- advances on hereditarily free variables: now "frees" is primitive

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / substitution / cpys_alt.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
index 929007118c8ec26e2eaee90a189f6fc3a7c3462d..e297be69878f28f2aa0a2d88caa8aafb332620ab 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/extpsubststaralt_6.ma".
+include "basic_2/notation/relations/psubststaralt_6.ma".
 include "basic_2/substitution/cpys_lift.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
@@ -21,10 +21,10 @@ include "basic_2/substitution/cpys_lift.ma".
 inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
 | cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
 | cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
-               ⇩[0, i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(d+e-i)) G K V1 V2 →
+               ⇩[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(d+e-i)) G K V1 V2 →
                ⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
 | cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
-               cpysa d e G L V1 V2 → cpysa (⫯d) e G (L.ⓑ{I}V2) T1 T2 →
+               cpysa d e G L V1 V2 → cpysa (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
                cpysa d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
 | cpysa_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
                cpysa d e G L V1 V2 → cpysa d e G L T1 T2 →
@@ -33,7 +33,7 @@ inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
 
 interpretation
    "context-sensitive extended multiple substritution (term) alternative"
-   'ExtPSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
+   'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
 
 (* Basic properties *********************************************************)
 
@@ -41,63 +41,62 @@ lemma lsuby_cpysa_trans: ∀G,d,e. lsub_trans … (cpysa d e G) (lsuby d e).
 #G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
 [ //
 | #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
-  elim (lsuby_fwd_ldrop2_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
+  elim (lsuby_ldrop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
 | /4 width=1 by lsuby_succ, cpysa_bind/
 | /3 width=1 by cpysa_flat/
 ]
 qed-.
 
-lemma cpysa_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶▶*×[d, e] T.
+lemma cpysa_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶▶*[d, e] T.
 #G #T elim T -T //
 #I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
 qed.
 
-lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T →
-                       ∀T2. ⦃G, L⦄ ⊢ T ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T2.
+lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T →
+                       ∀T2. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
 #G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
 [ #I #G #L #d #e #X #H
   elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
 | #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
-  lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
+  lapply (ldrop_fwd_drop2 … HLK) #H0LK
   lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
   elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
   /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
 | #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
   elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
-  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1 by lsuby_succ/ #HT2
-  lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
-  lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
-  lapply (lsuby_cpysa_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1 by lsuby_succ, cpysa_bind/
+  /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
 | #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
   elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
 ]
 qed-.
 
-lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T2.
+lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
 /3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpysa_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2.
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 /2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
 qed-.
 
+(* Advanced eliminators *****************************************************)
+
 lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
                     (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
                     (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
-                     ⇩[O, i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*×[O, ⫰(d+e-i)] V2 →
+                     ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 →
                      ⇧[O, i+1] V2 ≡ W2 → R O (⫰(d+e-i)) G K V1 V2 → R d e G L (#i) W2
                     ) →
-                    (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 →
-                     ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶*×[⫯d, e] T2 → R d e G L V1 V2 →
-                     R (⫯d) e G (L.ⓑ{I}V2) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+                    (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+                     ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 → R d e G L V1 V2 →
+                     R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
                     ) →
-                    (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 →
-                     ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → R d e G L V1 V2 →
+                    (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+                     ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L V1 V2 →
                      R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
                     ) →
-                    ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → R d e G L T1 T2.
+                    ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
 #R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpysa_cpys/
+/3 width=8 by cpysa_inv_cpys/
 qed-.