X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcnr.ma;h=1a6ce5d319d6f2828c4005864c3b5fcbaac52ecf;hb=52e675f555f559c047d5449db7fc89a51b977d35;hp=821d74c7a2b83eaf7c96f62793ac6e0a5fba655b;hpb=65008df95049eb835941ffea1aa682c9253c4c2b;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
index 821d74c7a..1a6ce5d31 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
@@ -12,91 +12,103 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/normal_2.ma".
+include "basic_2/notation/relations/prednormal_3.ma".
 include "basic_2/reduction/cpr.ma".
 
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
 
-definition cnr: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
+definition cnr: relation3 genv lenv term ≝ λG,L. NF … (cpr G L) (eq …).
 
 interpretation
-   "context-sensitive normality (term)"
-   'Normal L T = (cnr L T).
+   "normality for context-sensitive reduction (term)"
+   'PRedNormal G L T = (cnr G L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cnr_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥.
-#L #K #V #i #HLK #H
+lemma cnr_inv_delta: ∀G,L,K,V,i. ⇩[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄ → ⊥.
+#G #L #K #V #i #HLK #H
 elim (lift_total V 0 (i+1)) #W #HVW
-lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct
+lapply (H W ?) -H [ /3 width=6 by cpr_delta/ ] -HLK #H destruct
 elim (lift_inv_lref2_be … HVW) -HVW //
 qed-.
 
-lemma cnr_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄.
-#a #L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
+lemma cnr_inv_abst: ∀a,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}V.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡ 𝐍⦃T⦄.
+#a #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnr_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄.
-#L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
+lemma cnr_inv_abbr: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃-ⓓV.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡ 𝐍⦃T⦄.
+#G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnr_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
-#L #V #T #H elim (is_lift_dec T 0 1)
+lemma cnr_inv_zeta: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃+ⓓV.T⦄ → ⊥.
+#G #L #V #T #H elim (is_lift_dec T 0 1)
 [ * #U #HTU
-  lapply (H U ?) -H /2 width=3/ #H destruct
+  lapply (H U ?) -H /2 width=3 by cpr_zeta/ #H destruct
   elim (lift_inv_pair_xy_y … HTU)
 | #HT
-  elim (cpr_delift (⋆) V T (⋆. ⓓV) 0) // #T2 #T1 #HT2 #HT12
-  lapply (H (+ⓓV.T2) ?) -H /4 width=1/ -HT2 #H destruct /3 width=2/
+  elim (cpr_delift G (⋆) V T (⋆. ⓓV) 0) //
+  #T2 #T1 #HT2 #HT12 lapply (H (+ⓓV.T2) ?) -H /4 width=1 by tpr_cpr, cpr_bind/ -HT2
+  #H destruct /3 width=2 by ex_intro/
 ]
 qed-.
 
-lemma cnr_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#L #V1 #T1 #HVT1 @and3_intro
-[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
+lemma cnr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#G #L #V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
 | generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
   [ elim (lift_total V1 0 1) #V2 #HV12
-    lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct
-  | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1/ #H destruct
+    lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by tpr_cpr, cpr_theta/ -HV12 #H destruct
+  | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by tpr_cpr, cpr_beta/ #H destruct
 ]
 qed-.
 
-lemma cnr_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
-#L #V #T #H lapply (H T ?) -H /2 width=1/ #H
-@discr_tpair_xy_y //
+lemma cnr_inv_eps: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓝV.T⦄ → ⊥.
+#G #L #V #T #H lapply (H T ?) -H
+/2 width=4 by cpr_eps, discr_tpair_xy_y/
 qed-.
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: nf2_sort *)
-lemma cnr_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄.
-#L #k #X #H
+lemma cnr_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆k⦄.
+#G #L #k #X #H
 >(cpr_inv_sort1 … H) //
 qed.
 
+lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
+#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // *
+#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+(* Basic_1: was only: nf2_csort_lref *)
+lemma cnr_lref_atom: ∀G,L,i. ⇩[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
+#G #L #i #HL @cnr_lref_free >(drop_fwd_length … HL) -HL //
+qed.
+
 (* Basic_1: was: nf2_abst *)
-lemma cnr_abst: ∀a,L,W,T. L ⊢ 𝐍⦃W⦄ → L.ⓛW ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄.
-#a #L #W #T #HW #HT #X #H
+lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄.
+#a #G #L #W #T #HW #HT #X #H
 elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
 >(HW … HW0) -W0 >(HT … HT0) -T0 //
 qed.
 
 (* Basic_1: was only: nf2_appl_lref *)
-lemma cnr_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄.
-#L #V #T #HV #HT #HS #X #H
+lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄.
+#G #L #V #T #HV #HT #HS #X #H
 elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
 >(HV … HV0) -V0 >(HT … HT0) -T0 //
 qed.
 
 (* Basic_1: was: nf2_dec *)
-axiom cnr_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨
-               ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
+axiom cnr_dec: ∀G,L,T1. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T1⦄ ∨
+               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
 
 (* Basic_1: removed theorems 1: nf2_abst_shift *)