛌.u ⇔ t
 .
 #t #q #Ht #Hq
 elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
@@ -106,8 +109,8 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_L:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗟◗q ϵ ▵t →
-      â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ ▵u & 𝛌.u ⇔ t.
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗟◗q ϵ ▵t →
+      â\88\83â\88\83u. u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ ▵u & 𝛌.u ⇔ t.
 #t #q #Ht * #r #Hq
 elim (bdd_inv_in_comp_L … Ht Hq) -Ht -Hq
 #u #Hu #Hq #H0 destruct
@@ -115,8 +118,8 @@ elim (bdd_inv_in_comp_L … Ht Hq) -Ht -Hq
 qed-.
 
 lemma bdd_inv_in_comp_A:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗔◗q ϵ t →
-      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ u & @v.u ⇔ t
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗔◗q ϵ t →
+      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ u & @v.u ⇔ t
 .
 #t #q #Ht #Hq
 elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
@@ -129,8 +132,8 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_A:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗔◗q ϵ ▵t →
-      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ ▵u & @v.u ⇔ t
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗔◗q ϵ ▵t →
+      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ ▵u & @v.u ⇔ t
 .
 #t #q #Ht * #r #Hq
 elim (bdd_inv_in_comp_A … Ht Hq) -Ht -Hq
@@ -139,8 +142,8 @@ elim (bdd_inv_in_comp_A … Ht Hq) -Ht -Hq
 qed-.
 
 lemma bdd_inv_in_comp_S:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗦◗q ϵ t →
-      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ v & @v.u ⇔ t
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗦◗q ϵ t →
+      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ v & @v.u ⇔ t
 .
 #t #q #Ht #Hq
 elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
@@ -153,8 +156,8 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_S:
-      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\97 → 𝗦◗q ϵ ▵t →
-      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\97 & u ϵ ð\9d\90\83ð\9d\9b\97 & q ϵ ▵v & @v.u ⇔ t
+      â\88\80t,q. t ϵ ð\9d\90\83ð\9d\9b\95 → 𝗦◗q ϵ ▵t →
+      â\88\83â\88\83v,u. v ϵ ð\9d\90\83ð\9d\9b\95 & u ϵ ð\9d\90\83ð\9d\9b\95 & q ϵ ▵v & @v.u ⇔ t
 .
 #t #q #Ht * #r #Hq
 elim (bdd_inv_in_comp_S … Ht Hq) -Ht -Hq
@@ -165,18 +168,18 @@ qed-.
 (* Advanced inversions ******************************************************)
 
 lemma bbd_mono_in_root_d:
-      â\88\80l,n,p,t. t ϵ ð\9d\90\83ð\9d\9b\97 → p◖𝗱n ϵ ▵t → p◖l ϵ ▵t → 𝗱n = l.
+      â\88\80l,n,p,t. t ϵ ð\9d\90\83ð\9d\9b\95 → p◖𝗱n ϵ ▵t → p◖l ϵ ▵t → 𝗱n = l.
 #l #n #p elim p -p
 [ #t #Ht <list_cons_comm <list_cons_comm #Hn #Hl
   elim (bdd_inv_in_root_d … Ht Hn) -Ht -Hn *
   [ #H0 #_
-    lapply (preterm_root_eq_repl … H0) -H0 #H0
+    lapply (prototerm_root_eq_repl … H0) -H0 #H0
     lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl
-    elim (preterm_in_root_inv_lcons_oref … Hl) -Hl //
+    elim (prototerm_in_root_inv_lcons_oref … Hl) -Hl //
   | #u #_ #_ #H0
-    lapply (preterm_root_eq_repl … H0) -H0 #H0
+    lapply (prototerm_root_eq_repl … H0) -H0 #H0
     lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl
-    elim (preterm_in_root_inv_lcons_iref … Hl) -Hl //
+    elim (prototerm_in_root_inv_lcons_iref … Hl) -Hl //
   ]
 | * [ #m ] #p #IH #t #Ht
   <list_cons_shift <list_cons_shift #Hn #Hl
@@ -184,29 +187,30 @@ lemma bbd_mono_in_root_d:
     [ #_ #H0
       elim (eq_inv_list_empty_rcons ??? H0)
     | #u #Hu #Hp #H0
-      lapply (preterm_root_eq_repl … H0) -H0 #H0
+      lapply (prototerm_root_eq_repl … H0) -H0 #H0
       lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl
-      elim (preterm_in_root_inv_lcons_iref … Hl) -Hl #_ #Hl
+      elim (prototerm_in_root_inv_lcons_iref … Hl) -Hl #_ #Hl
       /2 width=4 by/
     ]
   | elim (bdd_inv_in_root_L … Ht Hn) -Ht -Hn
     #u #Hu #Hp #H0
-    lapply (preterm_root_eq_repl … H0) -H0 #H0
+    lapply (prototerm_root_eq_repl … H0) -H0 #H0
     lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl  
-    elim (preterm_in_root_inv_lcons_abst … Hl) -Hl #_ #Hl
+    elim (prototerm_in_root_inv_lcons_abst … Hl) -Hl #_ #Hl
     /2 width=4 by/
   | elim (bdd_inv_in_root_A … Ht Hn) -Ht -Hn
     #v #u #_ #Hu #Hp #H0
-    lapply (preterm_root_eq_repl … H0) -H0 #H0
+    lapply (prototerm_root_eq_repl … H0) -H0 #H0
     lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl
-    elim (preterm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
+    elim (prototerm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
     /2 width=4 by/
   | elim (bdd_inv_in_root_S … Ht Hn) -Ht -Hn
     #v #u #Hv #_ #Hp #H0
-    lapply (preterm_root_eq_repl … H0) -H0 #H0
+    lapply (prototerm_root_eq_repl … H0) -H0 #H0
     lapply (subset_in_eq_repl_fwd ?? … Hl … H0) -H0 -Hl #Hl
-    elim (preterm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
+    elim (prototerm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
     /2 width=4 by/
   ]
 ]
 qed-.
+*)