(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
definition cnx: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. NF … (cpx h g G L) (eq …).
+ λh,o,G,L. NF … (cpx h o G L) (eq …).
interpretation
"normality for context-sensitive extended reduction (term)"
- 'PRedNormal h g L T = (cnx h g L T).
+ 'PRedNormal h o L T = (cnx h o L T).
(* Basic inversion lemmas ***************************************************)
-lemma cnx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄ → deg h g k 0.
-#h #g #G #L #k #H elim (deg_total h g k)
+lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄ → deg h o s 0.
+#h #o #G #L #s #H elim (deg_total h o s)
#d @(nat_ind_plus … d) -d // #d #_ #Hkd
-lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H
+lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_st/ -L -d #H
lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *)
lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *)
-lapply (next_lt h k) >H -H #H elim (lt_refl_false … H)
+lapply (next_lt h s) >H -H #H elim (lt_refl_false … H)
qed-.
-lemma cnx_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄ → ⊥.
-#h #g #I #G #L #K #V #i #HLK #H
+lemma cnx_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄ → ⊥.
+#h #o #I #G #L #K #V #i #HLK #H
elim (lift_total V 0 (i+1)) #W #HVW
lapply (H W ?) -H [ /3 width=7 by cpx_delta/ ] -HLK #H destruct
elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/
qed-.
-lemma cnx_inv_abst: ∀h,g,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}V.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-#h #g #a #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abst: ∀h,o,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}V.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
+#h #o #a #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
]
qed-.
-lemma cnx_inv_abbr: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃-ⓓV.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-#h #g #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abbr: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃-ⓓV.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
+#h #o #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
]
qed-.
-lemma cnx_inv_zeta: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃+ⓓV.T⦄ → ⊥.
-#h #g #G #L #V #T #H elim (is_lift_dec T 0 1)
+lemma cnx_inv_zeta: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃+ⓓV.T⦄ → ⊥.
+#h #o #G #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
lapply (H U ?) -H /2 width=3 by cpx_zeta/ #H destruct
elim (lift_inv_pair_xy_y … HTU)
]
qed-.
-lemma cnx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#h #g #G #L #V1 #T1 #HVT1 @and3_intro
+lemma cnx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#h #o #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2 #H destruct //
| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
]
qed-.
-lemma cnx_inv_eps: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓝV.T⦄ → ⊥.
-#h #g #G #L #V #T #H lapply (H T ?) -H
+lemma cnx_inv_eps: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #o #G #L #V #T #H lapply (H T ?) -H
/2 width=4 by cpx_eps, discr_tpair_xy_y/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cnx_fwd_cnr: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
-#h #g #G #L #T #H #U #HTU
+lemma cnx_fwd_cnr: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+#h #o #G #L #T #H #U #HTU
@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *)
qed-.
(* Basic properties *********************************************************)
-lemma cnx_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄.
-#h #g #G #L #k #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_
+lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄.
+#h #o #G #L #s #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_
lapply (deg_mono … Hkd Hk) -h -L <plus_n_Sm #H destruct
qed.
-lemma cnx_sort_iter: ∀h,g,G,L,k,d. deg h g k d → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆((next h)^d k)⦄.
-#h #g #G #L #k #d #Hkd
+lemma cnx_sort_iter: ∀h,o,G,L,s,d. deg h o s d → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆((next h)^d s)⦄.
+#h #o #G #L #s #d #Hkd
lapply (deg_iter … d Hkd) -Hkd <minus_n_n /2 width=6 by cnx_sort/
qed.
-lemma cnx_lref_free: ∀h,g,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
-#h #g #G #L #i #Hi #X #H elim (cpx_inv_lref1 … H) -H // *
+lemma cnx_lref_free: ∀h,o,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄.
+#h #o #G #L #i #Hi #X #H elim (cpx_inv_lref1 … H) -H // *
#I #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.
-lemma cnx_lref_atom: ∀h,g,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
-#h #g #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
+lemma cnx_lref_atom: ∀h,o,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄.
+#h #o #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
qed.
-lemma cnx_abst: ∀h,g,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}W.T⦄.
-#h #g #a #G #L #W #T #HW #HT #X #H
+lemma cnx_abst: ∀h,o,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}W.T⦄.
+#h #o #a #G #L #W #T #HW #HT #X #H
elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
>(HW … HW0) -W0 >(HT … HT0) -T0 //
qed.
-lemma cnx_appl_simple: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄.
-#h #g #G #L #V #T #HV #HT #HS #X #H
+lemma cnx_appl_simple: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄.
+#h #o #G #L #V #T #HV #HT #HS #X #H
elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
>(HV … HV0) -V0 >(HT … HT0) -T0 //
qed.
-axiom cnx_dec: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T1⦄ ∨
- ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & (T1 = T2 → ⊥).
+axiom cnx_dec: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T1⦄ ∨
+ ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & (T1 = T2 → ⊥).
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