| rtm_gtype : ∀G,V,u,E,S,p. p = |G| →
rtm_step (mk_rtm (G. ⓛV) u E S (§p))
(mk_rtm G u E S V)
-| rtm_tau : ∀G,u,E,S,W,T.
+| rtm_eps : ∀G,u,E,S,W,T.
rtm_step (mk_rtm G u E S (ⓝW. T))
(mk_rtm G u E S T)
| rtm_appl : ∀G,u,E,S,V,T.
/3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/
qed.
-lemma cprs_tau: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
+lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
#G #L #T1 #T2 #H @(cprs_ind … H) -T2
-/3 width=3 by cprs_strap1, cpr_cprs, cpr_tau/
+/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
qed.
lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
/4 width=6 by cprs_strap2, cprs_delta/
|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
-|4,6: /3 width=1 by cprs_flat, cprs_tau/
+|4,6: /3 width=1 by cprs_flat, cprs_eps/
|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
]
qed-.
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
qed.
-lemma cpxs_tau: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_eps: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_tau/
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
qed.
-lemma cpxs_ti: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+lemma cpxs_ct: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
#h #g #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
-/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ti/
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ct/
qed.
lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
lemma lpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpx h g G).
#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
[ /2 width=3 by/
-| /3 width=2 by cpx_cpxs, cpx_sort/
+| /3 width=2 by cpx_cpxs, cpx_st/
| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
/4 width=7 by cpxs_delta, cpxs_strap2/
|4,9: /4 width=1 by cpxs_beta, cpxs_bind, lpx_pair/
-|5,7,8: /3 width=1 by cpxs_flat, cpxs_ti, cpxs_tau/
+|5,7,8: /3 width=1 by cpxs_flat, cpxs_ct, cpxs_eps/
| /4 width=3 by cpxs_zeta, lpx_pair/
| /4 width=3 by cpxs_theta, cpxs_strap1, lpx_pair/
]
elim (IHn … Hnl) -IHn
[ #H lapply (tstc_inv_atom1 … H) -H #H >H -H /2 width=1 by or_intror/
| generalize in match Hnl; -Hnl @(nat_ind_plus … n) -n
- /4 width=3 by cpxs_strap2, cpx_sort, or_intror/
+ /4 width=3 by cpxs_strap2, cpx_st, or_intror/
| >iter_SO >iter_n_Sm //
]
]
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lrsubeq_4.ma".
+include "basic_2/notation/relations/lrsubeqc_4.ma".
include "basic_2/static/aaa.ma".
include "basic_2/computation/acp_cr.ma".
interpretation
"local environment refinement (abstract candidates of reducibility)"
- 'LRSubEq RP G L1 L2 = (lsubc RP G L1 L2).
+ 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
(* Basic inversion lemmas ***************************************************)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_max.ma".
-include "basic_2/notation/relations/psubst_6.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/lsuby.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* activate genv *)
-inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
-| cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I})
-| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e →
- ⇩[i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W
-| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
- cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
- cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
- cpy d e G L V1 V2 → cpy d e G L T1 T2 →
- cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation "context-sensitive extended ordinary substritution (term)"
- 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpy_trans: ∀G,d,e. lsub_trans … (cpy d e G) (lsuby d e).
-#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_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.
-#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.
-#I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK
- /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i d) #Hid
- /3 width=4 by 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)
- /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
-| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L.ⓑ{J}W1) (d+1)) -IHU1
- /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK
- /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
- ]
-]
-qed-.
-
-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 #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/
-| /3 width=1 by cpy_flat/
-]
-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 #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)
- /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
-| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
- /2 width=1 by cpy_bind/
-| /2 width=1 by cpy_flat/
-]
-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 #T2 #d #e #HT12
-lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
-/2 width=2 by cpy_weak_top/
-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.
-#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
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
- /3 width=5 by ex2_intro, cpy_flat/
-]
-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.
-#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
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-(* 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} →
- T2 = ⓪{J} ∨
- ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
- ⇩[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-.
-
-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 &
- ⇧[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: ∀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 →
- T2 = #i ∨
- ∃∃I,K,V. d ≤ i & i < d + e &
- ⇩[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: ∀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 →
- ∀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 &
- 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-.
-
-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 →
- ∀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-.
-
-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.
-#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
- elim (ylt_yle_false … Hdi) -Hdi //
-| /3 width=1 by eq_f2/
-| /3 width=1 by eq_f2/
-]
-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_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_fwd_up … HU12 … HTU1) -HU12 -HTU1
-/2 width=4 by cpy_inv_refl_O2/
-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
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶[d1, e1] T1 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶[d1, e1] T.
-#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (cpy_inv_lref1 … H) -H
- [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
- | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
- ]
-| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
- /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶[d1, e1] T1 →
- ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶[d1, e1] T.
-#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
- elim (cpy_inv_lref1 … H1) -H1
- [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
- [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
- | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
- ]
- ]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0
- [ -H #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
- ]
-| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Note: the constant 1 comes from cpy_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T0 →
- ∀T2. ⦃G, L⦄ ⊢ T0 ▶[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶[d, e] T2.
