(* avtivate genv *)
inductive cpx (h) (g): relation4 genv lenv term term ≝
| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
-| cpx_sort : ∀G,L,k,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
+| cpx_st : ∀G,L,k,d. deg h g k (d+1) → cpx h g G L (⋆k) (⋆(next h k))
| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
- â\87©[0, i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
- â\87§[0, i + 1] V2 ≡ W2 → cpx h g G L (#i) W2
+ â¬\87[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
+ â¬\86[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
| cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
cpx h g G L V1 V2 → cpx h g G L T1 T2 →
cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
- â\87§[0, 1] T2 ≡ T → cpx h g G L (+ⓓV.T1) T2
-| cpx_tau : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
-| cpx_ti : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
+ â¬\86[0, 1] T2 ≡ T → cpx h g G L (+ⓓV.T1) T2
+| cpx_eps : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
+| cpx_ct : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
| cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
| cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- cpx h g G L V1 V â\86\92 â\87§[0, 1] V ≡ V2 → cpx h g G L W1 W2 →
+ cpx h g G L V1 V â\86\92 â¬\86[0, 1] V ≡ V2 → cpx h g G L W1 W2 →
cpx h g G (L.ⓓW1) T1 T2 →
cpx h g G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
.
lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
[ //
-| /2 width=2 by cpx_sort/
+| /2 width=2 by cpx_st/
| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 *
- /4 width=7 by cpx_delta, cpx_ti/
-|4,9: /4 width=1 by cpx_bind, cpx_beta, lsubr_bind/
-|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
-|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_bind/
+ elim (lsubr_fwd_drop2_pair … HL12 … HLK1) -HL12 -HLK1 *
+ /4 width=7 by cpx_delta, cpx_ct/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsubr_pair/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_pair/
]
qed-.
lemma cpr_cpx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2
-/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_beta, cpx_theta/
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
qed.
lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
#h #g * /2 width=1 by cpx_bind, cpx_flat/
qed.
-lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
- â\88\83â\88\83T2,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, g] T2 & â\87§[d, 1] T ≡ T2.
+lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
+ â\88\83â\88\83T2,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, g] T2 & â¬\86[l, 1] T ≡ T2.
#h #g #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i l) #Hil [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
destruct
elim (lift_total V 0 (i+1)) #W #HVW
elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /3 width=9 by cpx_bind, ldrop_ldrop, lift_bind, ex2_2_intro/
+ [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, drop_drop, lift_bind, ex2_2_intro/
| elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
]
]
qed-.
-lemma cpx_append: ∀h,g,G. l_appendable_sn … (cpx h g G).
-#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpx_sort, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_ti, cpx_beta, cpx_theta/
-#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpx_delta … I … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
(* Basic inversion lemmas ***************************************************)
fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J}
- | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
- | â\88\83â\88\83I,K,V,V2,i. â\87©[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i + 1] V2 ≡ T2 & J = LRef i.
+ | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
+ | â\88\83â\88\83I,K,V,V2,i. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2 & J = LRef i.
#G #h #g #L #T1 #T2 * -L -T1 -T2
[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
-| #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #G #L #k #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
∨∨ T2 = ⓪{J}
- | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
- | â\88\83â\88\83I,K,V,V2,i. â\87©[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i + 1] V2 ≡ T2 & J = LRef i.
+ | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
+ | â\88\83â\88\83I,K,V,V2,i. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2 & J = LRef i.
/2 width=3 by cpx_inv_atom1_aux/ qed-.
lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
- ∃∃l. deg h g k (l+1) & T2 = ⋆(next h k).
+ ∃∃d. deg h g k (d+1) & T2 = ⋆(next h k).
#h #g #G #L #T2 #k #H
elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k0 #l0 #Hkl0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
+[ #k0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
T2 = #i ∨
- â\88\83â\88\83I,K,V,V2. â\87©[O, i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i + 1] V2 ≡ T2.
+ â\88\83â\88\83I,K,V,V2. â¬\87[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2.
#h #g #G #L #T2 #i #H
elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k #l #_ #_ #H destruct
+[ #k #d #_ #_ #H destruct
| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
lemma cpx_inv_lref1_ge: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 → |L| ≤ i → T2 = #i.
#h #g #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
-#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (ldrop_fwd_length_lt2 … HLK) -K -I -V1
+#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed-.
lemma cpx_inv_gref1: ∀h,g,G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
#h #g #G #L #T2 #p #H
elim (cpx_inv_atom1 … H) -H // *
-[ #k #l #_ #_ #H destruct
+[ #k #d #_ #_ #H destruct
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,J}V2.T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T &
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T &
a = true & J = Abbr.
#h #g #G #L #U1 #U2 * -L -U1 -U2
[ #I #G #L #b #J #W #U1 #H destruct
-| #G #L #k #l #_ #b #J #W #U1 #H destruct
+| #G #L #k #d #_ #b #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,I} V2. T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T &
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T &
a = true & I = Abbr.
/2 width=3 by cpx_inv_bind1_aux/ qed-.
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓓ{a} V2. T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T & a = true.
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T & a = true.
#h #g #a #G #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
qed-.
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
#h #g #G #L #U #U2 * -L -U -U2
[ #I #G #L #J #W #U1 #H destruct
-| #G #L #k #l #_ #J #W #U1 #H destruct
+| #G #L #k #d #_ #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
| ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
| #T2 #_ #_ #H destruct
]
qed-.
-
-lemma cpx_fwd_shift1: ∀h,g,G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#h #g #G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpx_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.