+++ /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/computation/ltprs.ma".
-include "basic_2/dynamic/lsubsv.ma".
-
-(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
-
-axiom yprs: ∀h. sd h → bi_relation lenv term.
-
-interpretation "'big tree' parallel computation (closure)"
- 'BTPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2).
-
-axiom cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
-
-axiom sstas_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
- h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
-
-axiom lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
-
-axiom ltpr_cprs_yprs: ∀h,g,L1,L2,T1,T2. L1 ➡ L2 → L2 ⊢ T1 ➡* T2 →
- h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
-
-axiom ygt: ∀h. sd h → bi_relation lenv term.
-
-interpretation "'big tree' proper parallel computation (closure)"
- 'BTPRedStarProper h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
-
-axiom fw_ygt: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
-
-axiom ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
s: substitution
u: supclosure
x: extended reduction
+y: extended substitution
- forth letter (if present)
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* 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 @{ 'ExtPSubst $G $L $T1 $d $e $T2 }.
include "basic_2/grammar/genv.ma".
include "basic_2/grammar/cl_shift.ma".
include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/substitution/lsubr.ma".
+include "basic_2/relocation/lsubr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
--- /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/extpsubst_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/relocation/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: nat → nat → 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 ≤ i → i < d + e →
+ ⇩[0, 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 + 1) e G (L.ⓑ{I} V2) 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 parallel substritution (term)"
+ 'ExtPSubst 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.
+#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
+[ //
+| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+ elim (lsuby_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_pair, cpy_bind/
+| /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.
+
+lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[0, 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)
+ /3 width=5 by cpy_subst, le_n, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L.ⓑ{J}W2) (d+1)) -IHU1
+ /3 width=9 by cpy_bind, ldrop_ldrop, 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, transitive_le/
+| /4 width=3 by cpy_bind, le_to_lt_to_lt, le_S_S/
+| /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)
+ /3 width=5 by cpy_subst, lt_to_le_to_lt/
+| normalize /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. 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 elim H -G -L -T1 -T2 -d -e
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j) [ -Hide -Hjde | -Hdi -Hdj ]
+ [ >(plus_minus_m_m j d) in ⊢ (%→?);
+ /3 width=5 by cpy_subst, ex2_intro/
+ | #Hij lapply (plus_minus_m_m … Hjde) -Hjde
+ /3 width=9 by cpy_atom, cpy_subst, ex2_intro/
+ ]
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I} V) ?) -HT1 /3 width=5 by cpy_bind, ex2_intro/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma cpy_split_down: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d + e - i] T &
+ ⦃G, L⦄ ⊢ T ▶×[d, i - d] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
+| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j)
+ [ -Hide -Hjde >(plus_minus_m_m j d) in⦄ ⊢ (% → ?); // -Hdj /3 width=8/
+ | -Hdi -Hdj
+ >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (cpy_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma cpy_append: ∀K,T1,T2,d,e. K⦄ ⊢ T1 ▶×[d, e] T2 →
+ ∀L. L @@ K⦄ ⊢ T1 ▶×[d, e] T2.
+#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
+#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_ldrop2_length … HK0) #H
+@(cpy_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpy_inv_atom1_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
+[ #L #I #d #e #J #H destruct /2 width=1/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+]
+qed.
+
+lemma cpy_inv_atom1: ∀L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+/2 width=3/ qed-.
+
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma cpy_inv_sort1: ∀L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H
+elim (cpy_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma cpy_inv_lref1: ∀L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
+ T2 = #i ∨
+ ∃∃K,V. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2.
+#L #T2 #i #d #e #H
+elim (cpy_inv_atom1 … H) -H /2 width=1/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+qed-.
+
+lemma cpy_inv_gref1: ∀L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
+#L #T2 #p #d #e #H
+elim (cpy_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact cpy_inv_bind1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+ L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #a #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
+| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
+]
+qed.
+
+lemma cpy_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+ L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+/2 width=3/ qed-.
+
+fact cpy_inv_flat1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma cpy_inv_flat1: ∀d,e,L,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+/2 width=3/ qed-.
+
+fact cpy_inv_refl_O2_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
+ elim (lt_refl_false … Hdd)
+| /3 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma cpy_inv_refl_O2: ∀L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
+/2 width=6/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_tw: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
+/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+qed-.