-#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
-[ #I #G #L #d #e #T2 #H #He
- elim (cpy_inv_atom1 … H) -H
- [ #H destruct //
- | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
- ]
-| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
- >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- lapply (IHT10 … HT02 He) -T0 /3 width=1 by cpy_bind/
-| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
-]
-qed-.
-
-theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
-#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
- lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
- >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 … HV02) -IHV10 -HV02 // #V
- elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 … HV02) -V0 //
- elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Properties on relocation *************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
- ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt + et ≤ d → ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
- lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
- lapply (ylt_inv_inj … Hid) -Hid #Hid
- lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
- ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt ≤ d → d ≤ dt + et → ⦃G, L⦄ ⊢ U1 ▶[dt, et + e] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
- elim (lift_inv_lref1 … H) -H * #Hid #H destruct
- [ -Hdtd
- lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
- elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
- | -Hdti
- elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
- lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
- /4 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
- ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
- ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
- lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
- elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
- lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
- /3 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=6 by cpy_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
- lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
- lapply (ylt_inv_inj … Hid) -Hid #Hid
- lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
- elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -IHU12 -HTU1
- /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et-e] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
- [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
- [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
- elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
- /3 width=5 by cpy_subst, ex2_intro/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
- >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1
- /3 width=6 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK //
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- yinj d + e ≤ dt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt-e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
- lapply (yle_trans … Hdedt … Hdti) #Hdei
- elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
- elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
- elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
- [ /2 width=1 by monotonic_yle_minus_dx/
- | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
- >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Advancd inversion lemmas on relocation ***********************************)
-
-lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[d, dt + et - (yinj d + e)] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
-lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ yinj d + e →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d-dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
-[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
-elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak … HU2 d e ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
-lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubststar_6.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-definition cpys: ynat → ynat → relation4 genv lenv term term ≝
- λd,e,G. LTC … (cpy d e G).
-
-interpretation "context-sensitive extended multiple substritution (term)"
- 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma cpys_ind: ∀G,L,T1,d,e. ∀R:predicate term. R T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → R T → R T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T2.
-#G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma cpys_ind_dx: ∀G,L,T2,d,e. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → R T → R T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T1.
-#G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma cpy_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-/2 width=1 by inj/ qed.
-
-lemma cpys_strap1: ∀G,L,T1,T,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-normalize /2 width=3 by step/ qed-.
-
-lemma cpys_strap2: ∀G,L,T1,T,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-lemma lsuby_cpys_trans: ∀G,d,e. lsub_trans … (cpys d e G) (lsuby d e).
-/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
-qed-.
-
-lemma cpys_refl: ∀G,L,d,e. reflexive … (cpys d e G L).
-/2 width=1 by cpy_cpys/ qed.
-
-lemma cpys_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] ⓑ{a,I}V2.T2.
-#G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
-[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
-| /3 width=5 by cpys_strap1, cpy_bind/
-]
-qed.
-
-lemma cpys_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] ⓕ{I}V2.T2.
-#G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
-[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
-| /3 width=5 by cpys_strap1, cpy_flat/
-qed.
-
-lemma cpys_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 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind … H) -T2
-/3 width=7 by cpys_strap1, cpy_weak/
-qed-.
-
-lemma cpys_weak_top: ∀G,L,T1,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, |L| - d] T2.
-#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
-/3 width=4 by cpys_strap1, cpy_weak_top/
-qed-.
-
-lemma cpys_weak_full: ∀G,L,T1,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
-#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
-/3 width=5 by cpys_strap1, cpy_weak_full/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpys_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 #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
- elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
-/2 width=3 by transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] T2 → T2 = ⋆k.
-#G #L #T2 #k #d #e #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_sort1 … HT2) -HT2 //
-qed-.
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶*[d, e] T2 → T2 = §p.
-#G #L #T2 #p #d #e #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_gref1 … HT2) -HT2 //
-qed-.
-
-lemma cpys_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.
-#a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
- /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_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.
-#I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_flat1 … HU2) -HU2
- /3 width=5 by cpys_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 0] T2 → T1 = T2.
-#G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
- ⦃G, L⦄ ⊢ U1 ▶*[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2
-/2 width=7 by cpy_inv_lift1_eq/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/psubststaralt_6.ma".
-include "basic_2/substitution/cpys_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* alternative definition of cpys *)
-inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
-| cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
-| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(d+e-i)) G K V1 V2 →
- ⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
-| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
- cpysa d e G L V1 V2 → cpysa (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
- cpysa d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
- cpysa d e G L V1 V2 → cpysa d e G L T1 T2 →
- cpysa d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation
- "context-sensitive extended multiple substritution (term) alternative"
- 'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpysa_trans: ∀G,d,e. lsub_trans … (cpysa d e G) (lsuby d e).
-#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
-[ //
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_ldrop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
-| /4 width=1 by lsuby_succ, cpysa_bind/
-| /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpysa_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶▶*[d, e] T.
-#G #T elim T -T //
-#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
-qed.
-
-lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T →
- ∀T2. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
-#G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
-[ #I #G #L #d #e #X #H
- elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
-| #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
- lapply (ldrop_fwd_drop2 … HLK) #H0LK
- lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
- elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
- /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
-| #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
-| #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
-/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
- (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
- (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 →
- ⇧[O, i+1] V2 ≡ W2 → R O (⫰(d+e-i)) G K V1 V2 → R d e G L (#i) W2
- ) →
- (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 → R d e G L V1 V2 →
- R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L V1 V2 →
- R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
-#R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpysa_inv_cpys/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_cpy.ma".
