- some renaming according to the written version of basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
index 28b5982c4c10d05127ab6bbab81e62671510eeb5..ef48b05afeb068fef8ce035b6bbd80808f7b9b35 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
@@ -19,26 +19,26 @@ include "basic_2/substitution/cpy_lift.ma".
 (* Main properties **********************************************************)
 
 (* Basic_1: was: subst1_confluence_eq *)
-theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶[d1, e1] T1 →
-                     ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
-                     ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶[d1, e1] T.
-#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
+theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 →
+                     ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
 [ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
   elim (cpy_inv_lref1 … H) -H
   [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
-  | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+  | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
     lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
     >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
   ]
-| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
   elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
   elim (IHT01 … HT02) -T0 #T #HT1 #HT2
   lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
   lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
   /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
   elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02) -V0
   elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
@@ -46,21 +46,21 @@ theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶[d1, e1] T1 →
 qed-.
 
 (* Basic_1: was: subst1_confluence_neq *)
-theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶[d1, e1] T1 →
-                      ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
-                      (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
-                      ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶[d1, e1] T.
-#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
+theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 →
+                      ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+                      (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+                      ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
 [ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
   elim (cpy_inv_lref1 … H1) -H1
   [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
-  | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
-    [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
-    | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
+  | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
+    [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
+    | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
     ]
   ]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
   elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
   elim (IHT01 … HT02) -T0
@@ -69,7 +69,7 @@ theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶[d1, e1] T1 →
     lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
   | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
   ]
-| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
   elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV01 … HV02 H) -V0
   elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
@@ -78,43 +78,43 @@ qed-.
 
 (* Note: the constant 1 comes from cpy_subst *)
 (* Basic_1: was: subst1_trans *)
-theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T0 →
-                      ∀T2. ⦃G, L⦄ ⊢ T0 ▶[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶[d, e] T2.
-#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
-[ #I #G #L #d #e #T2 #H #He
+theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
+                      ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
+#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
+[ #I #G #L #l #m #T2 #H #Hm
   elim (cpy_inv_atom1 … H) -H
   [ #H destruct //
-  | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
-    lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+  | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
+    lapply (ylt_yle_trans … (l+m) … Hilm2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
   ]
-| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
   lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
   >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
   elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
   lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
-  lapply (IHT10 … HT02 He) -T0 /3 width=1 by cpy_bind/
-| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+  lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
   elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
 ]
 qed-.
 
-theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
-                        ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
-                        ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
-#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
+theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+                        ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+                        ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
 [ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
-  lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
-  lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
+  lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
+  lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
   >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
   elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
   elim (IHV10 … HV02) -IHV10 -HV02 // #V
   elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
   lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
   elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   elim (IHV10 … HV02) -V0 //
   elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/