(* *)
(**************************************************************************)
-include "basic_2/relocation/lift.ma".
+include "basic_2/substitution/lift.ma".
include "apps_2/functional/notation.ma".
(* FUNCTIONAL RELOCATION ****************************************************)
(**************************************************************************)
include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/ldrops.ma".
+include "basic_2/multiple/ldrops.ma".
(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/lifts_lifts.ma".
-include "basic_2/substitution/ldrops_ldrops.ma".
+include "basic_2/multiple/lifts_lifts.ma".
+include "basic_2/multiple/ldrops_ldrops.ma".
include "basic_2/static/aaa_lifts.ma".
include "basic_2/static/aaa_aaa.ma".
include "basic_2/computation/lsubc_ldrops.ma".
include "basic_2/notation/relations/ineint_5.ma".
include "basic_2/grammar/aarity.ma".
-include "basic_2/substitution/gr2_gr2.ma".
-include "basic_2/substitution/lifts_lift_vector.ma".
-include "basic_2/substitution/ldrops_ldrop.ma".
+include "basic_2/multiple/gr2_gr2.ma".
+include "basic_2/multiple/lifts_lift_vector.ma".
+include "basic_2/multiple/ldrops_ldrop.ma".
include "basic_2/computation/acp.ma".
(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/fqus_fqus.ma".
+include "basic_2/multiple/fqus_fqus.ma".
include "basic_2/unfold/lsstas_lift.ma".
include "basic_2/reduction/cpx_lift.ma".
include "basic_2/computation/cpxs.ma".
(**************************************************************************)
include "basic_2/grammar/tstc_vector.ma".
-include "basic_2/relocation/lift_vector.ma".
+include "basic_2/substitution/lift_vector.ma".
include "basic_2/computation/cpxs_tstc.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
(**************************************************************************)
include "basic_2/notation/relations/lazybtpredproper_8.ma".
-include "basic_2/substitution/fleq.ma".
+include "basic_2/multiple/fleq.ma".
include "basic_2/computation/fpbu.ma".
(* SINGLE-STEP "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********)
(* *)
(**************************************************************************)
-include "basic_2/substitution/fleq_fleq.ma".
+include "basic_2/multiple/fleq_fleq.ma".
include "basic_2/computation/fpbu_fleq.ma".
include "basic_2/computation/fpbc.ma".
(**************************************************************************)
include "basic_2/notation/relations/btpredstar_8.ma".
-include "basic_2/substitution/fqus.ma".
+include "basic_2/multiple/fqus.ma".
include "basic_2/reduction/fpb.ma".
include "basic_2/computation/cpxs.ma".
include "basic_2/computation/lpxs.ma".
(**************************************************************************)
include "basic_2/notation/relations/btpredstaralt_8.ma".
-include "basic_2/substitution/lleq_fqus.ma".
-include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/multiple/lleq_fqus.ma".
+include "basic_2/multiple/lleq_lleq.ma".
include "basic_2/computation/cpxs_lleq.ma".
include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/fpbs.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/fleq.ma".
+include "basic_2/multiple/fleq.ma".
include "basic_2/computation/fpbs.ma".
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/fleq.ma".
+include "basic_2/multiple/fleq.ma".
include "basic_2/computation/fpbs_alt.ma".
include "basic_2/computation/fpbu_lleq.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_fqus.ma".
-include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/multiple/lleq_fqus.ma".
+include "basic_2/multiple/lleq_lleq.ma".
include "basic_2/computation/cpxs_lleq.ma".
include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/fpbu.ma".
(**************************************************************************)
include "basic_2/notation/relations/predsnstar_3.ma".
-include "basic_2/relocation/lpx_sn_tc.ma".
+include "basic_2/substitution/lpx_sn_tc.ma".
include "basic_2/reduction/lpr.ma".
(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
(**************************************************************************)
include "basic_2/notation/relations/sn_6.ma".
-include "basic_2/substitution/lleq.ma".
+include "basic_2/multiple/lleq.ma".
include "basic_2/reduction/lpx.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
(**************************************************************************)
include "basic_2/notation/relations/snalt_6.ma".
-include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/multiple/lleq_lleq.ma".
include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/lsx.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
include "basic_2/reduction/lpx_ldrop.ma".
include "basic_2/computation/lsx.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/multiple/lleq_lleq.ma".
include "basic_2/reduction/lpx_lleq.ma".
include "basic_2/computation/lsx.ma".
--- /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/cofreestar_4.ma".