-include "basic_2/substitution/cpys_alt.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpys_inv_SO2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[d, 1] T2.
-#G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_strip_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶*[d1, e1] T1 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
-
-lemma cpys_strip_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶*[d1, e1] T1 →
- ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
-
-lemma cpys_strap1_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T0 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶*[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
-
-lemma cpys_strap2_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
-
-lemma cpys_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
- ∀i. d ≤ 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 #i #Hdi #Hide @(cpys_ind … H) -T2
-[ /2 width=3 by ex2_intro/
-| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (cpy_split_up … HT12 … Hide) -HT12 -Hide #T0 #HT0 #HT02
- elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
- >ymax_pre_sn_comm //
-]
-qed-.
-
-lemma cpys_inv_lift1_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (cpys_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
-lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
->yplus_minus_inj /2 width=3 by ex2_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem cpys_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶*[d1, e1] T1 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
-
-theorem cpys_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶*[d1, e1] T1 →
- ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶*[d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
-
-theorem cpys_trans_eq: ∀G,L,T1,T,T2,d,e.
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 →
- ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
-normalize /2 width=3 by trans_TC/ qed-.
-
-theorem cpys_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T0 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T ▶*[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
-
-theorem cpys_antisym_eq: ∀G,L1,T1,T2,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T2 →
- ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] T1 → T1 = T2.
-#G #L1 #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
-[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #_ #_ #HVW2 #_ #L2 #HW2
- elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hdi -Hide | ]
- [ lapply (cpys_weak_full … HW2) -HW2 #HW2
- lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
- [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
- #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
- | elim (ldrop_O1_le … Hi) -Hi #K2 #HLK2
- elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
- /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hdi -Hide
- #X #_ #H elim (lift_inv_lref2_be … H) -H //
- ]
-| #a #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
- #V #T #HV2 #HT2 #H destruct
- lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
-| #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
- #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/cpy_lift.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
- d ≤ yinj i → i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(d+e-i)] U1 →
- ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] U2.
-#I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
-[ /3 width=5 by cpy_cpys, cpy_subst/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU … HU0) -IHU #H
- lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
- lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
- lapply (cpy_weak … HU02 d e ? ?) -HU02
- [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
- >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
-]
-qed.
-
-lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,d.
- d ≤ yinj i →
- ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
- ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, ∞] U2.
-#I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12
-@(cpys_subst … HLK … HU12) >yminus_Y_inj //
-qed.
-
-(* Advanced inverion lemmas *************************************************)
-
-lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*[d, e] T2 →
- T2 = ⓪{I} ∨
- ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
- ⇩[i] L ≡ K.ⓑ{J}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
- ⇧[O, i+1] V2 ≡ T2 &
- I = LRef i.
-#I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
-[ /2 width=1 by or_introl/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
- | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (ldrop_fwd_drop2 … HLK) #H
- elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
- [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
- /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
- ]
-]
-qed-.
-
-lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
- T2 = #i ∨
- ∃∃I,K,V1,V2. d ≤ i & i < d + e &
- ⇩[i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
- ⇧[O, i+1] V2 ≡ T2.
-#G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
-* #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
-qed-.
-
-lemma cpys_inv_lref1_ldrop: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
- ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⇧[O, i+1] V2 ≡ T2 →
- ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2
- & d ≤ i
- & i < d + e.
-#G #L #T2 #i #d #e #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
-[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
-| * #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
- lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
- lapply (ldrop_mono … HLY … HLK) -L #H destruct
- /2 width=1 by and3_intro/
-]
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
- ∀L,U1,s,d,e. dt + et ≤ yinj d → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
- ∀L,U1,s,d,e. dt ≤ yinj d → d ≤ dt + et →
- ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[dt, et + e] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
- ∀L,U1,s,d,e. yinj d ≤ dt → ⇩[s, d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ▶*[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
-[ #U2 #H >(lift_mono … HTU1 … H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT … HTU) -IHT #HU1
- lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-(* Inversion lemmas for relocation ******************************************)
-
-lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- yinj d + e ≤ dt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ yinj d + e →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] W2 →
- ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
- d ≤ yinj i → i < d + e →
- ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ W2.
-#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
-elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
->yplus_O1 <yplus_inj >yplus_SO2
-[ >yminus_succ2 /2 width=3 by ex2_intro/
-| /2 width=1 by ylt_fwd_le_succ1/
-| /2 width=3 by yle_trans/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 break ⊑ × [ term 46 d , break term 46 e ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'ExtLRSubEq $L1 $d $e $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/extlrsubeq_4.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-inductive lsuby: relation4 ynat ynat lenv lenv ≝
-| lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
-| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
- lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-| lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
- lsuby 0 (⫯e) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
-| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
- lsuby d e L1 L2 → lsuby (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
-.
-
-interpretation
- "local environment refinement (extended substitution)"
- 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, ⫰e] L2 → 0 < e →
- L1.ⓑ{I1}V ⊑×[0, e] L2.ⓑ{I2}V.
