include "basic_2/notation/relations/extpsubst_6.ma".
include "basic_2/grammar/genv.ma".
+include "basic_2/grammar/cl_shift.ma".
include "basic_2/relocation/ldrop_append.ma".
include "basic_2/relocation/lsuby.ma".
(* Basic properties *********************************************************)
-lemma lsuby_cpy_trans: ∀G,d,e.lsub_trans … (cpy d e G) lsuby.
+lemma lsuby_cpy_trans: ∀G,L1,T1,T2,d,e. ⦃G, L1⦄ ⊢ T1 ▶×[d, e] T2 →
+ ∀L2. L2 ⊑×[d, e] L1 → ⦃G, L2⦄ ⊢ T1 ▶×[d, e] T2.
#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
[ //
| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (lsuby_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
-| /4 width=1 by lsuby_pair, cpy_bind/
+ elim (lsuby_fwd_ldrop2_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
| /3 width=1 by cpy_flat/
]
qed-.
elim (IHV12 i) -IHV12 // #V #HV1 #HV2
elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
-Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I} V) ?) -HT1 /3 width=5 by cpy_bind, ex2_intro/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
+ -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
]
-qed.
+qed-.
-lemma cpy_split_down: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
+lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
∀i. d ≤ i → i ≤ d + e →
∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d + e - i] T &
⦃G, L⦄ ⊢ T ▶×[d, i - d] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
elim (lt_or_ge i j)
- [ -Hide -Hjde >(plus_minus_m_m j d) in⦄ ⊢ (% → ?); // -Hdj /3 width=8/
+ [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=9 by ex2_intro, cpy_atom, cpy_subst/
| -Hdi -Hdj
- >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
+ >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=5 by ex2_intro, cpy_subst/
]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
-Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (cpy_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
+ lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
+ -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
]
-qed.
+qed-.
-lemma cpy_append: ∀K,T1,T2,d,e. K⦄ ⊢ T1 ▶×[d, e] T2 →
- ∀L. L @@ K⦄ ⊢ T1 ▶×[d, e] T2.
-#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
-#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_ldrop2_length … HK0) #H
-@(cpy_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/
-qed.
+lemma cpy_append: ∀G,K,T1,T2,d,e. ⦃G, K⦄ ⊢ T1 ▶×[d, e] T2 →
+ ∀L. ⦃G, L @@ K⦄ ⊢ T1 ▶×[d, e] T2.
+#G #K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e
+/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
+#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
+qed-.
(* Basic inversion lemmas ***************************************************)
-fact cpy_inv_atom1_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
-[ #L #I #d #e #J #H destruct /2 width=1/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀J. T1 = ⓪{J} →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K.ⓑ{I}V &
+ ⇧[O, i + 1] V ≡ T2 &
+ J = LRef i.
+#G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e
+[ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
]
-qed.
+qed-.
-lemma cpy_inv_atom1: ∀L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
+lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-/2 width=3/ qed-.
+ ∃∃J,K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K.ⓑ{J}V &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: ∀L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H
+lemma cpy_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
+#G #L #T2 #k #d #e #H
elim (cpy_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
+* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
-(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
+lemma cpy_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
T2 = #i ∨
- ∃∃K,V. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2.
-#L #T2 #i #d #e #H
-elim (cpy_inv_atom1 … H) -H /2 width=1/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+ ∃∃I,K,V. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K.ⓑ{I}V &
+ ⇧[O, i + 1] V ≡ T2.
+#G #L #T2 #i #d #e #H
+elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
qed-.
-lemma cpy_inv_gref1: ∀L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H
+lemma cpy_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
+#G #L #T2 #p #d #e #H
elim (cpy_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
+* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
-fact cpy_inv_bind1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
- L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #a #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
-| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
+ ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct
]
-qed.
+qed-.
-lemma cpy_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
+lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
- L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-/2 width=3/ qed-.
+ ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
-fact cpy_inv_flat1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+ U2 = ⓕ{I}V2.T2.
+#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
-qed.
+qed-.
+
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+ U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
-lemma cpy_inv_flat1: ∀d,e,L,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-/2 width=3/ qed-.
-fact cpy_inv_refl_O2_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
elim (lt_refl_false … Hdd)
-| /3 width=1/
-| /3 width=1/
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
]
-qed.
+qed-.
-lemma cpy_inv_refl_O2: ∀L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
-/2 width=6/ qed-.
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma cpy_fwd_tw: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
qed-.
-lemma cpy_fwd_shift1: ∀L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
+lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶×[d, e] T →
∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
[ #L #T1 #T #d #e #HT1
@(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
| #I #L1 #V1 #IH #L #T1 #X #d #e
#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/ (**) (* explicit constructor *)
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
+ /2 width=3 by trans_eq/
]
qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
projects / helm.git / blobdiff