- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / substitution / tps_tps.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_tps.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_tps.ma
index 3155fe64f85a30d040f78dea52116f41ed4d295e..3882750bd5f9c1f86b8bf6202fecd4c60a9adf25 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_tps.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/tps_tps.ma
@@ -19,9 +19,9 @@ include "basic_2/substitution/tps_lift.ma".
 (* Main properties **********************************************************)
 
 (* Basic_1: was: subst1_confluence_eq *)
-theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ▶ T1 →
-                     ∀T2,d2,e2. L ⊢ T0 [d2, e2] ▶ T2 →
-                     ∃∃T. L ⊢ T1 [d2, e2] ▶ T & L ⊢ T2 [d1, e1] ▶ T.
+theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
+                     ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+                     ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
 #L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
 [ /2 width=3/
 | #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
@@ -31,13 +31,13 @@ theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ▶ T1 →
     lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
     >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
   ]
-| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
   elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
-  lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
+  lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
   elim (IHV01 … HV02) -V0 #V #HV1 #HV2
   elim (IHT01 … HT02) -T0 #T #HT1 #HT2
-  lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
-  lapply (tps_lsubs_conf … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
+  lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+  lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
 | #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
   elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02) -V0
@@ -46,10 +46,10 @@ theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ▶ T1 →
 qed.
 
 (* Basic_1: was: subst1_confluence_neq *)
-theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ▶ T1 →
-                      ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ▶ T2 →
+theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
+                      ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
                       (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
-                      ∃∃T. L2 ⊢ T1 [d2, e2] ▶ T & L1 ⊢ T2 [d1, e1] ▶ T.
+                      ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
 #L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
 [ /2 width=3/
 | #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
@@ -66,13 +66,13 @@ theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ▶ T1 →
       elim (lt_refl_false … H)
     ]
   ]
-| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
   elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
   elim (IHT01 … HT02 ?) -T0
   [ -H #T #HT1 #HT2
-    lapply (tps_lsubs_conf … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
-    lapply (tps_lsubs_conf … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
+    lapply (tps_lsubs_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
+    lapply (tps_lsubs_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
   | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
     [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
   ]
@@ -85,9 +85,9 @@ qed.
 
 (* Note: the constant 1 comes from tps_subst *)
 (* Basic_1: was: subst1_trans *)
-theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 [d, e] ▶ T0 →
-                      ∀T2. L ⊢ T0 [d, 1] ▶ T2 → 1 ≤ e →
-                      L ⊢ T1 [d, e] ▶ T2.
+theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
+                      ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
+                      L ⊢ T1 ▶ [d, e] T2.
 #L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
 [ #L #I #d #e #T2 #H #He
   elim (tps_inv_atom1 … H) -H
@@ -98,32 +98,32 @@ theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 [d, e] ▶ T0 →
 | #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
   lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
   <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
-| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
   elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
-  lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+  lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
   lapply (IHT10 … HT02 He) -T0 #HT12
-  lapply (tps_lsubs_conf … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
+  lapply (tps_lsubs_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
 | #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
   elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
 ]
 qed.
 
-theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ▶ T0 →
-                        ∀T2,d2,e2. L ⊢ T0 [d2, e2] ▶ T2 → d2 + e2 ≤ d1 →
-                        ∃∃T. L ⊢ T1 [d2, e2] ▶ T & L ⊢ T [d1, e1] ▶ T2.
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+                        ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+                        ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
 #L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
 [ /2 width=3/
 | #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
   lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
   lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
   <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=4/
-| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
   elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
-  lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+  lapply (tps_lsubs_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
   elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
   elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
-  lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
-  lapply (tps_lsubs_conf … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
+  lapply (tps_lsubs_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+  lapply (tps_lsubs_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
 | #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
   elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV10 … HV02 ?) -V0 //