-#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
-qed.
-
-lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[⫰d, e] L2 → 0 < d →
- L1.ⓑ{I1}V1 ⊑×[d, e] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
-qed.
-
-lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊑×[0, ∞] L2 →
- ∀I1,I2,V. L1.ⓑ{I1}V ⊑×[0,∞] L2.ⓑ{I2}V.
-#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
-qed.
-
-lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
-#Hd destruct /2 width=1 by lsuby_succ/
-#e elim (ynat_cases … e) [| * #x ]
-#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
-qed.
-
-lemma lsuby_O2: ∀L2,L1,d. |L2| ≤ |L1| → L1 ⊑×[d, yinj 0] L2.
-#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #d #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
-| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r … H) -H #HL12
- elim (ynat_cases d) /3 width=1 by lsuby_zero/
- * /3 width=1 by lsuby_succ/
-]
-qed.
-
-lemma lsuby_sym: ∀d,e,L1,L2. L1 ⊑×[d, e] L2 → |L1| = |L2| → L2 ⊑×[d, e] L1.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2
-[ #L1 #d #e #H >(length_inv_zero_dx … H) -L1 //
-| /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_succ/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuby_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #d #e * -L1 -L2 -d -e //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
-]
-qed-.
-
-lemma lsuby_inv_atom1: ∀L2,d,e. ⋆ ⊑×[d, e] L2 → L2 = ⋆.
-/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-
-fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊑×[0, 0] L2 →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=9 by lsuby_inv_zero1_aux/ qed-.
-
-fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
- L2 = ⋆ ∨
- ∃∃J2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
- /3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊑×[0, e] L2 → 0 < e →
- L2 = ⋆ ∨
- ∃∃I2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
-/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-
-fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5 by lsuby_inv_succ1_aux/ qed-.
-
-fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
- ∃∃J1,K1,W1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊑×[0, 0] K2.ⓑ{I2}V2 →
- ∃∃I1,K1,V1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=9 by lsuby_inv_zero2_aux/ qed-.
-
-fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
- ∃∃J1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
- /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
- ∃∃I1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V.
-/2 width=6 by lsuby_inv_pair2_aux/ qed-.
-
-fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
- ∃∃J1,K1,W1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W2 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=5 by lsuby_inv_succ2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuby_fwd_length: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
-qed-.
-
-(* Properties on basic slicing **********************************************)
-
-lemma lsuby_ldrop_trans_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
- d ≤ i → i < d + e →
- ∃∃I1,K1. K1 ⊑×[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #s #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct -I2 >ypred_succ
- /2 width=4 by ldrop_pair, ex2_2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H
- #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
- >yminus_succ <yminus_inj /3 width=4 by ldrop_drop_lt, ex2_2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hdi
- elim (yle_inv_succ1 … Hdi) -Hdi
- #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
- #Hide lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=4 by ylt_O, ldrop_drop_lt, ex2_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lsuby.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-(* Main properties **********************************************************)
-
-theorem lsuby_trans: ∀d,e. Transitive … (lsuby d e).
-#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
-[ #L1 #d #e #X #H lapply (lsuby_inv_atom1 … H) -H
- #H destruct //
-| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
- * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubst $G $L $T1 $d $e $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $G $L $T1 $d $e $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarAlt $G $L $T1 $d $e $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⫃ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEq $L1 $L2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⊢ break term 46 L1 ⫃ break [ term 46 R ] break term 46 L2 )"
+notation "hvbox( L1 break ⊆ [ term 46 d , break term 46 e ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEq $R $G $L1 $L2 }.
+ for @{ 'LRSubEq $L1 $d $e $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⫃ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqC $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⫃ break [ term 46 R ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqC $R $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubst $G $L $T1 $d $e $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $G $L $T1 $d $e $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarAlt $G $L $T1 $d $e $T2 }.
]
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 *********************************************************)
elim (cnr_inv_appl … H) -H /2 width=1/
| #I #L #V #T * #H1 #H2 destruct
[ elim (cnr_inv_zeta … H2)
- | elim (cnr_inv_tau … H2)
+ | elim (cnr_inv_eps … H2)
]
|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
[1,3: elim (cnr_inv_abbr … H2) -H2 /2 width=1/
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)
#l @(nat_ind_plus … l) -l // #l #_ #Hkl
-lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_sort/ -L -l #H destruct -H -e0 (**) (* destruct does not remove some premises *)
+lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_st/ -L -l #H destruct -H -e0 (**) (* destruct does not remove some premises *)
lapply (next_lt h k) >e1 -e1 #H elim (lt_refl_false … H)
qed-.
]
qed-.
-lemma cnx_inv_tau: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓝV.T⦄ → ⊥.
+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
-/2 width=4 by cpx_tau, discr_tpair_xy_y/
+/2 width=4 by cpx_eps, discr_tpair_xy_y/
qed-.