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+definition cofrees: relation4 ynat nat lenv term ≝
+ λd,i,L,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶*[d, ∞] U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
+
+interpretation
+ "context-sensitive exclusion from free variables (term)"
+ 'CoFreeStar L i d T = (cofrees d i L T).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cofrees_fwd_lift: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
+/2 width=1 by/ qed-.
+
+lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
+#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
+#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
+#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cofrees_inv_gen: ∀L,U,U0,d,i. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
+ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥.
+#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U -d /2 width=2 by/
+qed-.
+
+lemma cofrees_inv_lref_eq: ∀L,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃#i⦄ → ⊥.
+#L #d #i #H elim (H (#i)) -H //
+#X #H elim (lift_inv_lref2_be … H) -H //
+qed-.
+
+lemma cofrees_inv_bind: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
+/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
+
+lemma cofrees_inv_flat: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
+
+(* Basic Properties *********************************************************)
+
+lemma cofrees_lsuby_conf: ∀L1,U,d,i. L1 ⊢ i ~ϵ 𝐅*[d]⦃U⦄ →
+ ∀L2. L1 ⊆[d, ∞] L2 → L2 ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+/3 width=3 by lsuby_cpys_trans/ qed-.
+
+lemma cofrees_sort: ∀L,d,i,k. L ⊢ i ~ϵ 𝐅*[d]⦃⋆k⦄.
+#L #d #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_gref: ∀L,d,i,p. L ⊢ i ~ϵ 𝐅*[d]⦃§p⦄.
+#L #d #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_bind: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d] ⦃V⦄ →
+ ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ →
+ ∀a. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄.
+#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
+qed.
+
+lemma cofrees_flat: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ →
+ ∀I. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄.
+#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
+qed.
+
+lemma cofrees_cpy_trans: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
+ ∀i. L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄.
+/3 width=3 by cpys_strap2/ qed-.
+
+axiom cofrees_dec: ∀L,T,d,i. Decidable (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄).
+
+(* Basic negated properties *************************************************)
+
+lemma frees_cpy_div: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
+ ∀i. (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → ⊥).
+/3 width=7 by cofrees_cpy_trans/ qed-.
+
+(* Basic negated inversion lemmas *******************************************)
+
+lemma frees_inv_bind: ∀a,I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄ → ⊥) →
+ (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ → ⊥).
+#a #I #L #W #U #d #i #H elim (cofrees_dec L W d i)
+/4 width=9 by cofrees_bind, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_flat: ∀I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄ → ⊥) →
+ (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥).
+#I #L #W #U #d #H elim (cofrees_dec L W d)
+/4 width=8 by cofrees_flat, or_intror, or_introl/
+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_nlift.ma".
+include "basic_2/substitution/cofrees_lift.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Alternative definition of frees_ge ***************************************)
+
+lemma nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
+#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
+qed-.
+
+lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+ (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
+ ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
+ (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
+#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
+#U1 #U2 #HU12 #HU2 *
+[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
+ * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
+| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
+ * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
+ lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
+ #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
+ #HK12
+ @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
+ @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
+ >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
+]
+qed-.
+
+lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
+ (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
+ (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
+ ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
+#R #IH1 #IH2 #d #i #L #U
+generalize in match d; -d generalize in match i; -i
+@(f2_ind … rfw … L U) -L -U
+#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
+-IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
+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/cofrees.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cofrees_inv_lref_be: ∀L,d,i,j. L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → d ≤ yinj j → j < i →
+ ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #i #j #Hj #Hdj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
+#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
+#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
+>minus_plus <plus_minus_m_m //
+qed-.
+
+lemma cofrees_inv_be: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+ ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → d ≤ yinj j → j < i → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #d #i #_ #j #H elim (H (⋆k)) -H //
+| -IH #j #_ #d #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
+ /2 width=9 by cofrees_inv_lref_be/
+| -IH #p #_ #d #i #_ #j #H elim (H (§p)) -H //
+| #a #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
+ elim (cofrees_inv_bind … H1) -H1 #HW #HU
+ elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=9 by/ ]
+ -HW #HnU lapply (IH … HU … HnU I K V ? ? ?)
+ /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/ -a -I -J -L -W -U -d
+ >minus_plus_plus_l //
+| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
+ elim (cofrees_inv_flat … H1) -H1 #HW #HU
+ elim (nlift_inv_flat … H2) -H2 [ /3 width=9 by/ ]
+ #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cofrees_lref_skip: ∀L,d,i,j. j < i → yinj j < d → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false … Hdj) -i -I -L -K -W1 -W2 -X //
+]
+qed.