+
+lemma cpy_fwd_shift1: ∀L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #d #e #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X #d #e
+ >shift_append_assoc normalize #H
+ elim (cpy_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+]
+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/substitution/tps_lift.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
+#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (tps_inv_lref1 … H) -H
+ [ #HX destruct /3 width=6/
+ | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
+ ]
+| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
+| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
+#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+ elim (tps_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=6/
+ | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
+ [ -Hd1 -Hde2
+ lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
+ lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
+ elim (lt_refl_false … H)
+ | -Hd2 -Hde1
+ lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
+ lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
+ elim (lt_refl_false … H)
+ ]
+ ]
+| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02 ?) -T0
+ [ -H #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
+ | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
+ [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
+ ]
+| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5/
+]
+qed.
+
+(* Note: the constant 1 comes from tps_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
+ ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
+ L ⊢ T1 ▶ [d, e] T2.
+#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
+[ #L #I #d #e #T2 #H #He
+ elim (tps_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
+ ]
+| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
+ <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
+| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ lapply (IHT10 … HT02 He) -T0 #HT12
+ lapply (tps_lsubr_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
+| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
+]
+qed.
+
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
+[ /2 width=3/
+| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
+ lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=8/
+| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02 ?) -V0 //
+ elim (IHT10 … HT02 ?) -T0 // /3 width=6/
+]
+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/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
+#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
+[ //
+| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
+ elim (lt_or_ge i (d+1)) #HiSd
+ [ -Hide1 -HV0
+ lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct
+ | -V -Hdi /2 width=4/
+ ]
+| /4 width=3/
+| /3 width=3/
+]
+qed.
+
+lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
+/2 width=3/ qed-.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #HT12 #K #V #HLK
+lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
+lapply (tps_inv_refl_O2 … HT12) -HT12 //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt + et ≤ d →
+ L ⊢ U1 ▶ [dt, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #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/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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
+ @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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=6/
+]
+qed.
+
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt ≤ d → d ≤ dt + et →
+ L ⊢ U1 ▶ [dt, et + e] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (lt_to_le_to_lt … (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/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+ | -Hdti
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+ ]
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
+ ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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=6/
+]
+qed.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ d ≤ dt →
+ L ⊢ U1 ▶ [dt + e, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
+ lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
+ lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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
+ @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #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=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #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=4/
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -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=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
+ lapply (transitive_le … Hdedt … Hdti) #Hdei
+ elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+ elim (le_inv_plus_l … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
+ <plus_minus // /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
+ elim (lift_inv_lref2 … H) -H * #H
+ [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
+ elim (lt_refl_false … H)
+ | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
+ elim (lt_refl_false … H)
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+]
+qed.
+(*
+ Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le (plus d h) i) ->
+ (EX u1 | (subst0 (minus i h) v u1 u2) &
+ t1 = (lift h d u1)
+ ).
+
+
+ Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le d i) -> (lt i (plus d h)) ->
+ (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
+*)
+lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
+qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
+qed.
+
+lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
+elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
+lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
+lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
+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/lrsubeq_2.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+inductive lsubr: relation lenv ≝
+| lsubr_sort: ∀L. lsubr L (⋆)
+| lsubr_bind: ∀I,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubr_abst: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (restricted)"
+ 'LRSubEq L1 L2 = (lsubr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_refl: ∀L. L ⊑ L.
+#L elim L -L // /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
+/2 width=3 by lsubr_inv_atom1_aux/ qed-.
+
+fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓛW →
+ L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
+#L1 #L2 * -L1 -L2
+[ #L #K1 #W #H destruct /2 width=1/
+| #I #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #_ #K1 #W #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⊑ L2 →
+ L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
+/2 width=3 by lsubr_inv_abst1_aux/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+#L1 #L2 * -L1 -L2
+[ #L #K2 #W #H destruct
+| #I #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
+| #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
+ ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
+/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubr_fwd_ldrop2_bind: ∀L1,L2. L1 ⊑ L2 →
+ ∀I,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I}W →
+ (∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I}W) ∨
+ ∃∃K1,V. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #I #K2 #W #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /3 width=3/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
+ ]
+| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #i #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+ [ /3 width=4/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
+ ]
+]
+qed-.
+
+lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
+ ∀K2,V,i. ⇩[0, i] L2 ≡ K2.ⓓV →
+ ∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓV.
