some notation renamed and fixed

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
index 3aae4505cef663853cf69e39ea322d808166c404..422c568781c12d8c99970eaa948804ac20f9a6c2 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/extpsubststar_6.ma".
+include "basic_2/notation/relations/psubststar_6.ma".
 include "basic_2/relocation/cpy.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
@@ -21,35 +21,35 @@ definition cpys: ynat → ynat → relation4 genv lenv term term ≝
                  λd,e,G. LTC … (cpy d e G).
 
 interpretation "context-sensitive extended multiple substritution (term)"
-   'ExtPSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
+   'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
 
 (* Basic eliminators ********************************************************)
 
 lemma cpys_ind: ∀G,L,T1,d,e. ∀R:predicate term. R T1 →
-                (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T → ⦃G, L⦄ ⊢ T ▶×[d, e] T2 → R T → R T2) →
-                ∀T2. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → R T2.
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T2.
 #G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cpys_ind_dx: ∀G,L,T2,d,e. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T → ⦃G, L⦄ ⊢ T ▶*×[d, e] T2 → R T → R T1) →
-                   ∀T1. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → R T1.
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T1.
 #G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpy_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2.
+lemma cpy_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
 /2 width=1 by inj/ qed.
 
 lemma cpys_strap1: ∀G,L,T1,T,T2,d,e.
-                   ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T → ⦃G, L⦄ ⊢ T ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2.
+                   ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
 normalize /2 width=3 by step/ qed-.
 
 lemma cpys_strap2: ∀G,L,T1,T,T2,d,e.
-                   ⦃G, L⦄ ⊢ T1 ▶×[d, e] T → ⦃G, L⦄ ⊢ T ▶*×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2.
+                   ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
 normalize /2 width=3 by TC_strap/ qed-.
 
 lemma lsuby_cpys_trans: ∀G,d,e. lsub_trans … (cpys d e G) (lsuby d e).
@@ -59,46 +59,46 @@ qed-.
 lemma cpys_refl: ∀G,L,d,e. reflexive … (cpys d e G L).
 /2 width=1 by cpy_cpys/ qed.
 
-lemma cpys_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 →
-                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*×[⫯d, e] T2 →
-                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*×[d, e] ⓑ{a,I}V2.T2.
+lemma cpys_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] ⓑ{a,I}V2.T2.
 #G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
 [ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
 | /3 width=5 by cpys_strap1, cpy_bind/
 ]
 qed.
 
-lemma cpys_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 →
-                 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 →
-                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*×[d, e] ⓕ{I}V2.T2.
+lemma cpys_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+                 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] ⓕ{I}V2.T2.
 #G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
 [ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
 | /3 width=5 by cpys_strap1, cpy_flat/
 qed.
 
-lemma cpys_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*×[d1, e1] T2 →
+lemma cpys_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T2 →
                  ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
-                 ⦃G, L⦄ ⊢ T1 ▶*×[d2, e2] T2.
+                 ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T2.
 #G #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind … H) -T2
 /3 width=7 by cpys_strap1, cpy_weak/
 qed-.
 
 lemma cpys_weak_top: ∀G,L,T1,T2,d,e.
-                     ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[d, |L| - d] T2.
+                     ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, |L| - d] T2.
 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
 /3 width=4 by cpys_strap1, cpy_weak_top/
 qed-.
 
 lemma cpys_weak_full: ∀G,L,T1,T2,d,e.
-                      ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*×[0, |L|] T2.
+                      ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
 /3 width=5 by cpys_strap1, cpy_weak_full/
 qed-.
 
-lemma cpys_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
+lemma cpys_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
                ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
                d ≤ dt → d + e ≤ dt + et →
-               ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*×[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
+               ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #H #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
@@ -114,22 +114,22 @@ qed-.
 (* Basic inversion lemmas ***************************************************)
 
 (* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶*×[d, e] T2 → T2 = ⋆k.
+lemma cpys_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] T2 → T2 = ⋆k.
 #G #L #T2 #k #d #e #H @(cpys_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT1 destruct
 >(cpy_inv_sort1 … HT2) -HT2 //
 qed-.
 
 (* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶*×[d, e] T2 → T2 = §p.
+lemma cpys_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶*[d, e] T2 → T2 = §p.
 #G #L #T2 #p #d #e #H @(cpys_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT1 destruct
 >(cpy_inv_gref1 … HT2) -HT2 //
 qed-.
 
-lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*×[d, e] U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 &
-                               ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*×[⫯d, e] T2 &
+lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 &
+                               ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 &
                                U2 = ⓑ{a,I}V2.T2.
 #a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
 [ /2 width=5 by ex3_2_intro/
@@ -140,8 +140,8 @@ lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*
 ]
 qed-.
 
-lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*×[d, e] U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 &
+lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 &
                                U2 = ⓕ{I}V2.T2.
 #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
 [ /2 width=5 by ex3_2_intro/
@@ -151,26 +151,26 @@ lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*×[d
 ]
 qed-.
 
-lemma cpys_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*×[d, 0] T2 → T1 = T2.
+lemma cpys_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 0] T2 → T1 = T2.
 #G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
 qed-.
 
 lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
-                         ⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+                         ⦃G, L⦄ ⊢ U1 ▶*[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
 #G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2
 /2 width=7 by cpy_inv_lift1_eq/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
 /2 width=3 by transitive_le/
 qed-.
 
-lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*×[d, e] T →
+lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*[d, e] T →
                        ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
 #G #L #L1 #T1 #T #d #e #H @(cpys_ind … H) -T
 [ /2 width=4 by ex2_2_intro/