+
+lemma cofrees_lref_lt: ∀L,d,i,j. i < j → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
+| * #I #K #V1 #V2 #_ #_ #_ #H -I -L -K -V1 -d
+ elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
+]
+qed.
+
+lemma cofrees_lref_gt: ∀I,L,K,W,d,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
+ K ⊢ (i-j-1) ~ϵ 𝐅*[O]⦃W⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
+ #H destruct elim (HW1 … HW12) -I -K -W1 -d
+ #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
+ >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
+]
+qed.
+
+lemma cofrees_lref_free: ∀L,d,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
+#L #d #i #j #Hj #Hji #X #H elim (cpys_inv_lref1_Y2 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I #K #W1 #W2 #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
+ #H elim (lt_refl_false j) -d -i -K -W1 -W2 -X /2 width=3 by lt_to_le_to_lt/
+]
+qed.
+
+(* Advanced negated inversion lemmas ****************************************)
+
+lemma frees_inv_lref_gt: ∀L,d,i,j. j < i → (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) →
+ ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*[0]⦃W⦄ → ⊥) & d ≤ yinj j.
+#L #d #i #j #Hji #H elim (ylt_split j d) #Hjd
+[ elim H -H /2 width=6 by cofrees_lref_skip/
+| elim (lt_or_ge j (|L|)) #Hj
+ [ elim (ldrop_O1_lt … Hj) -Hj /4 width=10 by cofrees_lref_gt, ex3_3_intro/
+ | elim H -H /2 width=6 by cofrees_lref_free/
+ ]
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,d,i,j. (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
+#L #d #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
+#Hij elim H -H /2 width=6 by cofrees_lref_lt, cofrees_lref_free/
+qed-.
+
+lemma frees_inv_gen: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+ ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #d #i #H elim H -H //
+| #j #Hn #d #i #H elim (lt_or_eq_or_gt i j)
+ [ -n #Hij elim H -H /2 width=5 by cofrees_lref_lt/
+ | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
+ | #Hji elim (frees_inv_lref_gt … H) // -H
+ #I #K #W1 #HLK #H #Hdj elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
+ #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
+ #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
+ #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
+ >minus_plus <plus_minus_m_m //
+ ]
+| -IH #p #_ #d #i #H elim H -H //
+| #a #I #W #U #Hn #d #i #H elim (frees_inv_bind … H) -H
+ #H elim (IH … H) // -H -n
+ /4 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn, ex2_intro/
+| #I #W #U #Hn #d #i #H elim (frees_inv_flat … H) -H
+ #H elim (IH … H) // -H -n
+ /4 width=9 by cpys_flat, nlift_flat_dx, nlift_flat_sn, ex2_intro/
+]
+qed-.
+
+lemma frees_ind: ∀L,d,i. ∀R:predicate term.
+ (∀U1. (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1) →
+ (∀U1,U2. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → R U2 → R U1) →
+ ∀U. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U.
+#L #d #i #R #IH1 #IH2 #U1 #H elim (frees_inv_gen … H) -H
+#U2 #H #HnU2 @(cpys_ind_dx … H) -U1 /4 width=8 by cofrees_inv_gen/
+qed-.
+
+(* Advanced negated properties **********************************************)
+
+lemma frees_be: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
+ ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) →
+ ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+ ∀d. d ≤ yinj j → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
+/4 width=11 by cofrees_inv_be/ qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cofrees_lift_be: ∀d0,e0,i. d0 ≤ i → i ≤ d0 + e0 →
+ ∀L,K,s. ⇩[s, d0, e0+1] L ≡ K → ∀T,U. ⇧[d0, e0+1] T ≡ U →
+ ∀d. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hd0i #Hide0 #L #K #s #HLK #T1 #U1 #HTU1 #d #U2 #HU12
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0+1)) #H2
+ [ letin cpys_inv ≝ cpys_inv_lift1_ge_up
+ | letin cpys_inv ≝ cpys_inv_lift1_ge
+ ]
+| letin cpys_inv ≝ cpys_inv_lift1_be
+]
+elim (cpys_inv … HU12 … HLK … HTU1) // #T2 #_ #HTU2 -s -L -K -U1 -T1 -d
+elim (lift_split … HTU2 i e0) /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_lift_ge: ∀d0,e0,i. d0 + e0 ≤ i →
+ ∀L,K,s. ⇩[s, d0, e0] L ≡ K → ∀T,U. ⇧[d0, e0] T ≡ U →
+ ∀d. K ⊢ i-e0 ~ϵ 𝐅*[d-yinj e0]⦃T⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hde0i #L #K #s #HLK #T1 #U1 #HTU1 #d #HT1 #U2 #HU12
+elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0)) #H2
+ [ elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) // >yplus_inj >yminus_Y_inj #T2 #HT12
+ lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12
+ | elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) // #T2
+ ]
+| elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) // >yminus_Y_inj #T2 #HT12
+ lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) // -HT12
+]
+-s -L #HT12 #HTU2
+elim (HT1 … HT12) -T1 #V2 #HVT2
+elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L ⊢ break term 46 i ~ ϵ 𝐅 * [ break term 46 d ] ⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'CoFreeStar $L $i $d $T }.
