- extended multiple substitutions now uses bounds in ynat (ie. they

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
index 631ab202d88328b8fe1e54a09e6aeea3858464db..3cf6f8b31ff67ba044fdfd47221b9453b946e988 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
@@ -20,39 +20,41 @@ include "basic_2/substitution/cpys.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
-                  d ≤ i → i < d + e →
-                  ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, d+e-i-1] U1 →
-                  ∀U2. ⇧[0, i + 1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
+                  d ≤ yinj i → i < d + e →
+                  ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 →
+                  ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
 #I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
 [ /3 width=5 by cpy_cpys, cpy_subst/
 | #U #U1 #_ #HU1 #IHU #U2 #HU12
   elim (lift_total U 0 (i+1)) #U0 #HU0
   lapply (IHU … HU0) -IHU #H
   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
-  lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
-  lapply (cpy_weak … HU02 d e ? ?) -HU02 [2,3: /2 width=3 by cpys_strap1, le_S/ ]
-  >minus_plus >commutative_plus /2 width=1 by le_minus_to_plus_r/
+  lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+  lapply (cpy_weak … HU02 d e ? ?) -HU02
+  [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+  >yplus_O_sn <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
 ]
 qed.
 
 (* Advanced inverion lemmas *************************************************)
 
-lemma cpys_inv_atom1: ∀G,L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
+lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
                       T2 = ⓪{I} ∨
-                      ∃∃J,K,V1,V2,i. d ≤ i & i < d + e &
+                      ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
                                     ⇩[O, i] L ≡ K.ⓑ{J}V1 &
                                      ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
-                                     ⇧[O, i + 1] V2 ≡ T2 &
+                                     ⇧[O, i+1] V2 ≡ T2 &
                                      I = LRef i.
-#G #L #T2 #I #d #e #H @(cpys_ind … H) -T2
+#I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
 [ /2 width=1 by or_introl/
 | #T #T2 #_ #HT2 *
   [ #H destruct
     elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
   | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
     lapply (ldrop_fwd_ldrop2 … HLK) #H
-    elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) normalize -HT2 -H -HVT [2,3,4: /2 width=1 by le_S/ ]
-    <minus_plus /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+    elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+    [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+    /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
   ]
 ]
 qed-.
@@ -61,8 +63,8 @@ lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 →
                       T2 = #i ∨
                       ∃∃I,K,V1,V2. d ≤ i & i < d + e &
                                    ⇩[O, i] L ≡ K.ⓑ{I}V1 &
-                                   ⦃G, K⦄ ⊢ V1 ▶*×[0, d + e - i - 1] V2 &
-                                   ⇧[O, i + 1] V2 ≡ T2.
+                                   ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
+                                   ⇧[O, i+1] V2 ≡ T2.
 #G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
 * #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
 qed-.
@@ -70,7 +72,7 @@ qed-.
 (* Relocation properties ****************************************************)
 
 lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+                    ∀L,U1,d,e. dt + et ≤ yinj d → ⇩[d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
                     ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
@@ -83,7 +85,7 @@ lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 qed-.
 
 lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+                    ∀L,U1,d,e. dt ≤ yinj d → d ≤ dt + et →
                     ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
                     ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
@@ -96,9 +98,9 @@ lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 qed-.
 
 lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+                    ∀L,U1,d,e. yinj d ≤ dt → ⇩[d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                    ⦃G, L⦄ ⊢ U1 ▶*×[dt + e, et] U2.
+                    ⦃G, L⦄ ⊢ U1 ▶*×[dt+e, et] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
@@ -137,15 +139,15 @@ lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2
 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
-  elim (cpy_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+  elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,d,e.
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
                          ⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
 #G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2 //
 #U #U2 #_ #HU2 #IHU destruct
-<(cpy_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+<(cpy_inv_lift1_eq … HTU1 … HU2) -HU2 -HTU1 //
 qed-.
 
 lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →