closure property S4 added to abstract candidates of reducibility ...

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / substitution / tps.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma
index df69749cd69c6b11a3d4ac983573a10e7d8bab68..8a5dc13d083afa67e25924771bdb769e7b881d04 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma
@@ -20,7 +20,7 @@ include "Basic_2/substitution/ldrop.ma".
 inductive tps: nat → nat → lenv → relation term ≝
 | tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I})
 | tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
-             â\87\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\87\91[0, i + 1] V ≡ W → tps d e L (#i) W
+             â\87©[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\87§[0, i + 1] V ≡ W → tps d e L (#i) W
 | tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
              tps d e L V1 V2 → tps (d + 1) e (L. 𝕓{I} V2) T1 T2 →
              tps d e L (𝕓{I} V1. T1) (𝕓{I} V2. T2)
@@ -34,8 +34,8 @@ interpretation "parallel substritution (term)"
 
 (* Basic properties *********************************************************)
 
-lemma tps_lsubs_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\89« T2 →
-                      â\88\80L2. L1 [d, e] â\89¼ L2 â\86\92 L2 â\8a¢ T1 [d, e] â\89« T2.
+lemma tps_lsubs_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\96¶ T2 →
+                      â\88\80L2. L1 [d, e] â\89¼ L2 â\86\92 L2 â\8a¢ T1 [d, e] â\96¶ T2.
 #L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
 [ //
 | #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
@@ -45,14 +45,14 @@ lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
 ]
 qed.
 
-lemma tps_refl: â\88\80T,L,d,e. L â\8a¢ T [d, e] â\89« T.
+lemma tps_refl: â\88\80T,L,d,e. L â\8a¢ T [d, e] â\96¶ T.
 #T elim T -T //
 #I elim I -I /2 width=1/
 qed.
 
 (* Basic_1: was: subst1_ex *)
-lemma tps_full: â\88\80K,V,T1,L,d. â\87\93[0, d] L ≡ (K. 𝕓{Abbr} V) →
-                â\88\83â\88\83T2,T. L â\8a¢ T1 [d, 1] â\89« T2 & â\87\91[d, 1] T ≡ T2.
+lemma tps_full: â\88\80K,V,T1,L,d. â\87©[0, d] L ≡ (K. 𝕓{Abbr} V) →
+                â\88\83â\88\83T2,T. L â\8a¢ T1 [d, 1] â\96¶ T2 & â\87§[d, 1] T ≡ T2.
 #K #V #T1 elim T1 -T1
 [ * #i #L #d #HLK /2 width=4/
   elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
@@ -67,9 +67,9 @@ lemma tps_full: ∀K,V,T1,L,d. ⇓[0, d] L ≡ (K. 𝕓{Abbr} V) →
 ]
 qed.
 
-lemma tps_weak: â\88\80L,T1,T2,d1,e1. L â\8a¢ T1 [d1, e1] â\89« T2 →
+lemma tps_weak: â\88\80L,T1,T2,d1,e1. L â\8a¢ T1 [d1, e1] â\96¶ T2 →
                 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
-                L â\8a¢ T1 [d2, e2] â\89« T2.
+                L â\8a¢ T1 [d2, e2] â\96¶ T2.
 #L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
 [ //
 | #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
@@ -81,7 +81,7 @@ lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
 qed.
 
 lemma tps_weak_top: ∀L,T1,T2,d,e.
-                    L â\8a¢ T1 [d, e] â\89« T2 â\86\92 L â\8a¢ T1 [d, |L| - d] â\89« T2.
+                    L â\8a¢ T1 [d, e] â\96¶ T2 â\86\92 L â\8a¢ T1 [d, |L| - d] â\96¶ T2.
 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
 [ //
 | #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
@@ -93,14 +93,14 @@ lemma tps_weak_top: ∀L,T1,T2,d,e.
 qed.
 
 lemma tps_weak_all: ∀L,T1,T2,d,e.
-                    L â\8a¢ T1 [d, e] â\89« T2 â\86\92 L â\8a¢ T1 [0, |L|] â\89« T2.
+                    L â\8a¢ T1 [d, e] â\96¶ T2 â\86\92 L â\8a¢ T1 [0, |L|] â\96¶ T2.
 #L #T1 #T2 #d #e #HT12
 lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
 lapply (tps_weak_top … HT12) //
 qed.
 
-lemma tps_split_up: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → ∀i. d ≤ i → i ≤ d + e →
-                    â\88\83â\88\83T. L â\8a¢ T1 [d, i - d] â\89« T & L â\8a¢ T [i, d + e - i] â\89« T2.
+lemma tps_split_up: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → ∀i. d ≤ i → i ≤ d + e →
+                    â\88\83â\88\83T. L â\8a¢ T1 [d, i - d] â\96¶ T & L â\8a¢ T [i, d + e - i] â\96¶ T2.
 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
 [ /2 width=3/
 | #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
@@ -124,11 +124,11 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact tps_inv_atom1_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → ∀I. T1 = 𝕒{I} →
+fact tps_inv_atom1_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → ∀I. T1 = 𝕒{I} →
                         T2 = 𝕒{I} ∨
                         ∃∃K,V,i. d ≤ i & i < d + e &
-                                 â\87\93[O, i] L ≡ K. 𝕓{Abbr} V &
-                                 â\87\91[O, i + 1] V ≡ T2 &
+                                 â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+                                 â\87§[O, i + 1] V ≡ T2 &
                                  I = LRef i.
 #L #T1 #T2 #d #e * -L -T1 -T2 -d -e
 [ #L #I #d #e #J #H destruct /2 width=1/
@@ -138,43 +138,43 @@ fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 
 ]
 qed.
 