(* Basic forward lemmas *****************************************************)
elim (cnx_inv_appl … H) -H /2 width=1 by/
| #I #L #V #T * #H1 #H2 destruct
[ elim (cnx_inv_zeta … H2)
- | elim (cnx_inv_tau … H2)
+ | elim (cnx_inv_eps … H2)
]
|6,7: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
[1,3: elim (cnx_inv_abbr … H2) -H2 /2 width=1 by/
cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
-| cpr_tau : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
+| cpr_eps : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 *
/3 width=6 by cpr_delta/
|3,7: /4 width=1 by lsubr_bind, cpr_bind, cpr_beta/
-|4,6: /3 width=1 by cpr_flat, cpr_tau/
+|4,6: /3 width=1 by cpr_flat, cpr_eps/
|5,8: /4 width=3 by lsubr_bind, cpr_zeta, cpr_theta/
]
qed-.
pr2_gen_ctail pr2_ctail
*)
(* Basic_1: removed local theorems 4:
- pr0_delta_tau pr0_cong_delta
+ pr0_delta_eps pr0_cong_delta
pr2_free_free pr2_free_delta
*)
elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpr_zeta, ldrop_skip/
| #G #K #V #T1 #T2 #_ #IHT12 #L #s #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpr_tau/
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpr_eps/
| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_div_le … HU2 … HTU) -U /3 width=6 by cpr_zeta, ex2_intro/
| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpr_tau, ex2_intro/
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpr_eps, ex2_intro/
| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
(* 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,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
⇩[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
⇧[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
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 →
⇧[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
+| 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)
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 width=7 by cpx_delta, cpx_ct/
|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/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_bind/
]
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 →
lemma leq_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) (leq 0 (∞)).
#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 (leq_ldrop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
|4,9: /4 width=1 by cpx_bind, cpx_beta, leq_pair_O_Y/
-|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
|6,10: /4 width=3 by cpx_zeta, cpx_theta, leq_pair_O_Y/
]
qed-.
lemma ssta_cpx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] T2 →
⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
#h #g #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2
-[ /3 width=4 by cpx_sort, da_inv_sort/
+[ /3 width=4 by cpx_st, da_inv_sort/
| #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #H
elim (da_inv_lref … H) -H * #K0 #V0 [| #l0 ] #HLK0
lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /2 width=7 by cpx_delta/
| /4 width=2 by cpx_bind, da_inv_bind/
| /4 width=3 by cpx_flat, da_inv_flat/
-| /4 width=3 by cpx_tau, da_inv_flat/
+| /4 width=3 by cpx_eps, da_inv_flat/
]
qed.
>(lift_mono … H1 … H2) -H1 -H2 //
| #G #K #k #l #Hkl #L #s #d #e #_ #U1 #H1 #U2 #H2
>(lift_inv_sort1 … H1) -U1
- >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_sort/
+ >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_st/
| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
elim (lift_inv_lref1 … H) * #Hid #H destruct
[ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpx_zeta, ldrop_skip/
| #G #K #V #T1 #T2 #_ #IHT12 #L #s #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_tau/
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_eps/
| #G #K #V1 #V2 #T #_ #IHV12 #L #s #d #e #HLK #U1 #H #U2 #HVU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ti/
+ elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ct/
| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
]
| #G #L #k #l #Hkl #K #s #d #e #_ #T1 #H
- lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_sort, lift_sort, ex2_intro/
+ lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_st, lift_sort, ex2_intro/
| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
elim (lift_inv_lref2 … H) -H * #Hid #H destruct
[ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_tau, ex2_intro/
+ elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_eps, ex2_intro/
| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ti, ex2_intro/
+ elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ct, ex2_intro/
| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
∀L1. L1 ⋕[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
+#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
[ #H elim (ylt_yle_false … H) //
| * /3 width=7 by cpx_delta/
| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #H elim (lleq_inv_bind_O … H) -H
/3 width=3 by cpx_zeta/
| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by cpx_tau/
+ /3 width=1 by cpx_eps/
| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by cpx_ti/
+ /3 width=1 by cpx_ct/
| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
#HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=1 by cpx_beta/
| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
qed-.
-fact cpr_conf_lpr_flat_tau:
+fact cpr_conf_lpr_flat_eps:
∀G,L0,V0,T0. (
∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_eps, ex2_intro/
qed-.
-fact cpr_conf_lpr_tau_tau:
+fact cpr_conf_lpr_eps_eps:
∀G,L0,V0,T0. (
∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
|4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
] destruct
[ /3 width=10 by cpr_conf_lpr_flat_flat/
- | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
+ | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_eps/
| /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
| /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
- | /3 width=8 by cpr_conf_lpr_flat_tau/
- | /3 width=7 by cpr_conf_lpr_tau_tau/
+ | /3 width=8 by cpr_conf_lpr_flat_eps/
+ | /3 width=7 by cpr_conf_lpr_eps_eps/
| /3 width=12 by cpr_conf_lpr_flat_beta/
| /3 width=13 by cpr_conf_lpr_beta_beta/
| /3 width=14 by cpr_conf_lpr_flat_theta/
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_max.ma".
+include "basic_2/notation/relations/psubst_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/relocation/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
+| cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I})
+| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e →
+ ⇩[i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W
+| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
+ cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
+ cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
+ cpy d e G L V1 V2 → cpy d e G L T1 T2 →
+ cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation "context-sensitive extended ordinary substritution (term)"
+ 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpy_trans: ∀G,d,e. lsub_trans … (cpy d e G) (lsuby d e).
+#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_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.
+#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.
