- some renaming according to the written version of basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / unfold / lstas_lstas.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
index 1b584365748de6f474d59840b98bc4a9d0c1d974..0f1c9cb31f217756312502fce3c2a4ba98fd7932 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
@@ -18,24 +18,24 @@ include "basic_2/unfold/lstas_lift.ma".
 
 (* Main properties **********************************************************)
 
-theorem lstas_trans: ∀h,G,L,T1,T,l1. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
-                     ∀T2,l2. ⦃G, L⦄ ⊢ T •*[h, l2] T2 → ⦃G, L⦄ ⊢ T1 •*[h, l1+l2] T2.
-#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
-[ #G #L #l1 #k #X #l2 #H >(lstas_inv_sort1 … H) -X
+theorem lstas_trans: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                     ∀T2,d2. ⦃G, L⦄ ⊢ T •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*[h, d1+d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ #G #L #d1 #k #X #d2 #H >(lstas_inv_sort1 … H) -X
   <iter_plus /2 width=1 by lstas_sort/
-| #G #L #K #V1 #V #U #i #l1 #HLK #_ #HVU #IHV1 #U2 #l2 #HU2
+| #G #L #K #V1 #V #U #i #d1 #HLK #_ #HVU #IHV1 #U2 #d2 #HU2
   lapply (drop_fwd_drop2 … HLK) #H0
   elim (lstas_inv_lift1 … HU2 … H0 … HVU)
   /3 width=6 by lstas_ldef/
 | //
-| #G #L #K #W1 #W #U #i #l1 #HLK #_ #HWU #IHW1 #U2 #l2 #HU2
+| #G #L #K #W1 #W #U #i #d1 #HLK #_ #HWU #IHW1 #U2 #d2 #HU2
   lapply (drop_fwd_drop2 … HLK) #H0
   elim (lstas_inv_lift1 … HU2 … H0 … HWU)
   /3 width=6 by lstas_succ/
-| #a #I #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+| #a #I #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
   elim (lstas_inv_bind1 … H) -H #T2 #HT2 #H destruct
   /3 width=1 by lstas_bind/
-| #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+| #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
   elim (lstas_inv_appl1 … H) -H #T2 #HT2 #H destruct
   /3 width=1 by lstas_appl/
 | /3 width=1 by lstas_cast/
@@ -43,12 +43,12 @@ theorem lstas_trans: ∀h,G,L,T1,T,l1. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
 qed-.
 
