(**************************************************************************)
include "basic_2/unfold/lsstas_lsstas.ma".
-include "basic_2/computation/fpbs_lift.ma".
include "basic_2/computation/fpbg_fpns.ma".
include "basic_2/equivalence/cpes_cpds.ma".
include "basic_2/dynamic/snv.ma".
(* *)
(**************************************************************************)
-include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/dynamic/snv_lift.ma".
include "basic_2/dynamic/snv_cpcs.ma".
include "basic_2/dynamic/lsubsv_snv.ma".
(**************************************************************************)
include "basic_2/computation/cpds_cpds.ma".
-include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/dynamic/snv_aaa.ma".
include "basic_2/dynamic/snv_cpcs.ma".
include "basic_2/dynamic/lsubsv_lsstas.ma".
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⋕ break [ term 46 d , break term 46 T ] break term 46 L2 )"
+notation "hvbox( L1 ⋕ break [ term 46 T , break term 46 d ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LazyEq $d $T $L1 $L2 }.
+ for @{ 'LazyEq $T $d $L1 $L2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⋕ ⋕ break [ term 46 d , break term 46 T ] break term 46 L2 )"
+notation "hvbox( L1 ⋕ ⋕ break [ term 46 T , break term 46 d ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LazyEqAlt $d $T $L1 $L2 }.
+ for @{ 'LazyEqAlt $T $d $L1 $L2 }.
>yminus_Y_inj #J #Y #X #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
#H destruct /3 width=7 by cpx_delta/
| #J #K1 #V #HLK1 #_ #HV1 #Hid elim (ldrop_leq_conf_lt … HL12 … HLK1) -HL12 /2 width=1 by ylt_inj/
- #Y #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
+ <yminus_SO2 >yminus_inj #Y #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
#H destruct /3 width=7 by cpx_delta/
]
| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_bind … H) -H
elim (cpx_inv_sort1 … H2) -H2 [| * #l #_ ]
#H destruct /2 width=1 by lleq_sort/
| #Is #Id #L1s #L1d #K1s #K1d #V1s #V1d #d #i #Hid #HLK1s #HLK1d #_ #_ #_ #IHV1d #H1 #X #H2 #L2s #H1s #H2s #L2d #H1d #H2d #H3
- elim (ldrop_leq_conf_lt … H1 … HLK1s) -H1 /2 width=1 by ylt_inj/ #Y #H1 #HY
+ elim (ldrop_leq_conf_lt … H1 … HLK1s) -H1 /2 width=1 by ylt_inj/
+ <yminus_SO2 >yminus_inj #Y #H1 #HY
lapply (ldrop_mono … HY … HLK1d) -HY #H destruct
elim (lpx_ldrop_conf … HLK1s … H1s) -H1s #Y #H #HLK2s
elim (lpx_inv_pair1 … H) -H #K2s #V2s #H1s #HV12s #H destruct
elim (lpx_ldrop_conf … HLK1d … H1d) -H1d #Y #H #HLK2d
elim (lpx_inv_pair1 … H) -H #K2d #V2d #H1d #HV12d #H destruct
- elim (ldrop_leq_conf_be … H2s … HLK1s) -H2s /2 width=1 by ylt_inj/ #Z #Y #X #HK12s #H
+ elim (ldrop_leq_conf_be … H2s … HLK1s) -H2s /2 width=1 by ylt_inj/
+ >yplus_O1 <yminus_SO2 >yminus_inj #Z #Y #X #HK12s #H
lapply (ldrop_mono … H … HLK2s) -H #H destruct
- elim (ldrop_leq_conf_be … H2d … HLK1d) -H2d /2 width=1 by ylt_inj/ #Z #Y #X #HK12d #H
+ elim (ldrop_leq_conf_be … H2d … HLK1d) -H2d /2 width=1 by ylt_inj/
+ >yplus_O1 <yminus_SO2 >yminus_inj #Z #Y #X #HK12d #H
lapply (ldrop_mono … H … HLK2d) -H #H destruct
- elim (ldrop_leq_conf_lt … H3 … HLK2s) -H3 /2 width=1 by ylt_inj/ #Y #H3 #HY
+ elim (ldrop_leq_conf_lt … H3 … HLK2s) -H3 /2 width=1 by ylt_inj/
+ <yminus_SO2 >yminus_inj #Y #H3 #HY
lapply (ldrop_mono … HY … HLK2d) -HY #H destruct
elim (cpx_inv_lref1 … H2) -H2 -L1s
[ -L1d #H destruct /3 width=15 by lleq_skip/
[ #d #e #J1 #K1 #W1 #i #H elim (ldrop_inv_atom1 … H) -H
#H destruct
| #I #L1 #L2 #V #HL12 #IHL12 #J1 #K1 #W1 #i #_ #_ #H elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 >yplus_O_sn >yplus_O_sn
+| #I1 #I2 #L1 #L2 #V1 #V2 #e #HL12 >yplus_O1 >yplus_O1
#IHL12 #J1 #K1 #W1 #i #H #_ elim (eq_or_gt i) #Hi destruct [ -IHL12 | -HL12 ]
[ #_ lapply (ldrop_inv_O2 … H) -H
#H destruct >ypred_succ /2 width=5 by ldrop_pair, ex2_3_intro/
elim (ldrop_inv_atom1 … H) -H #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ >yplus_O_sn
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #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/
(**************************************************************************)
include "basic_2/relocation/cpy_cpy.ma".