+#I #G #K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK
+ /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i d) #Hid
+ /3 width=4 by 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)
+ /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L.ⓑ{J}W1) (d+1)) -IHU1
+ /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK
+ /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+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 #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/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+(* Note: lemma 1250 *)
+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 #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)
+ /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
+| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
+ /2 width=1 by cpy_bind/
+| /2 width=1 by cpy_flat/
+]
+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 #T2 #d #e #HT12
+lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
+/2 width=2 by cpy_weak_top/
+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.
+#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
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+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.
+#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
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+(* 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} →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
+ ⇩[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-.
+
+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 &
+ ⇧[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: ∀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 →
+ T2 = #i ∨
+ ∃∃I,K,V. d ≤ i & i < d + e &
+ ⇩[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: ∀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 →
+ ∀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 &
+ 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-.
+
+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 →
+ ∀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-.
+
+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.
+#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
+ elim (ylt_yle_false … Hdi) -Hdi //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
+]
+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_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_fwd_up … HU12 … HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+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
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶[d1, e1] T1 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶[d1, e1] T.
+#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (cpy_inv_lref1 … H) -H
+ [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
+ ]
+| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
+ /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶[d1, e1] T1 →
+ ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶[d1, e1] T.
+#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+ elim (cpy_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
+ [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
+ ]
+ ]
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0
+ [ -H #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
+ ]
+| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Note: the constant 1 comes from cpy_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T0 →
+ ∀T2. ⦃G, L⦄ ⊢ T0 ▶[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶[d, e] T2.
+#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
+[ #I #G #L #d #e #T2 #H #He
+ elim (cpy_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+ ]
+| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
+| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ lapply (IHT10 … HT02 He) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
+]
+qed-.
+
+theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
+#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
+ lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ elim (IHV10 … HV02) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02) -V0 //
+ elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/relocation/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
+ ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt + et ≤ d → ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
+ lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj … Hid) -Hid #Hid
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
+ ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt ≤ d → d ≤ dt + et → ⦃G, L⦄ ⊢ U1 ▶[dt, et + e] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
+ | -Hdti
+ elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
+ /4 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
+ ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
+ ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶[dt+e, et] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
+ lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
+ elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
+ lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
+ /3 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=6 by cpy_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
+ lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj … Hid) -Hid #Hid
+ lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+ elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -IHU12 -HTU1
+ /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et-e] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
+ [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+ /3 width=5 by cpy_subst, ex2_intro/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+ >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1
+ /3 width=6 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK //
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ yinj d + e ≤ dt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt-e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
+ lapply (yle_trans … Hdedt … Hdti) #Hdei
+ elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
+ elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+ elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
+ >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Advancd inversion lemmas on relocation ***********************************)
+
+lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[d, dt + et - (yinj d + e)] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ yinj d + e →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d-dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
+lapply (cpy_weak … HU2 d e ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_plus.ma".
+include "basic_2/notation/relations/lrsubeq_4.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
+| lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
+| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
+ lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
+ lsuby 0 (⫯e) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
+ lsuby d e L1 L2 → lsuby (⫯d) e (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+.
+
+interpretation
+ "local environment refinement (extended substitution)"
+ 'LRSubEq L1 d e L2 = (lsuby d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊆[0, ⫰e] L2 → 0 < e →
+ L1.ⓑ{I1}V ⊆[0, e] L2.ⓑ{I2}V.
+#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊆[⫰d, e] L2 → 0 < d →
+ L1.ⓑ{I1}V1 ⊆[d, e] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
+qed.
+
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+ ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: ∀L,d,e. L ⊆[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
+#Hd destruct /2 width=1 by lsuby_succ/
+#e elim (ynat_cases … e) [| * #x ]
+#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
+qed.
+
+lemma lsuby_O2: ∀L2,L1,d. |L2| ≤ |L1| → L1 ⊆[d, yinj 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
+[ #d #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
+| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r … H) -H #HL12
+ elim (ynat_cases d) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
+]
+qed.
+
+lemma lsuby_sym: ∀d,e,L1,L2. L1 ⊆[d, e] L2 → |L1| = |L2| → L2 ⊆[d, e] L1.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2
+[ #L1 #d #e #H >(length_inv_zero_dx … H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_succ/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #d #e * -L1 -L2 -d -e //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
+]
+qed-.
+
+lemma lsuby_inv_atom1: ∀L2,d,e. ⋆ ⊆[d, e] L2 → L2 = ⋆.
+/2 width=5 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by lsuby_inv_zero1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
+ L2 = ⋆ ∨
+ ∃∃J2,K2. K1 ⊆[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
+ /3 width=4 by ex2_2_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊆[0, e] L2 → 0 < e →
+ L2 = ⋆ ∨
+ ∃∃I2,K2. K1 ⊆[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
+/2 width=6 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊆[d, e] L2 → 0 < d →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by lsuby_inv_succ1_aux/ qed-.
+
+fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
+ ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
+ ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=9 by lsuby_inv_zero2_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
+ ∃∃J1,K1. K1 ⊆[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
+ /2 width=4 by ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊆[0, e] K2.ⓑ{I2}V → 0 < e →
+ ∃∃I1,K1. K1 ⊆[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V.
+/2 width=6 by lsuby_inv_pair2_aux/ qed-.