 (* Note: apparently this was missing in basic_1 *)
-theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h l G L).
-#h #G #L #l #T #T1 #H elim H -G -L -T -T1 -l
-[ #G #L #l #k #X #H >(lstas_inv_sort1 … H) -X //
-| #G #L #K #V #V1 #U1 #i #l #HLK #_ #HVU1 #IHV1 #X #H
+theorem lstas_mono: ∀h,G,L,d. singlevalued … (lstas h d G L).
+#h #G #L #d #T #T1 #H elim H -G -L -T -T1 -d
+[ #G #L #d #k #X #H >(lstas_inv_sort1 … H) -X //
+| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H
   elim (lstas_inv_lref1 … H) -H *
-  #K0 #V0 #W0 [3: #l0 ] #HLK0
+  #K0 #V0 #W0 [3: #d0 ] #HLK0
   lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
   #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct
   /2 width=5 by lift_mono/
@@ -56,16 +56,16 @@ theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h l G L).
   elim (lstas_inv_lref1_O … H) -H *
   #K0 #V0 #W0 #HLK0
   lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct //
-| #G #L #K #W #W1 #U1 #i #l #HLK #_ #HWU1 #IHWV #X #H
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H
   elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0
   lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
   #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct
   /2 width=5 by lift_mono/
-| #a #I #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H
   elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+| #G #L #V #T #U1 #d #_ #IHTU1 #X #H
   elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #W #T #U1 #l #_ #IHTU1 #U2 #H
+| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H
   lapply (lstas_inv_cast1 … H) -H /2 width=1 by/
 ]
 qed-.
@@ -73,51 +73,51 @@ qed-.
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_1: was just: sty0_correct *)
-lemma lstas_correct: ∀h,G,L,T1,T,l1. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
-                     ∀l2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, l2] T2.
-#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
+lemma lstas_correct: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                     ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
 [ /2 width=2 by lstas_sort, ex_intro/
-| #G #L #K #V1 #V #U #i #l #HLK #_ #HVU #IHV1 #l2
-  elim (IHV1 l2) -IHV1 #V2
+| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2
+  elim (IHV1 d2) -IHV1 #V2
   elim (lift_total V2 0 (i+1))
   lapply (drop_fwd_drop2 … HLK) -HLK
   /3 width=11 by ex_intro, lstas_lift/
-| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #l2
-  @(nat_ind_plus … l2) -l2 /3 width=5 by lstas_zero, ex_intro/
-  #l2 #_ elim (IHW1 l2) -IHW1 #W2 #HW2
+| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2
+  @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+  #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2
   lapply (lstas_trans … HW1 … HW2) -W
   elim (lift_total W2 0 (i+1))
   /3 width=7 by lstas_succ, ex_intro/
-| #G #L #K #W1 #W #U #i #l #HLK #_ #HWU #IHW1 #l2
-  elim (IHW1 l2) -IHW1 #W2
+| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2
+  elim (IHW1 d2) -IHW1 #W2
   elim (lift_total W2 0 (i+1))
   lapply (drop_fwd_drop2 … HLK) -HLK
   /3 width=11 by ex_intro, lstas_lift/
-| #a #I #G #L #V #T #U #l #_ #IHTU #l2
-  elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
-| #G #L #V #T #U #l #_ #IHTU #l2
-  elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
-| #G #L #W #T #U #l #_ #IHTU #l2
-  elim (IHTU l2) -IHTU /2 width=2 by ex_intro/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /2 width=2 by ex_intro/
 ]
 qed-.
 
 (* more main properties *****************************************************)
 
-theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
-                       ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
-                       ⦃G, L⦄ ⊢ U1 •*[h, l2-l1] U2.
-#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
->(plus_minus_m_m … Hl12) in ⊢ (%→?); -Hl12 >commutative_plus #H
+theorem lstas_conf_le: ∀h,G,L,T,U1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] U1 →
+                       ∀U2,d2. d1 ≤ d2 → ⦃G, L⦄ ⊢ T •*[h, d2] U2 →
+                       ⦃G, L⦄ ⊢ U1 •*[h, d2-d1] U2.
+#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12
+>(plus_minus_m_m … Hd12) in ⊢ (%→?); -Hd12 >commutative_plus #H
 elim (lstas_split … H) -H #U #HTU
 >(lstas_mono … HTU … HTU1) -T //
 qed-.
 
-theorem lstas_conf: ∀h,G,L,T0,T1,l1. ⦃G, L⦄ ⊢ T0 •*[h, l1] T1 →
-                    ∀T2,l2. ⦃G, L⦄ ⊢ T0 •*[h, l2] T2 →
-                    ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l2] T & ⦃G, L⦄ ⊢ T2 •*[h, l1] T.
-#h #G #L #T0 #T1 #l1 #HT01 #T2 #l2 #HT02
-elim (lstas_lstas … HT01 (l1+l2)) #T #HT0
+theorem lstas_conf: ∀h,G,L,T0,T1,d1. ⦃G, L⦄ ⊢ T0 •*[h, d1] T1 →
+                    ∀T2,d2. ⦃G, L⦄ ⊢ T0 •*[h, d2] T2 →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T2 •*[h, d1] T.
+#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02
+elim (lstas_lstas … HT01 (d1+d2)) #T #HT0
 lapply (lstas_conf_le … HT01 … HT0) // -HT01 <minus_plus_m_m_commutative
 lapply (lstas_conf_le … HT02 … HT0) // -T0 <minus_plus_m_m
 /2 width=3 by ex2_intro/