-include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/cpys_alt.ma".
(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
∀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-.
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_O_sn <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+ >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
]
qed.
* #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. ⇩[O, i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⇧[O, i+1] V2 ≡ T2 →
+ ∧∧ ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] 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-.
+
(* Relocation properties ****************************************************)
lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazyeq_4.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition lleq: relation4 nat term lenv lenv ≝
+ λd,T,L1,L2. |L1| = |L2| ∧
+ (∀U. ⦃⋆, L1⦄ ⊢ T ▶*×[d, ∞] U ↔ ⦃⋆, L2⦄ ⊢ T ▶*×[d, ∞] U).
+
+interpretation
+ "lazy equivalence (local environment)"
+ 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ⋕[⋆k, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_sort1 … H) -H //
+qed.
+
+lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ⋕[§p, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_gref1 … H) -H //
+qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1.ⓑ{I}V ⋕[T, d+1] L2.ⓑ{I}V →
+ L1 ⋕[ⓑ{a,I}V.T, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_bind1 … H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV #H1VW #H1WV
+elim (IHT U) -IHT #H1TU #H1UT
+@cpys_bind /2 width=1 by/ -HVW -H1VW -H1WV
+[ @(lsuby_cpys_trans … (L2.ⓑ{I}V))
+| @(lsuby_cpys_trans … (L1.ⓑ{I}V))
+] /4 width=5 by lsuby_cpys_trans, lsuby_succ/ (**) (* full auto too slow *)
+qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1 ⋕[T, d] L2 → L1 ⋕[ⓕ{I}V.T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_flat1 … H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV elim (IHT U) -IHT
+/3 width=1 by cpys_flat/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → |L1| = |L2|.
+#L1 #L2 #T #d * //
+qed-.
+
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1 ⋕[V, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓑ{a,I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1.ⓑ{I}V ⋕[T, d+1] L2.ⓑ{I}V.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12
+#U elim (H (ⓑ{a,I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
+ L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[V, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓕ{I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
+ L1 ⋕[ⓕ{I}V.T, d] L2 → L1 ⋕[T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (ⓕ{I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 … H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,d. L1 ⋕[ⓑ{a,I}V.T, d] L2 →
+ L1 ⋕[V, d] L2 ∧ L1.ⓑ{I}V ⋕[T, d+1] L2.ⓑ{I}V.
+/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-.
+
+lemma lleq_inv_flat: ∀I,L1,L2,V,T,d. L1 ⋕[ⓕ{I}V.T, d] L2 →
+ L1 ⋕[V, d] L2 ∧ L1 ⋕[T, d] L2.
+/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ 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/lazyeqalt_4.ma".
+include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/substitution/lleq_lleq.ma".
+
+inductive lleqa: relation4 nat term lenv lenv ≝
+| lleqa_sort: ∀L1,L2,d,k. |L1| = |L2| → lleqa d (⋆k) L1 L2
+| lleqa_skip: ∀L1,L2,d,i. |L1| = |L2| → i < d → lleqa d (#i) L1 L2
+| lleqa_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ i →
+ ⇩[0, i] L1 ≡ K1.ⓑ{I1}V → ⇩[0, i] L2 ≡ K2.ⓑ{I2}V →
+ lleqa 0 V K1 K2 → lleqa d (#i) L1 L2
+| lleqa_free: ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → lleqa d (#i) L1 L2
+| lleqa_gref: ∀L1,L2,d,p. |L1| = |L2| → lleqa d (§p) L1 L2
+| lleqa_bind: ∀a,I,L1,L2,V,T,d.
+ lleqa d V L1 L2 → lleqa (d+1) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ lleqa d (ⓑ{a,I}V.T) L1 L2
+| lleqa_flat: ∀I,L1,L2,V,T,d.
+ lleqa d V L1 L2 → lleqa d T L1 L2 → lleqa d (ⓕ{I}V.T) L1 L2
+.