+
+fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
+ ∃∃J1,K1,W1. K1 ⊆[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W2 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
+ /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊆[d, e] K2.ⓑ{I2}V2 → 0 < d →
+ ∃∃I1,K1,V1. K1 ⊆[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5 by lsuby_inv_succ2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: ∀L1,L2,d,e. L1 ⊆[d, e] L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
+qed-.
+
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_ldrop_trans_be: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+ d ≤ i → i < d + e →
+ ∃∃I1,K1. K1 ⊆[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct -I2 >ypred_succ
+ /2 width=4 by ldrop_pair, ex2_2_intro/
+ | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+ #H <H -H #H lapply (ylt_inv_succ … H) -H
+ #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
+ >yminus_succ <yminus_inj /3 width=4 by ldrop_drop_lt, ex2_2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hdi
+ elim (yle_inv_succ1 … Hdi) -Hdi
+ #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
+ #Hide lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+ #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+ /4 width=4 by ylt_O, ldrop_drop_lt, ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: ∀d,e. Transitive … (lsuby d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
+[ #L1 #d #e #X #H lapply (lsuby_inv_atom1 … H) -H
+ #H destruct //
+| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
+| #I1 #I2 #L1 #L2 #V #e #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
+ * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
+]
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lrsubeq_2.ma".
+include "basic_2/notation/relations/lrsubeqc_2.ma".
include "basic_2/relocation/ldrop.ma".
(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
interpretation
"local environment refinement (restricted)"
- 'LRSubEq L1 L2 = (lsubr L1 L2).
+ 'LRSubEqC L1 L2 = (lsubr L1 L2).
(* Basic properties *********************************************************)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubststar_6.ma".
+include "basic_2/relocation/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+definition cpys: ynat → ynat → relation4 genv lenv term term ≝
+ λd,e,G. LTC … (cpy d e G).
+
+interpretation "context-sensitive extended multiple substritution (term)"
+ 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpys_ind: ∀G,L,T1,d,e. ∀R:predicate term. R T1 →
+ (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → R T → R T2) →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T2.
+#G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cpys_ind_dx: ∀G,L,T2,d,e. ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → R T → R T1) →
+ ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T1.
+#G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cpy_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+/2 width=1 by inj/ qed.
+
+lemma cpys_strap1: ∀G,L,T1,T,T2,d,e.
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+normalize /2 width=3 by step/ qed-.
+
+lemma cpys_strap2: ∀G,L,T1,T,T2,d,e.
+ ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+lemma lsuby_cpys_trans: ∀G,d,e. lsub_trans … (cpys d e G) (lsuby d e).
+/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
+qed-.
+
+lemma cpys_refl: ∀G,L,d,e. reflexive … (cpys d e G L).
+/2 width=1 by cpy_cpys/ qed.
+
+lemma cpys_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] ⓑ{a,I}V2.T2.
+#G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
+[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
+| /3 width=5 by cpys_strap1, cpy_bind/
+]
+qed.
+
+lemma cpys_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] ⓕ{I}V2.T2.
+#G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
+[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
+| /3 width=5 by cpys_strap1, cpy_flat/
+qed.
+
+lemma cpys_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 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind … H) -T2
+/3 width=7 by cpys_strap1, cpy_weak/
+qed-.
+
+lemma cpys_weak_top: ∀G,L,T1,T2,d,e.
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, |L| - d] T2.
+#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
+/3 width=4 by cpys_strap1, cpy_weak_top/
+qed-.
+
+lemma cpys_weak_full: ∀G,L,T1,T2,d,e.
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
+#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
+/3 width=5 by cpys_strap1, cpy_weak_full/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_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 #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
+ elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
+/2 width=3 by transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] T2 → T2 = ⋆k.
+#G #L #T2 #k #d #e #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_sort1 … HT2) -HT2 //
+qed-.
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶*[d, e] T2 → T2 = §p.
+#G #L #T2 #p #d #e #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_gref1 … HT2) -HT2 //
+qed-.
+
+lemma cpys_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.
+#a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
+ /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_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.
+#I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_flat1 … HU2) -HU2
+ /3 width=5 by cpys_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 0] T2 → T1 = T2.
+#G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
+ ⦃G, L⦄ ⊢ U1 ▶*[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2
+/2 width=7 by cpy_inv_lift1_eq/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubststaralt_6.ma".
+include "basic_2/substitution/cpys_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* alternative definition of cpys *)
+inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
+| cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
+| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(d+e-i)) G K V1 V2 →
+ ⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
+| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
+ cpysa d e G L V1 V2 → cpysa (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
+ cpysa d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
+ cpysa d e G L V1 V2 → cpysa d e G L T1 T2 →
+ cpysa d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "context-sensitive extended multiple substritution (term) alternative"
+ 'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpysa_trans: ∀G,d,e. lsub_trans … (cpysa d e G) (lsuby d e).
+#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
+[ //
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_ldrop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
+| /4 width=1 by lsuby_succ, cpysa_bind/
+| /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpysa_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶▶*[d, e] T.
+#G #T elim T -T //
+#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
+qed.
