universary milestone in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reduction / cnr.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
index 34d95faaeed1a272d803e89a44fa588495002186..1a307432d59574dd8c8172094a5b4762ca002fa0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
@@ -12,41 +12,41 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/normal_3.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: relation3 genv lenv term ≝ λG,L. NF … (cpr G L) (eq …).
 
 interpretation
-   "context-sensitive normality (term)"
-   'Normal G L T = (cnr G L T).
+   "normality for context-sensitive reduction (term)"
+   'PRedNormal G L T = (cnr G L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cnr_inv_delta: â\88\80G,L,K,V,i. â\87©[i] L â\89¡ K.â\93\93V â\86\92 â¦\83G, Lâ¦\84 â\8a¢ 𝐍⦃#i⦄ → ⊥.
+lemma cnr_inv_delta: â\88\80G,L,K,V,i. â¬\87[i] L â\89¡ K.â\93\93V â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\9e¡ 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 by cpr_delta/ ] -HLK #H destruct
-elim (lift_inv_lref2_be … HVW) -HVW //
+elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/
 qed-.
 
-lemma cnr_inv_abst: ∀a,G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓛ{a}V.T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ 𝐍⦃T⦄.
+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: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃-ⓓV.T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ 𝐍⦃T⦄.
+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: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
+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 by cpr_zeta/ #H destruct
@@ -58,7 +58,7 @@ lemma cnr_inv_zeta: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
 ]
 qed-.
 
-lemma cnr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
+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 //
@@ -69,35 +69,46 @@ lemma cnr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G,
 ]
 qed-.
 
-lemma cnr_inv_tau: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
+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_tau, discr_tpair_xy_y/
+/2 width=4 by cpr_eps, discr_tpair_xy_y/
 qed-.
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: nf2_sort *)
-lemma cnr_sort: ∀G,L,k. ⦃G, L⦄ ⊢ 𝐍⦃⋆k⦄.
-#G #L #k #X #H
+lemma cnr_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆s⦄.
+#G #L #s #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,G,L,W,T. ⦃G, L⦄ ⊢ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃ⓛ{a}W.T⦄.
+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: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃ⓐV.T⦄.
+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: ∀G,L,T1. ⦃G, L⦄ ⊢ 𝐍⦃T1⦄ ∨
+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 *)