X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Flleq_lleq.ma;h=a3a276ff367e31236fb25bd784f58d40ab966ea4;hb=1555848a5546d0154964286d3400114481d78962;hp=ca8035680133d2490a74e2bc9e159a0852e32154;hpb=e4be4188d549da5fde54cdc37a6fb4eb2469c15b;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma
index ca8035680..a3a276ff3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma
@@ -12,133 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/cpys_cpys.ma".
 include "basic_2/substitution/lleq_ldrop.ma".
 
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_lref: ∀L1,L2. ∀d:ynat. ∀i:nat. L1 ⋕[#i, d] L2 →
-                     ∨∨ |L1| ≤ i ∧ |L2| ≤ i
-                      | yinj i < d
-                      | ∃∃I1,I2,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V &
-                                         ⇩[i] L2 ≡ K2.ⓑ{I2}V &
-                                         K1 ⋕[V, yinj 0] K2 & d ≤ yinj i.
-#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
-elim (ylt_split i d) /2 width=1 by or3_intro1/ #Hdi #Hi
-elim (ldrop_O1_lt … Hi) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
-lapply (ldrop_fwd_length_minus2 … HLK2) #H
-lapply (ldrop_fwd_length_minus2 … HLK1) >HL12 <H -HL12 -H
-#H lapply (injective_plus_l … H) -H #HK12
-elim (lift_total V1 0 (i+1)) #W1 #HVW1
-elim (lift_total V2 0 (i+1)) #W2 #HVW2
-elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
-elim (cpys_inv_lref1_ldrop … (H1 ?) … HLK2 … HVW1) -H1
-[ elim (cpys_inv_lref1_ldrop … (H2 ?) … HLK1 … HVW2) -H2 ]
-/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
-lapply (cpys_antisym_eq … H12 … H21) -H12 -H21 #H destruct
-@or3_intro2 @(ex4_5_intro … HLK1 HLK2) // @conj // -HK12
-#V elim (lift_total V 0 (i+1))
-#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
-[ elim (cpys_inv_lref1_ldrop … (H12 ?) … HLK2 … HVW) -H12 -H21
-| elim (cpys_inv_lref1_ldrop … (H21 ?) … HLK1 … HVW) -H21 -H12
-] [1,3: >yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/
-qed-.
-
-lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
-                        ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
-                        i < d ∨
-                        ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref … H) -H [ * || * ]
-[ #_ #H elim (lt_refl_false i)
-  lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
-  /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I1 #I2 #K11 #K22 #V0 #HLK11 #HLK22 #HV0 #Hdi lapply (ldrop_mono … HLK22 … HLK2) -L2
-  #H destruct /3 width=5 by ex3_2_intro, or_intror/
-]
-qed-.
-
-lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
-                        ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
-                        i < d ∨
-                        ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 … d … HLK1) -HLK1
-[2: * ] /4 width=6 by lleq_sym, ex3_2_intro, or_introl, or_intror/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
-                           ∀I2,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I2}V →
-                           ∃∃I1,K1. ⇩[i] L1 ≡ K1.ⓑ{I1}V & K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx … H … HLK2) -L2
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
-                           ∀I1,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I1}V →
-                           ∃∃I2,K2. ⇩[i] L2 ≡ K2.ⓑ{I2}V & K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 … Hdi … HLK1) -Hdi -HLK1
-/3 width=4 by lleq_sym, ex2_2_intro/
-qed-.
-
-lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
-                        ∀I1,I2,K1,K2,V.
-                        ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V → K1 ⋕[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V #HLK1 #HLK2
-elim (lleq_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
-#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ⋕[T, d] L2).
-#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
-| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
-  [ #HL12 #d elim (ylt_split i d) /3 width=1 by lleq_skip, or_introl/
-    #Hdi elim (lt_or_ge i (|L1|)) #HiL1
-    elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, lleq_free/
-    elim (ldrop_O1_lt … HiL2) #I2 #K2 #V2 #HLK2
-    elim (ldrop_O1_lt … HiL1) #I1 #K1 #V1 #HLK1
-    elim (eq_term_dec V2 V1)
-    [ #H3 elim (IH K1 V1 … K2 0) destruct
-      /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/
-    ]
-    -IH #H3 @or_intror
-    #H elim (lleq_fwd_lref … H) -H [1,3,4,6: * ]
-    [1,3: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
-    |5,6: /2 width=4 by ylt_yle_false/
-    ]
-    #Z1 #Z2 #Y1 #Y2 #X #HLY1 #HLY2 #HX #_
-    lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
-    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
-    #H2 #H1 destruct /2 width=1 by/
-  ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
-| #a #I #V #T #Hn #L2 #d destruct
-  elim (IH L1 V … L2 d) /2 width=1 by/
-  elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/
-  #H1 #H2 @or_intror
-  #H elim (lleq_inv_bind … H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 #d destruct
-  elim (IH L1 V … L2 d) /2 width=1 by/
-  elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, lleq_flat/
-  #H1 #H2 @or_intror
-  #H elim (lleq_inv_flat … H) -H /2 width=1 by/
-]
--n /4 width=3 by lleq_fwd_length, or_intror/
-qed-.
-
 (* Main properties **********************************************************)
 
 theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
-#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/
-qed-.
+/2 width=3 by lleq_llpx_sn_trans/ qed-.
 
 theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ⋕[d, T] L1→ L ⋕[d, T] L2 → L1 ⋕[d, T] L2.
 /3 width=3 by lleq_trans, lleq_sym/ qed-.
@@ -146,20 +25,6 @@ theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ⋕[d, T] L1→ L ⋕[d, T] L2 → L1 
 theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ⋕[d, T] L → L2 ⋕[d, T] L → L1 ⋕[d, T] L2.
 /3 width=3 by lleq_trans, lleq_sym/ qed-.
 
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,d. (L1 ⋕[ⓑ{a,I}V.T, d] L2 → ⊥) →
-                      (L1 ⋕[V, d] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V → ⊥).
-#a #I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_bind, or_intror, or_introl/
-qed-.
-
-lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ⋕[ⓕ{I}V.T, d] L2 → ⊥) →
-                      (L1 ⋕[V, d] L2 → ⊥) ∨ (L1 ⋕[T, d] L2 → ⊥).
-#I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_flat, or_intror, or_introl/
-qed-.
-
 (* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1⋕[T, d] L →
                            ∀L2. (L ⋕[T, d] L2 → ⊥) → (L1 ⋕[T, d] L2 → ⊥).
 /3 width=3 by lleq_canc_sn/ qed-.