+
+lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T →
+ ∀T2. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
+#G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
+[ #I #G #L #d #e #X #H
+ elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
+| #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
+ lapply (ldrop_fwd_drop2 … HLK) #H0LK
+ lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
+ elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
+ /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
+| #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
+/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+ (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
+ (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 →
+ ⇧[O, i+1] V2 ≡ W2 → R O (⫰(d+e-i)) G K V1 V2 → R d e G L (#i) W2
+ ) →
+ (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 → R d e G L V1 V2 →
+ R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L V1 V2 →
+ R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
+#R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
+/3 width=8 by cpysa_inv_cpys/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_cpy.ma".
+include "basic_2/substitution/cpys_alt.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpys_inv_SO2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[d, 1] T2.
+#G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶*[d1, e1] T1 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶*[d1, e1] T1 →
+ ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T0 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶*[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
+ ∀i. d ≤ 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 #i #Hdi #Hide @(cpys_ind … H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (cpy_split_up … HT12 … Hide) -HT12 -Hide #T0 #HT0 #HT02
+ elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //
+]
+qed-.
+
+lemma cpys_inv_lift1_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpys_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpys_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶*[d1, e1] T1 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
+
+theorem cpys_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶*[d1, e1] T1 →
+ ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶*[d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
+
+theorem cpys_trans_eq: ∀G,L,T1,T,T2,d,e.
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+normalize /2 width=3 by trans_TC/ qed-.
+
+theorem cpys_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T0 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶*[d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T & ⦃G, L⦄ ⊢ T ▶*[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
+
+theorem cpys_antisym_eq: ∀G,L1,T1,T2,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T2 →
+ ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] T1 → T1 = T2.
+#G #L1 #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #_ #_ #HVW2 #_ #L2 #HW2
+ elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hdi -Hide | ]
+ [ lapply (cpys_weak_full … HW2) -HW2 #HW2
+ lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
+ [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
+ #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
+ | elim (ldrop_O1_le … Hi) -Hi #K2 #HLK2
+ elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
+ /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hdi -Hide
+ #X #_ #H elim (lift_inv_lref2_be … H) -H //
+ ]
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
+ #V #T #HV2 #HT2 #H destruct
+ lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
+| #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
+ #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/cpy_lift.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
+ d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(d+e-i)] U1 →
+ ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] U2.
+#I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
+[ /3 width=5 by cpy_cpys, cpy_subst/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU … HU0) -IHU #H
+ lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
+ lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+ lapply (cpy_weak … HU02 d e ? ?) -HU02
+ [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+ >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+]
+qed.
+
+lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,d.
+ d ≤ yinj i →
+ ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
+ ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, ∞] U2.
+#I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12
+@(cpys_subst … HLK … HU12) >yminus_Y_inj //
+qed.
+
+(* Advanced inverion lemmas *************************************************)
+
+lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*[d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
+ ⇩[i] L ≡ K.ⓑ{J}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
+ ⇧[O, i+1] V2 ≡ T2 &
+ I = LRef i.
+#I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
+[ /2 width=1 by or_introl/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
+ | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
+ lapply (ldrop_fwd_drop2 … HLK) #H
+ elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+ [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+ /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V1,V2. d ≤ i & i < d + e &
+ ⇩[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
+qed-.
+
+lemma cpys_inv_lref1_ldrop: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
+ ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⇧[O, i+1] V2 ≡ T2 →
+ ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2
+ & d ≤ i
+ & i < d + e.
+#G #L #T2 #i #d #e #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
+[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
+| * #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
+ lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
+ lapply (ldrop_mono … HLY … HLK) -L #H destruct
+ /2 width=1 by and3_intro/
+]
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+ ∀L,U1,s,d,e. dt + et ≤ yinj d → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+ ∀L,U1,s,d,e. dt ≤ yinj d → d ≤ dt + et →
+ ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[dt, et + e] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+ ∀L,U1,s,d,e. yinj d ≤ dt → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ▶*[dt+e, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+(* Inversion lemmas for relocation ******************************************)
+
+lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ yinj d + e ≤ dt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ yinj d + e →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] W2 →
+ ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
+ d ≤ yinj i → i < d + e →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.
[ "lleq ( ? ⋕[?,?] ? )" "lleq_alt" + "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
}
]
- [ { "iterated structural successor for closures" * } {
+ [ { "contxt-sensitive extended multiple substitution" * } {
+ [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
+ }
+ ]
+ [ { "iterated structural successor for closures" * } {
[ "fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_fqus" * ]
[ "fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_fqup" * ]
}
[ "lpx_sn" "lpx_sn_alt" + "lpx_sn_tc" + "lpx_sn_ldrop" + "lpx_sn_lpx_sn" * ]
}
]
- [ { "basic local env. slicing" * } {
+ [ { "contxt-sensitive extended ordinary substitution" * } {
+ [ "cpy ( ⦃?,?⦄ ⊢ ? ▶[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
+ }
+ ]
+ [ { "local env. ref. for extended substitution" * } {
+ [ "lsuby ( ? ⊑×[?,?] ? )" "lsuby_lsuby" * ]
+ }
+ ]
+ [ { "basic local env. slicing" * } {
[ "ldrop ( ⇩[?,?,?] ? ≡ ? )" "ldrop_leq" + "ldrop_ldrop" * ]
}
]
projects / helm.git / commitdiff