+++ /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/cofreestar_3.ma".
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-definition cofrees: relation3 lenv nat term ≝
- λL,i,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶* U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
-
-interpretation
- "context-sensitive exclusion from free variables (term)"
- 'CoFreeStar L i U = (cofrees L i U).
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cofrees_fwd_lift: ∀L,U,i. L ⊢ i ~ϵ 𝐅*⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
-/2 width=1 by/ qed-.
-
-lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*⦃W⦄.
-#a #I #L #W1 #U #i #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
- L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*⦃U⦄.
-#a #I #L #W #U1 #i #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*⦃W⦄.
-#I #L #W1 #U #i #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*⦃U⦄.
-#I #L #W #U1 #i #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cofrees_inv_gen: ∀L,U,U0,i. ⦃⋆, L⦄ ⊢ U ▶* U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
- L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥.
-#L #U #U0 #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U /2 width=2 by/
-qed-.
-
-lemma cofrees_inv_lref_eq: ∀L,i. L ⊢ i ~ϵ 𝐅*⦃#i⦄ → ⊥.
-#L #i #H elim (H (#i)) -H //
-#X #H elim (lift_inv_lref2_be … H) -H //
-qed-.
-
-lemma cofrees_inv_bind: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*⦃U⦄.
-/3 width=7 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
-
-lemma cofrees_inv_flat: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*⦃U⦄.
-/3 width=6 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
-
-(* Basic Properties *********************************************************)
-
-lemma cofrees_sort: ∀L,i,k. L ⊢ i ~ϵ 𝐅*⦃⋆k⦄.
-#L #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_gref: ∀L,i,p. L ⊢ i ~ϵ 𝐅*⦃§p⦄.
-#L #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_bind: ∀L,V,i. L ⊢ i ~ϵ 𝐅*⦃V⦄ →
- ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*⦃T⦄ →
- ∀a. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}V.T⦄.
-#L #W1 #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
-qed.
-
-lemma cofrees_flat: ∀L,V,i. L ⊢ i ~ϵ 𝐅*⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*⦃T⦄ →
- ∀I. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}V.T⦄.
-#L #W1 #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
-qed.
-
-axiom cofrees_dec: ∀L,T,i. Decidable (L ⊢ i ~ϵ 𝐅*⦃T⦄).
-
-(* Basic negated inversion lemmas *******************************************)
-
-lemma frees_inv_bind: ∀a,I,L,V,T,i. (L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*⦃T⦄ → ⊥).
-#a #I #L #W #U #i #H elim (cofrees_dec L W i)
-/4 width=8 by cofrees_bind, or_intror, or_introl/
-qed-.
-
-lemma frees_inv_flat: ∀I,L,V,T,i. (L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*⦃T⦄ → ⊥).
-#I #L #W #U #i #H elim (cofrees_dec L W i)
-/4 width=7 by cofrees_flat, or_intror, or_introl/
-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/lift_neg.ma".
-include "basic_2/relocation/lift_lift.ma".
-include "basic_2/substitution/cpys.ma".
-include "basic_2/substitution/cofrees_lift.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Alternative definition of frees_ge ***************************************)
-
-(*
-lemma cpys_fwd_nlift2: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶* U2 →
- ∀i. (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) →
- (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) ∨
- ∃∃I,K,W,j. j < i & ⇩[j]L ≡ K.ⓑ{I}W &
- (∀V. ⇧[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥).