+#L1 #L2 #HL12 #K2 #V #i #HLK2 elim (lsubr_fwd_ldrop2_bind … HL12 … HLK2) -L2 // *
+#K1 #W #_ #_ #H destruct
+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/lsubr.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+(* Auxiliary inversion lemmas ***********************************************)
+
+fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⊑ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓑ{I}X
+ | ∃∃K2,V,W. K1 ⊑ K2 & L2 = K2.ⓛW &
+ I = Abbr & X = ⓝW.V.
+#L1 #L2 * -L1 -L2
+[ #L #J #K1 #X #H destruct /2 width=1/
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
+| #L1 #L2 #V #W #HL12 #J #K1 #X #H destruct /3 width=6/
+]
+qed-.
+
+lemma lsubr_inv_bind1: ∀I,K1,L2,X. K1.ⓑ{I}X ⊑ L2 →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓑ{I}X
+ | ∃∃K2,V,W. K1 ⊑ K2 & L2 = K2.ⓛW &
+ I = Abbr & X = ⓝW.V.
+/2 width=3 by lsubr_inv_bind1_aux/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H
+ lapply (lsubr_inv_atom1 … H) -H //
+| #I #L1 #L #V #_ #IHL1 #X #H
+ elim (lsubr_inv_bind1 … H) -H // *
+ #L2 [2: #V2 #W2 ] #HL2 #H1 [ #H2 #H3 ] destruct /3 width=1/
+| #L1 #L #V1 #W #_ #IHL1 #X #H
+ elim (lsubr_inv_abst1 … H) -H // *
+ #L2 #HL2 #H destruct /3 width=1/
+]
+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/grammar/lenv_length.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+inductive lsubr: nat → nat → relation lenv ≝
+| lsubr_sort: ∀d,e. lsubr d e (⋆) (⋆)
+| lsubr_OO: ∀L1,L2. lsubr 0 0 L1 L2
+| lsubr_abbr: ∀L1,L2,V,e. lsubr 0 e L1 L2 →
+ lsubr 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
+| lsubr_abst: ∀L1,L2,I,V1,V2,e. lsubr 0 e L1 L2 →
+ lsubr 0 (e + 1) (L1. ⓑ{I}V1) (L2. ⓛV2)
+| lsubr_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
+ lsubr d e L1 L2 → lsubr (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
+.
+
+interpretation
+ "local environment refinement (substitution)"
+ 'SubEq L1 d e L2 = (lsubr d e L1 L2).
+
+definition lsubr_trans: ∀S. (lenv → relation S) → Prop ≝ λS,R.
+ ∀L2,s1,s2. R L2 s1 s2 →
+ ∀L1,d,e. L1 ⊑ [d, e] L2 → R L1 s1 s2.
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_bind_eq: ∀L1,L2,e. L1 ⊑ [0, e] L2 → ∀I,V.
+ L1. ⓑ{I} V ⊑ [0, e + 1] L2.ⓑ{I} V.
+#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
+qed.
+
+lemma lsubr_abbr_lt: ∀L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓓV ⊑ [0, e] L2.ⓓV.
+#L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma lsubr_abst_lt: ∀L1,L2,I,V1,V2,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓑ{I}V1 ⊑ [0, e] L2. ⓛV2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma lsubr_skip_lt: ∀L1,L2,d,e. L1 ⊑ [d - 1, e] L2 → 0 < d →
+ ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 ⊑ [d, e] L2. ⓑ{I2} V2.
+#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
+qed.
+
+lemma lsubr_bind_lt: ∀I,L1,L2,V,e. L1 ⊑ [0, e - 1] L2 → 0 < e →
+ L1. ⓓV ⊑ [0, e] L2. ⓑ{I}V.
+* /2 width=1/ qed.
+
+lemma lsubr_refl: ∀d,e,L. L ⊑ [d, e] L.
+#d elim d -d
+[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
+| #d #IHd #e #L elim L -L // /2 width=1/
+]
+qed.
+
+lemma TC_lsubr_trans: ∀S,R. lsubr_trans S R → lsubr_trans S (λL. (TC … (R L))).
+#S #R #HR #L1 #s1 #s2 #H elim H -s2
+[ /3 width=5/
+| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
+ lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L1 = ⋆ →
+ L2 = ⋆ ∨ (d = 0 ∧ e = 0).
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ /2 width=1/
+| /3 width=1/
+| #L1 #L2 #W #e #_ #H destruct
+| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+]
+qed.
+
+lemma lsubr_inv_atom1: ∀L2,d,e. ⋆ ⊑ [d, e] L2 →
+ L2 = ⋆ ∨ (d = 0 ∧ e = 0).
+/2 width=3/ qed-.
