(**************************************************************************)
include "ground_2/ynat/ynat_max.ma".
-include "basic_2/notation/relations/extpsubst_6.ma".
+include "basic_2/notation/relations/psubst_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".
(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
.
interpretation "context-sensitive extended ordinary substritution (term)"
- 'ExtPSubst G L T1 d e T2 = (cpy d e G L T1 T2).
+ 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
(* Basic properties *********************************************************)
#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_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+ elim (lsuby_ldrop_trans_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-.
-lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶×[d, e] T.
+lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T.
#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
qed.
(* Basic_1: was: subst1_ex *)
lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[d] L ≡ K.ⓑ{I}V →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶×[d, 1] T2 & ⇧[d, 1] T ≡ T2.
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[d, 1] T2 & ⇧[d, 1] T ≡ T2.
#I #G #K #V #T1 elim T1 -T1
[ * #i #L #d #HLK
/2 width=4 by lift_sort, lift_gref, ex2_2_intro/
]
qed-.
-lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T2 →
+lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T2 →
∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T2.
+ ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T2.
#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
qed-.
lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[d, |L| - d] T2.
+ ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[d, |L| - d] T2.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
[ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
lapply (ldrop_fwd_length_lt2 … HLK)
qed-.
lemma cpy_weak_full: ∀G,L,T1,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[0, |L|] T2.
+ ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
#G #L #T1 #T2 #d #e #HT12
lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
/2 width=2 by cpy_weak_top/
qed-.
-lemma cpy_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
- ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → d + e ≤ dt + et →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶×[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #T1 #d #e #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
- [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
- #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
- @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 //
- elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
- <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
- /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
- /3 width=2 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i-d] T & ⦃G, L⦄ ⊢ T ▶×[i, d+e-i] T2.
+lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀i. i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d, i-d] T & ⦃G, L⦄ ⊢ T ▶[i, d+e-i] T2.
#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 #Hjde
]
qed-.
-lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶×[d, i-d] T2.
+lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀i. i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶[d, i-d] T2.
#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 #Hjde
]
qed-.
-lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
-#G #d #e #K #T1 #T2 #H elim H -G -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/
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → d + e ≤ dt + et →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #T1 #d #e #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
+ [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
+ #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
+ @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 //
+ elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+ <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+ /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
+ /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+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-.
(* Basic inversion lemmas ***************************************************)
-fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀J. T1 = ⓪{J} →
+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 ≤ yinj i & i < d + e &
⇩[i] L ≡ K.ⓑ{I}V &
]
qed-.
-lemma cpy_inv_atom1: ∀I,G,L,T2,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} ∨
∃∃J,K,V,i. d ≤ yinj i & i < d + e &
⇩[i] L ≡ K.ⓑ{J}V &
/2 width=4 by cpy_inv_atom1_aux/ qed-.
(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
+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 //
* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀G,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 ∨
∃∃I,K,V. d ≤ i & i < d + e &
⇩[i] L ≡ K.ⓑ{I}V &
* #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
qed-.
-lemma cpy_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
+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 //
* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
-fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+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 &
- ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶×[⫯d, e] T2 &
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 &
+ ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯d, 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
]
qed-.
-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 &
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶×[⫯d, e] T2 &
+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 &
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯d, e] T2 &
U2 = ⓑ{a,I}V2.T2.
/2 width=3 by cpy_inv_bind1_aux/ qed-.
-fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+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 &
+ ∃∃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
]
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 &
+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-.
-fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
+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
[ //
| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
]
qed-.
-lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
+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_1: was: subst1_gen_lift_eq *)
lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → U1 = U2.
-#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_up … HU12 … HTU1) -HU12 -HTU1
+ ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → U1 = U2.
+#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
/2 width=4 by cpy_inv_refl_O2/
qed-.
-(* Basic forward lemmas *****************************************************)
-
-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: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶×[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#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
- >shift_append_assoc normalize #H
- elim (cpy_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 ] (**) (* 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
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