+
+interpretation
+ "lazy equivalence (local environment) alternative"
+ 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lleqa_inv_lleq: ∀L1,L2,T,d. L1 ⋕⋕[T, d] L2 → L1 ⋕[T, d] L2.
+#L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lleq_lleqa: ∀L1,T,L2,d. L1 ⋕[T, d] L2 → L1 ⋕⋕[T, d] L2.
+#L1 #T @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
+[ #i #Hn #L2 #d #H elim (lleq_fwd_lref … H) [ * || * ]
+ /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
+| #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind … H) -H /3 width=1 by lleqa_bind/
+| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat … H) -H /3 width=1 by lleqa_flat/
+]
+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/cpys_lift.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_skip: ∀L1,L2,d,i. i < d → |L1| = |L2| → L1 ⋕[#i, d] L2.
+#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 … H) -H // *
+#I #Z #Y #X #H elim (ylt_yle_false … H) -H
+/2 width=1 by ylt_inj/
+qed.
+
+lemma lleq_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ i →
+ ⇩[0, i] L1 ≡ K1.ⓑ{I1}V → ⇩[0, i] L2 ≡ K2.ⓑ{I2}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2.
+#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
+[ lapply (ldrop_fwd_length … HLK1) -HLK1 #H1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H2
+ >H1 >H2 -H1 -H2 normalize //
+| #U @conj #H elim (cpys_inv_lref1 … H) -H // *
+ #I #K #X #W #_ #_ #H #HVW #HWU
+ [ lapply (ldrop_mono … H … HLK1) | lapply (ldrop_mono … H … HLK2) ] -H
+ #H destruct elim (IH W) /3 width=7 by cpys_subst, yle_inj/
+]
+qed.
+
+lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2.
+#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 … H) -H // *
+#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 … H) -H
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+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/cpys_cpys.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | i < d
+ | ∃∃I1,I2,K1,K2,V. ⇩[0, i] L1 ≡ K1.ⓑ{I1}V &
+ ⇩[0, i] L2 ≡ K2.ⓑ{I2}V &
+ K1 ⋕[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
+elim (lt_or_ge i d) /2 width=1 by or3_intro1/ #Hdi #Hi
+elim (ldrop_O1_lt … Hi) #I1 #K1 #V1 #HLK1
+elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
+lapply (ldrop_fwd_length_minus2 … HLK2) #H
+lapply (ldrop_fwd_length_minus2 … HLK1) >HL12 <H -HL12 -H
+#H lapply (injective_plus_l … H) -H #HK12
+elim (lift_total V1 0 (i+1)) #W1 #HVW1
+elim (lift_total V2 0 (i+1)) #W2 #HVW2
+elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
+elim (cpys_inv_lref1_ldrop … (H1 ?) … HLK2 … HVW1) -H1
+[ elim (cpys_inv_lref1_ldrop … (H2 ?) … HLK1 … HVW2) -H2 ]
+/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
+lapply (cpys_antisym_eq … H12 … H21) -H12 -H21 #H destruct
+@or3_intro2 @(ex4_5_intro … HLK1 HLK2) // @conj // -HK12
+#V elim (lift_total V 0 (i+1))
+#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
+[ elim (cpys_inv_lref1_ldrop … (H12 ?) … HLK2 … HVW) -H12 -H21
+| elim (cpys_inv_lref1_ldrop … (H21 ?) … HLK1 … HVW) -H21 -H12
+] /3 width=7 by cpys_subst, yle_inj/
+qed-.
]
class "yellow"
[ { "substitution" * } {
+ [ { "lazy equivalence for local environments" * } {
+ [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_lleq" * ]
+ }
+ ]
[ { "contxt-sensitive extended multiple substitution" * } {
[ "cpys ( ⦃?,?⦄ ⊢ ? ▶*×[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*×[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
}
]
class "orange"
[ { "relocation" * } {
- [ { "lazy equivalence for local environments" * } {
- [ "lleq ( ? ⋕[?,?] ? )" "lleq_lleq" * ]
- }
- ]
[ { "contxt-sensitive extended ordinary substitution" * } {
[ "cpy ( ⦃?,?⦄ ⊢ ? ▶×[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
}
(* inversion and forward lemmas on yle **************************************)
-lemma lt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
+lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
#m #n * -m -n /3 width=1 by yle_pred_sn, yle_inj, yle_Y/
qed-.
]
qed.
-lemma yplus_O_sn: ∀n:ynat. 0 + n = n.
+lemma yplus_O1: ∀n:ynat. 0 + n = n.
#n >yplus_comm // qed.
(* Basic inversion lemmas ***************************************************)
projects / helm.git / commitdiff