+
+fact lsubr_inv_skip1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀I1,K1,V1. L1 = K1.ⓑ{I1}V1 → 0 < d →
+ ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #I1 #K1 #V1 #H destruct
+| #L1 #L2 #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
+]
+qed.
+
+lemma lsubr_inv_skip1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑ [d, e] L2 → 0 < d →
+ ∃∃I2,K2,V2. K1 ⊑ [d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5/ qed-.
+
+fact lsubr_inv_atom2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 → L2 = ⋆ →
+ L1 = ⋆ ∨ (d = 0 ∧ e = 0).
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ /2 width=1/
+| /3 width=1/
+| #L1 #L2 #W #e #_ #H destruct
+| #L1 #L2 #I #W1 #W2 #e #_ #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #H destruct
+]
+qed.
+
+lemma lsubr_inv_atom2: ∀L1,d,e. L1 ⊑ [d, e] ⋆ →
+ L1 = ⋆ ∨ (d = 0 ∧ e = 0).
+/2 width=3/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀K2,V. L2 = K2.ⓓV → d = 0 → 0 < e →
+ ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #K1 #V #H destruct
+| #L1 #L2 #K1 #V #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #HL12 #K1 #V #H #_ #_ destruct /2 width=3/
+| #L1 #L2 #I #W1 #W2 #e #_ #K1 #V #H destruct
+| #L1 #L2 #I1 #I2 #W1 #W2 #d #e #_ #K1 #V #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,V,e. L1 ⊑ [0, e] K2.ⓓV → 0 < e →
+ ∃∃K1. K1 ⊑ [0, e - 1] K2 & L1 = K1.ⓓV.
+/2 width=5/ qed-.
+
+fact lsubr_inv_skip2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ ∀I2,K2,V2. L2 = K2.ⓑ{I2}V2 → 0 < d →
+ ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #d #e #I1 #K1 #V1 #H destruct
+| #L1 #L2 #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #W #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #W1 #W2 #e #_ #I1 #K1 #V1 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #J1 #J2 #W1 #W2 #d #e #HL12 #I1 #K1 #V1 #H #_ destruct /2 width=5/
+]
+qed.
+
+lemma lsubr_inv_skip2: ∀I2,L1,K2,V2,d,e. L1 ⊑ [d, e] K2.ⓑ{I2}V2 → 0 < d →
+ ∃∃I1,K1,V1. K1 ⊑ [d - 1, e] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+fact lsubr_fwd_length_full1_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ d = 0 → e = |L1| → |L1| ≤ |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_fwd_length_full1: ∀L1,L2. L1 ⊑ [0, |L1|] L2 → |L1| ≤ |L2|.
+/2 width=5/ qed-.
+
+fact lsubr_fwd_length_full2_aux: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
+ d = 0 → e = |L2| → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubr_fwd_length_full2: ∀L1,L2. L1 ⊑ [0, |L2|] L2 → |L2| ≤ |L1|.
+/2 width=5/ qed-.
(**************************************************************************)
include "basic_2/notation/relations/lrsubeqa_3.ma".
-include "basic_2/substitution/lsubr.ma".
+include "basic_2/relocation/lsubr.ma".
include "basic_2/static/aaa.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
(**************************************************************************)
include "basic_2/notation/relations/lrsubeqd_5.ma".
-include "basic_2/substitution/lsubr.ma".
+include "basic_2/relocation/lsubr.ma".
include "basic_2/static/da.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+++ /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/lrsubeq_2.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
-
-inductive lsubr: relation lenv ≝
-| lsubr_sort: ∀L. lsubr L (⋆)
-| lsubr_bind: ∀I,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubr_abst: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (restricted)"
- 'LRSubEq L1 L2 = (lsubr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsubr_refl: ∀L. L ⊑ L.
-#L elim L -L // /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom1: ∀L2. ⋆ ⊑ L2 → L2 = ⋆.
-/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-
-fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⊑ L2 → ∀K1,W. L1 = K1.ⓛW →
- L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
-#L1 #L2 * -L1 -L2
-[ #L #K1 #W #H destruct /2 width=1/
-| #I #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #V2 #_ #K1 #W #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⊑ L2 →
- L2 = ⋆ ∨ ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓛW.
-/2 width=3 by lsubr_inv_abst1_aux/ qed-.