-#G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ /3 width=2 by or_introl/
-| #I #G #L #K #V1 #V2 #W2 #j #HLK #_ #HVW2 #IHV12 #i #HnW2
- elim (lt_or_ge j i) #Hij
- [ @or_intror (**) @(ex4_4_intro … HLK) //
- [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
- #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
- | -HnW2 /2 width=7 by lift_inv_lref2_be/
- ]
- | elim (lift_split … HVW2 i j) -HVW2 //
- #X #_ #H elim HnW2 -HnW2 //
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #_ #_ #IHW12 #IHU12 #i #H elim (nlift_inv_bind … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12
- [ /4 width=9 by nlift_bind_sn, or_introl/
- | * /5 width=9 by nlift_bind_sn, ex4_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12
- [ /4 width=9 by nlift_bind_dx, or_introl/
- | * #J #K #W #j @(nat_ind_plus … j) -j
- [ #_ #H lapply (ldrop_inv_pair1 … H) -H
- #H destruct /4 width=9 by nlift_bind_sn, or_introl/
- | #j #_ #Hji #HLK #HnW
- lapply (ldrop_inv_drop1_lt … HLK ?) // -HLK #HLK #HnU1
- <minus_le_minus_minus_comm in HnW;
- /5 width=9 by nlift_bind_dx, monotonic_lt_pred, ex4_4_intro, or_intror/
- ]
- ]
- ]
-| #I #G #L #W1 #W2 #U1 #U2 #_ #_ #IHW12 #IHU12 #i #H elim (nlift_inv_flat … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12
- [ /4 width=9 by nlift_flat_sn, or_introl/
- | * /5 width=9 by nlift_flat_sn, ex4_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12
- [ /4 width=9 by nlift_flat_dx, or_introl/
- | * /5 width=9 by nlift_flat_dx, ex4_4_intro, or_intror/
- ]
-]
-qed-.
-*)
-lemma nlift_frees: ∀L,U,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥).
-#L #U #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
-qed-.
-(*
-lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
- (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
- (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
-#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
-#U1 #U2 #HU12 #HU2 *
-[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
- * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
-| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
- * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
- lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
- #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
- #HK12
- @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
- @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
- >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
-]
-qed-.
-
-lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
- (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
- (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
- ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
-#R #IH1 #IH2 #d #i #L #U
-generalize in match d; -d generalize in match i; -i
-@(f2_ind … rfw … L U) -L -U
-#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
--IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
-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/cofrees.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cofrees_inv_lref_lt: ∀L,i,j. L ⊢ i ~ϵ 𝐅*⦃#j⦄ → j < i →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄.
-#L #i #j #Hj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
-#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_delta/ -I -L -K -W1
-#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
->minus_plus <plus_minus_m_m //
-qed-.
-
-lemma cofrees_inv_lt: ∀L,U,i. L ⊢ i ~ϵ 𝐅*⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → j < i → K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄.
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #i #_ #j #H elim (H (⋆k)) -H //
-| -IH #j #_ #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
- /2 width=7 by cofrees_inv_lref_lt/
-| -IH #p #_ #i #_ #j #H elim (H (§p)) -H //
-| #a #J #W #U #Hn #i #H1 #j #H2 #I #K #V #HLK #Hji destruct
- elim (cofrees_inv_bind … H1) -H1 #HW #HU
- elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=7 by/ ]
- -HW #HnU lapply (IH … HU … HnU I K V ? ?)
- /2 width=1 by ldrop_drop, lt_minus_to_plus/ -a -I -J -L -W -U
- >minus_plus_plus_l //
-| #J #W #U #Hn #i #H1 #j #H2 #I #K #V #HLK #Hji destruct
- elim (cofrees_inv_flat … H1) -H1 #HW #HU
- elim (nlift_inv_flat … H2) -H2 [ /3 width=7 by/ ]
- #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cofrees_lref_gt: ∀L,i,j. i < j → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
-#L #i #j #Hij #X #H elim (cpys_inv_lref1 … H) -H
-[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
-| * #I #K #V1 #V2 #_ #_ #H -I -L -K -V1
- elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
-]
-qed.
-
-lemma cofrees_lref_lt: ∀I,L,K,W,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
- K ⊢ (i-j-1) ~ϵ 𝐅*⦃W⦄ → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
-#I #L #K #W1 #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I0 #K0 #W0 #W2 #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
- #H destruct elim (HW1 … HW12) -I -K -W1
- #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
- >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
-]
-qed.