-lemma tps_inv_atom1: â\88\80L,T2,I,d,e. L â\8a¢ ð\9d\95\92{I} [d, e] â\89« T2 →
+lemma tps_inv_atom1: â\88\80L,T2,I,d,e. L â\8a¢ ð\9d\95\92{I} [d, e] â\96¶ T2 →
                      T2 = 𝕒{I} ∨
                      ∃∃K,V,i. d ≤ i & i < d + e &
-                              â\87\93[O, i] L ≡ K. 𝕓{Abbr} V &
-                              â\87\91[O, i + 1] V ≡ T2 &
+                              â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+                              â\87§[O, i + 1] V ≡ T2 &
                               I = LRef i.
 /2 width=3/ qed-.
 
 
 (* Basic_1: was: subst1_gen_sort *)
-lemma tps_inv_sort1: â\88\80L,T2,k,d,e. L â\8a¢ â\8b\86k [d, e] â\89« T2 → T2 = ⋆k.
+lemma tps_inv_sort1: â\88\80L,T2,k,d,e. L â\8a¢ â\8b\86k [d, e] â\96¶ T2 → T2 = ⋆k.
 #L #T2 #k #d #e #H
 elim (tps_inv_atom1 … H) -H //
 * #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
 (* Basic_1: was: subst1_gen_lref *)
-lemma tps_inv_lref1: â\88\80L,T2,i,d,e. L â\8a¢ #i [d, e] â\89« T2 →
+lemma tps_inv_lref1: â\88\80L,T2,i,d,e. L â\8a¢ #i [d, e] â\96¶ T2 →
                      T2 = #i ∨
                      ∃∃K,V. d ≤ i & i < d + e &
-                            â\87\93[O, i] L ≡ K. 𝕓{Abbr} V &
-                            â\87\91[O, i + 1] V ≡ T2.
+                            â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+                            â\87§[O, i + 1] V ≡ T2.
 #L #T2 #i #d #e #H
 elim (tps_inv_atom1 … H) -H /2 width=1/
 * #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
 qed-.
 
-lemma tps_inv_gref1: â\88\80L,T2,p,d,e. L â\8a¢ §p [d, e] â\89« T2 → T2 = §p.
+lemma tps_inv_gref1: â\88\80L,T2,p,d,e. L â\8a¢ §p [d, e] â\96¶ T2 → T2 = §p.
 #L #T2 #p #d #e #H
 elim (tps_inv_atom1 … H) -H //
 * #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
-fact tps_inv_bind1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\89« U2 →
+fact tps_inv_bind1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\96¶ U2 →
                         ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
-                        â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & 
-                                 L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\89« T2 &
+                        â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & 
+                                 L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\96¶ T2 &
                                  U2 =  𝕓{I} V2. T2.
 #d #e #L #U1 #U2 * -d -e -L -U1 -U2
 [ #L #k #d #e #I #V1 #T1 #H destruct
@@ -184,15 +184,15 @@ fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma tps_inv_bind1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\93{I} V1. T1 [d, e] â\89« U2 →
-                     â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & 
-                              L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\89« T2 &
+lemma tps_inv_bind1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\93{I} V1. T1 [d, e] â\96¶ U2 →
+                     â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & 
+                              L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\96¶ T2 &
                               U2 =  𝕓{I} V2. T2.
 /2 width=3/ qed-.
 
-fact tps_inv_flat1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\89« U2 →
+fact tps_inv_flat1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\96¶ U2 →
                         ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
-                        â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & L â\8a¢ T1 [d, e] â\89« T2 &
+                        â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & L â\8a¢ T1 [d, e] â\96¶ T2 &
                                  U2 =  𝕗{I} V2. T2.
 #d #e #L #U1 #U2 * -d -e -L -U1 -U2
 [ #L #k #d #e #I #V1 #T1 #H destruct
@@ -202,12 +202,12 @@ fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma tps_inv_flat1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\97{I} V1. T1 [d, e] â\89« U2 →
-                     â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & L â\8a¢ T1 [d, e] â\89« T2 &
+lemma tps_inv_flat1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\97{I} V1. T1 [d, e] â\96¶ U2 →
+                     â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & L â\8a¢ T1 [d, e] â\96¶ T2 &
                               U2 =  𝕗{I} V2. T2.
 /2 width=3/ qed-.
 
-fact tps_inv_refl_O2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → e = 0 → T1 = T2.
+fact tps_inv_refl_O2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → e = 0 → T1 = T2.
 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
 [ //
 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
@@ -218,12 +218,12 @@ fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T
 ]
 qed.
 
-lemma tps_inv_refl_O2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 0] â\89« T2 → T1 = T2.
+lemma tps_inv_refl_O2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 0] â\96¶ T2 → T1 = T2.
 /2 width=6/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma tps_fwd_tw: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → #[T1] ≤ #[T2].
+lemma tps_fwd_tw: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → #[T1] ≤ #[T2].
 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
 /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
 qed-.