-
-fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⊑ L2 → ∀K2,W. L2 = K2.ⓓW →
- ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
-#L1 #L2 * -L1 -L2
-[ #L #K2 #W #H destruct
-| #I #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3/
-| #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⊑ K2.ⓓW →
- ∃∃K1. K1 ⊑ K2 & L1 = K1.ⓓW.
-/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubr_fwd_length: ∀L1,L2. L1 ⊑ L2 → |L2| ≤ |L1|.
-#L1 #L2 #H elim H -L1 -L2 // /2 width=1/
-qed-.
-
-lemma lsubr_fwd_ldrop2_bind: ∀L1,L2. L1 ⊑ L2 →
- ∀I,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I}W →
- (∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I}W) ∨
- ∃∃K1,V. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
-#L1 #L2 #H elim H -L1 -L2
-[ #L #I #K2 #W #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=3/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
- ]
-| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #i #H
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=4/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 * /4 width=3/ /4 width=4/
- ]
-]
-qed-.
-
-lemma lsubr_fwd_ldrop2_abbr: ∀L1,L2. L1 ⊑ L2 →
- ∀K2,V,i. ⇩[0, i] L2 ≡ K2.ⓓV →
- ∃∃K1. K1 ⊑ K2 & ⇩[0, i] L1 ≡ K1.ⓓV.
-#L1 #L2 #HL12 #K2 #V #i #HLK2 elim (lsubr_fwd_ldrop2_bind … HL12 … HLK2) -L2 // *
-#K1 #W #_ #_ #H destruct
-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/substitution/lsubr.ma".
-
-(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
-
-(* Auxiliary inversion lemmas ***********************************************)
-
-fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⊑ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- ∨∨ L2 = ⋆
- | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓑ{I}X
- | ∃∃K2,V,W. K1 ⊑ K2 & L2 = K2.ⓛW &
- I = Abbr & X = ⓝW.V.
-#L1 #L2 * -L1 -L2
-[ #L #J #K1 #X #H destruct /2 width=1/
-| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
-| #L1 #L2 #V #W #HL12 #J #K1 #X #H destruct /3 width=6/
-]
-qed-.
-
-lemma lsubr_inv_bind1: ∀I,K1,L2,X. K1.ⓑ{I}X ⊑ L2 →
- ∨∨ L2 = ⋆
- | ∃∃K2. K1 ⊑ K2 & L2 = K2.ⓑ{I}X
- | ∃∃K2,V,W. K1 ⊑ K2 & L2 = K2.ⓛW &
- I = Abbr & X = ⓝW.V.
-/2 width=3 by lsubr_inv_bind1_aux/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem lsubr_trans: Transitive … lsubr.
-#L1 #L #H elim H -L1 -L
-[ #L1 #X #H
- lapply (lsubr_inv_atom1 … H) -H //
-| #I #L1 #L #V #_ #IHL1 #X #H
- elim (lsubr_inv_bind1 … H) -H // *
- #L2 [2: #V2 #W2 ] #HL2 #H1 [ #H2 #H3 ] destruct /3 width=1/
-| #L1 #L #V1 #W #_ #IHL1 #X #H
- elim (lsubr_inv_abst1 … H) -H // *
- #L2 #HL2 #H destruct /3 width=1/
-]
-qed-.
]
class "yellow"
[ { "substitution" * } {
- [ { "restricted local env. ref." * } {
- [ "lsubr ( ? ⊑ ? )" "lsubr_lsubr" * ]
- }
- ]
[ { "iterated structural successor for closures" * } {
[ "fqus ( ⦃?,?,?⦄ ⊃* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_lleq" + "fqus_fqus" * ]
[ "fqup ( ⦃?,?,?⦄ ⊃+ ⦃?,?,?⦄ )" "fqup_lleq" + "fqup_fqup" * ]
[ "lleq ( ? ⋕[?,?] ? )" "lleq_lleq" * ]
}
]
+ [ { "contxt-sensitive extended substitution" * } {
+ [ "cpy ( ⦃?,?⦄ ⊢ ? ×▶[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
+ }
+ ]
+ [ { "local env. ref. for extended substitution" * } {
+ [ "lsuby ( ? ×⊑ ? )" "lsuby_lsuby" * ]
+ }
+ ]
+ [ { "restricted local env. ref." * } {
+ [ "lsubr ( ? ⊑ ? )" "lsubr_lsubr" * ]
+ }
+ ]
[ { "global env. slicing" * } {
[ "gdrop ( ⇩[?] ? ≡ ? )" "gdrop_gdrop" * ]
}
projects / helm.git / commitdiff