- we introduced the pointer_step rc in the perspective of proving

[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 f1cc2d9eedb2726d757008ce095be2eb3d410ffa..ab11536bdf4db831547641e2ffadce07e5338283 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps.ma
@@ -12,8 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/cl_weight.ma".
-include "basic_2/substitution/ldrop.ma".
+include "basic_2/substitution/ldrop_append.ma".
 
 (* PARALLEL SUBSTITUTION ON TERMS *******************************************)
 
@@ -21,9 +20,9 @@ inductive tps: nat → nat → lenv → relation term ≝
 | 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. ⓓV → ⇧[0, i + 1] V ≡ W → tps d e L (#i) W
-| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
+| tps_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
              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 d e L (ⓑ{a,I} V1. T1) (ⓑ{a,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 (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
@@ -34,12 +33,12 @@ 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_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 →
+                       ∀L2. L2 ≼ [d, e] L1 → 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
-  elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
+  elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
 | /4 width=1/
 | /3 width=1/
 ]
@@ -59,9 +58,9 @@ lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
   destruct
   elim (lift_total V 0 (i+1)) #W #HVW
   elim (lift_split … HVW i i ? ? ?) // /3 width=4/
-| * #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. ⓑ{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=9/
   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
   ]
 ]
@@ -111,11 +110,11 @@ lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀i. d ≤ i →
     generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
     >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 width=4/
   ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
   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. ⓑ{I} V) ?) -HT1 /3 width=5/
+  lapply (tps_lsubs_trans … 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/
@@ -134,17 +133,26 @@ lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
   | -Hdi -Hdj
     >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
   ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
   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. ⓑ{I} V) ?) -HT1 /3 width=5/
+  lapply (tps_lsubs_trans … 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/
 ]
 qed.
 
+lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 →
+                  ∀L. L @@ K ⊢ T1 ▶ [d, e] T2.
+#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
+#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_ldrop2_length … HK0) #H
+@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} →
@@ -156,7 +164,7 @@ fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = 
 #L #T1 #T2 #d #e * -L -T1 -T2 -d -e
 [ #L #I #d #e #J #H destruct /2 width=1/
 | #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
 ]
 qed.
@@ -195,22 +203,22 @@ elim (tps_inv_atom1 … H) -H //
 qed-.
 
 fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
-                        ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
+                        ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
                         ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & 
                                  L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
-                                 U2 =  ⓑ{I} V2. T2.
+                                 U2 = ⓑ{a,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
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+[ #L #k #d #e #a #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
+| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
 ]
 qed.
 
-lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 ▶ [d, e] U2 →
+lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,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.
+                              U2 = ⓑ{a,I} V2. T2.
 /2 width=3/ qed-.
 
 fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
@@ -220,7 +228,7 @@ fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
 #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
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
 ]
 qed.
@@ -246,11 +254,26 @@ lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 0] T2 → T1 = T2.
 
 (* 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-.
 
+lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶[d, e] T →
+                      ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #d #e #HT1
+  @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X #d #e
+  >shift_append_assoc normalize #H
+  elim (tps_inv_bind1 … H) -H
+  #V0 #T0 #_ #HT10 #H destruct
+  elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+  >append_length >HL12 -HL12
+  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+]
+qed-.
+
 (* Basic_1: removed theorems 25:
             subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
             subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans