X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Ftps.ma;h=11edc90b10b800e78d8c672e3af12d73c5781bb6;hb=ef3bdc4be26f6518a82a79c64e986253f7aeaa3c;hp=969e674859a50cf8da5770140d7ecfab2130cc16;hpb=9aa9a54946719d3fdb4cadb7c7d33fd13956c083;p=helm.git

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 969e67485..11edc90b1 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma
@@ -18,15 +18,15 @@ include "Basic_2/substitution/ldrop.ma".
 (* PARALLEL SUBSTITUTION ON TERMS *******************************************)
 
 inductive tps: nat → nat → lenv → relation term ≝
-| tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I})
+| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
 | tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
-             ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps d e L (#i) W
+             ⇩[0, i] L ≡ K. ⓓV → ⇧[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)
+             tps d e L V1 V2 → tps (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+             tps d e L (ⓑ{I} V1. T1) (ⓑ{I} V2. T2)
 | tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
              tps d e L V1 V2 → tps d e L T1 T2 →
-             tps d e L (𝕗{I} V1. T1) (𝕗{I} V2. T2)
+             tps d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
 .
 
 interpretation "parallel substritution (term)"
@@ -34,8 +34,8 @@ interpretation "parallel substritution (term)"
 
 (* Basic properties *********************************************************)
 
-lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
-                      ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≫ T2.
+lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶ T2 →
+                      ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ▶ 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: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
+lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ▶ T.
 #T elim T -T //
 #I elim I -I /2 width=1/
 qed.
 
 (* Basic_1: was: subst1_ex *)
-lemma tps_full: ∀K,V,T1,L,d. ↓[0, d] L ≡ (K. 𝕓{Abbr} V) →
-                ∃∃T2,T. L ⊢ T1 [d, 1] ≫ T2 & ↑[d, 1] T ≡ T2.
+lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
+                ∃∃T2,T. L ⊢ T1 [d, 1] ▶ T2 & ⇧[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/
@@ -61,15 +61,15 @@ lemma tps_full: ∀K,V,T1,L,d. ↓[0, d] L ≡ (K. 𝕓{Abbr} V) →
   elim (lift_split … HVW i i ? ? ?) // /3 width=4/
 | * #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. 𝕓{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=8/
+  [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=8/
   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
   ]
 ]
 qed.
 
-lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ▶ T2 →
                 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
-                L ⊢ T1 [d2, e2] ≫ T2.
+                L ⊢ T1 [d2, e2] ▶ 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 ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
+                    L ⊢ T1 [d, e] ▶ T2 → L ⊢ T1 [d, |L| - d] ▶ 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 ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
+                    L ⊢ T1 [d, e] ▶ T2 → L ⊢ T1 [0, |L|] ▶ 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: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
-                    ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
+lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → ∀i. d ≤ i → i ≤ d + e →
+                    ∃∃T. L ⊢ T1 [d, i - d] ▶ T & L ⊢ T [i, d + e - i] ▶ 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
@@ -115,7 +115,7 @@ lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i →
   elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
   elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
   -Hdi -Hide >arith_c1x #T #HT1 #HT2
-  lapply (tps_lsubs_conf … HT1 (L. 𝕓{I} V) ?) -HT1 /3 width=5/
+  lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
   elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
   -Hdi -Hide /3 width=5/
@@ -124,11 +124,11 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} →
-                        T2 = 𝕒{I} ∨
+fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → ∀I. T1 = ⓪{I} →
+                        T2 = ⓪{I} ∨
                         ∃∃K,V,i. d ≤ i & i < d + e &
-                                 ↓[O, i] L ≡ K. 𝕓{Abbr} V &
-                                 ↑[O, i + 1] V ≡ T2 &
+                                 ⇩[O, i] L ≡ K. ⓓV &
+                                 ⇧[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/
@@ -138,38 +138,44 @@ fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 
 ]
 qed.
 
-lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 →
-                     T2 = 𝕒{I} ∨
+lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} [d, e] ▶ T2 →
+                     T2 = ⓪{I} ∨
                      ∃∃K,V,i. d ≤ i & i < d + e &
-                              ↓[O, i] L ≡ K. 𝕓{Abbr} V &
-                              ↑[O, i + 1] V ≡ T2 &
+                              ⇩[O, i] L ≡ K. ⓓV &
+                              ⇧[O, i + 1] V ≡ T2 &
                               I = LRef i.
 /2 width=3/ qed-.
 
 
 (* Basic_1: was: subst1_gen_sort *)
-lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k.
+lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ▶ 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: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
+lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ▶ T2 →
                      T2 = #i ∨
                      ∃∃K,V. d ≤ i & i < d + e &
-                            ↓[O, i] L ≡ K. 𝕓{Abbr} V &
-                            ↑[O, i + 1] V ≡ T2.
+                            ⇩[O, i] L ≡ K. ⓓV &
+                            ⇧[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-.
 
-fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                        ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
-                        ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
-                                 L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 &
-                                 U2 =  𝕓{I} V2. T2.
+lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶ 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: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
+                        ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & 
+                                 L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
+                                 U2 =  ⓑ{I} V2. T2.
 #d #e #L #U1 #U2 * -d -e -L -U1 -U2
 [ #L #k #d #e #I #V1 #T1 #H destruct
 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
@@ -178,16 +184,16 @@ fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
-                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
-                              L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 &
-                              U2 =  𝕓{I} V2. T2.
+lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 [d, e] ▶ U2 →
+                     ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & 
+                              L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
+                              U2 =  ⓑ{I} V2. T2.
 /2 width=3/ qed-.
 
-fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                        ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
-                        ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
-                                 U2 =  𝕗{I} V2. T2.
+fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
+                        ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
+                                 U2 =  ⓕ{I} V2. T2.
 #d #e #L #U1 #U2 * -d -e -L -U1 -U2
 [ #L #k #d #e #I #V1 #T1 #H destruct
 | #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
@@ -196,12 +202,12 @@ fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
-                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
-                              U2 =  𝕗{I} V2. T2.
+lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 [d, e] ▶ U2 →
+                     ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
+                              U2 =  ⓕ{I} V2. T2.
 /2 width=3/ qed-.
 
-fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2.
+fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ 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
@@ -212,12 +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: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2.
+lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ▶ T2 → T1 = T2.
 /2 width=6/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → #[T1] ≤ #[T2].
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ 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-.