update 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 26fb80e561757d7ba359140d507c971467d8be38..ee69726dbb615fdd3fb80da45f03301900fd68d5 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 +12,38 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_equivalence.ma".
+include "delayed_updating/notation/functions/class_d_phi_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
-| bdd_abst: ∀t. bdd t → bdd 𝛌.t
-| bdd_appl: ∀u,t. bdd u → bdd t → bdd @u.t
+inductive bdd: 𝒫❨prototerm❩ ≝
+| bdd_oref: ∀n. bdd (#n)
+| bdd_iref: ∀t,n. bdd t → bdd (𝛗n.t)
+| bdd_abst: ∀t. bdd t → bdd (𝛌.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)"
+  "by-depth delayed (prototerm)"
   'ClassDPhi = (bdd).
 
 (* Basic inversions *********************************************************)
 
 lemma bdd_inv_in_comp_gen:
       ∀t,p. t ϵ 𝐃𝛗 → p ϵ t →
-      ∨∨ ∃∃n. #n ⇔ t & 𝗱❨n❩;𝐞 = p
-       | ∃∃u,q,n. u ϵ 𝐃𝛗 & q ϵ u & 𝛗n.u ⇔ t & 𝗱❨n❩;q = p
-       | ∃∃u,q. u ϵ 𝐃𝛗 & q ϵ u & 𝛌.u ⇔ t & 𝗟;q = p
-       | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ u & @v.u ⇔ t & 𝗔;q = p
-       | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ v & @v.u ⇔ t & 𝗦;q = p
+      ∨∨ ∃∃n. #n ⇔ t & 𝗱n◗𝐞 = p
+       | ∃∃u,q,n. u ϵ 𝐃𝛗 & q ϵ u & 𝛗n.u ⇔ t & 𝗱n◗q = p
+       | ∃∃u,q. u ϵ 𝐃𝛗 & q ϵ u & 𝛌.u ⇔ t & 𝗟◗q = p
+       | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ u & @v.u ⇔ t & 𝗔◗q = p
+       | ∃∃v,u,q. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ v & @v.u ⇔ t & 𝗦◗q = p
 .
 #t #p #H elim H -H
 [ #n * /3 width=3 by or5_intro0, ex2_intro/
@@ -62,7 +62,7 @@ lemma bdd_inv_in_comp_gen:
 qed-.
 
 lemma bdd_inv_in_comp_d:
-      ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ t →
+      ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱n◗q ϵ t →
       ∨∨ ∧∧ #n ⇔ t & 𝐞 = q
        | ∃∃u. u ϵ 𝐃𝛗 & q ϵ u & 𝛗n.u ⇔ t
 .
@@ -77,7 +77,7 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_d:
-      ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ ▵t →
+      ∀t,q,n. t ϵ 𝐃𝛗 → 𝗱n◗q ϵ ▵t →
       ∨∨ ∧∧ #n ⇔ t & 𝐞 = q
        | ∃∃u. u ϵ 𝐃𝛗 & q ϵ ▵u & 𝛗n.u ⇔ t
 .
@@ -92,7 +92,7 @@ elim (bdd_inv_in_comp_d … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_comp_L:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗟;q ϵ t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗟◗q ϵ t →
       ∃∃u. u ϵ 𝐃𝛗 & q ϵ u & 𝛌.u ⇔ t
 .
 #t #q #Ht #Hq
@@ -106,7 +106,7 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_L:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗟;q ϵ ▵t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗟◗q ϵ ▵t →
       ∃∃u. u ϵ 𝐃𝛗 & q ϵ ▵u & 𝛌.u ⇔ t.
 #t #q #Ht * #r #Hq
 elim (bdd_inv_in_comp_L … Ht Hq) -Ht -Hq
@@ -115,7 +115,7 @@ elim (bdd_inv_in_comp_L … Ht Hq) -Ht -Hq
 qed-.
 
 lemma bdd_inv_in_comp_A:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗔;q ϵ t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗔◗q ϵ t →
       ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ u & @v.u ⇔ t
 .
 #t #q #Ht #Hq
@@ -129,7 +129,7 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_A:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗔;q ϵ ▵t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗔◗q ϵ ▵t →
       ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ ▵u & @v.u ⇔ t
 .
 #t #q #Ht * #r #Hq
@@ -139,7 +139,7 @@ elim (bdd_inv_in_comp_A … Ht Hq) -Ht -Hq
 qed-.
 
 lemma bdd_inv_in_comp_S:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗦;q ϵ t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗦◗q ϵ t →
       ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ v & @v.u ⇔ t
 .
 #t #q #Ht #Hq
@@ -153,7 +153,7 @@ elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
 qed-.
 
 lemma bdd_inv_in_root_S:
-      ∀t,q. t ϵ 𝐃𝛗 → 𝗦;q ϵ ▵t →
+      ∀t,q. t ϵ 𝐃𝛗 → 𝗦◗q ϵ ▵t →
       ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ ▵v & @v.u ⇔ t
 .
 #t #q #Ht * #r #Hq
@@ -165,18 +165,18 @@ qed-.
 (* Advanced inversions ******************************************************)
 
 lemma bbd_mono_in_root_d:
-      ∀l,n,p,t. t ϵ 𝐃𝛗 → p,𝗱❨n❩ ϵ ▵t → p,l ϵ ▵t → 𝗱❨n❩ = l.
+      ∀l,n,p,t. t ϵ 𝐃𝛗 → 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
@@ -184,28 +184,28 @@ 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/
   ]
 ]