-
-lemma cofrees_lref_free: ∀L,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
-#L #i #j #Hj #Hji #X #H elim (cpys_inv_lref1 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I #K #W1 #W2 #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
- #H elim (lt_refl_false j) -K -W1 -W2 -X -i /2 width=3 by lt_to_le_to_lt/
-]
-qed.
-
-(* Advanced negated inversion lemmas ****************************************)
-
-lemma frees_inv_lref_lt: ∀L,i,j. j < i → (L ⊢ i ~ϵ 𝐅*⦃#j⦄ → ⊥) →
- ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*⦃W⦄ → ⊥).
-#L #i #j #Hji #H elim (lt_or_ge j (|L|)) #Hj
-[ elim (ldrop_O1_lt (Ⓕ) … Hj) -Hj /4 width=9 by cofrees_lref_lt, ex2_3_intro/
-| elim H -H /2 width=5 by cofrees_lref_free/
-]
-qed-.
-
-lemma frees_inv_lref_free: ∀L,i,j. (L ⊢ i ~ϵ 𝐅*⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
-#L #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
-#Hij elim H -H /2 width=5 by cofrees_lref_gt, cofrees_lref_free/
-qed-.
-
-lemma frees_inv_gen: ∀L,U,i. (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥) →
- ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶* U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #i #H elim H -H //
-| #j #Hn #i #H elim (lt_or_eq_or_gt i j)
- [ -n #Hij elim H -H /2 width=4 by cofrees_lref_gt/
- | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
- | #Hji elim (frees_inv_lref_lt … H) // -H
- #I #K #W1 #HLK #H elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
- #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
- #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_delta/ -I -L -K -W1
- #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
- >minus_plus <plus_minus_m_m //
- ]
-| -IH #p #_ #i #H elim H -H //
-| #a #I #W #U #Hn #i #H elim (frees_inv_bind … H) -H
- #H elim (IH … H) // -H -n
- #X #HX #HnX [ @(ex2_intro … (ⓑ{a,I}X.U)) | @(ex2_intro … (ⓑ{a,I}W.X)) ] (**) (* explicit constructor *)
- /3 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn/
-| #I #W #U #Hn #i #H elim (frees_inv_flat … H) -H
- #H elim (IH … H) // -H -n
- #X #HX #HnX [ @(ex2_intro … (ⓕ{I}X.U)) | @(ex2_intro … (ⓕ{I}W.X)) ] (**) (* explicit constructor *)
- /3 width=8 by cpys_flat, nlift_flat_dx, nlift_flat_sn/
-]
-qed-.
-
-(* Advanced negated properties **********************************************)
-
-lemma frees_lt: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
- ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄ → ⊥) →
- ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥).
-/4 width=9 by cofrees_inv_lt/ qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma cofrees_lift_be: ∀d,e,i. d ≤ i → i ≤ d + e →
- ∀L,K,s. ⇩[s, d, e+1] L ≡ K → ∀T,U. ⇧[d, e+1] T ≡ U →
- L ⊢ i ~ϵ 𝐅*⦃U⦄.
-#d #e #i #Hdi #Hide #L #K #s #HLK #T1 #U1 #HTU1 #U2 #HU12
-elim (cpys_inv_lift1 … HU12 … HLK … HTU1) #T2 #HTU2 #_ -s -L -K -U1 -T1
-elim (lift_split … HTU2 i e) /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_lift_ge: ∀d,e,i. d + e ≤ i →
- ∀L,K,s. ⇩[s, d, e] L ≡ K → ∀T,U. ⇧[d, e] T ≡ U →
- K ⊢ i-e ~ϵ 𝐅*⦃T⦄ → L ⊢ i ~ϵ 𝐅*⦃U⦄.
-#d #e #i #Hdei #L #K #s #HLK #T1 #U1 #HTU1 #HT1 #U2 #HU12
-elim (le_inv_plus_l … Hdei) -Hdei #Hdie #Hei
-elim (cpys_inv_lift1 … HU12 … HLK … HTU1) -s -L #T2 #HTU2 #HT12
-elim (HT1 … HT12) -T1 #V2 #HVT2
-elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L ⊢ break term 46 i ~ ϵ 𝐅 * ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'CoFreeStar $L $i $T }.
+++ /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/cofreestar_4.ma".
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-definition cofrees: relation4 ynat nat lenv term ≝
- λd,i,L,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶*[d, ∞] U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
-
-interpretation
- "context-sensitive exclusion from free variables (term)"
- 'CoFreeStar L i d T = (cofrees d i L T).
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cofrees_fwd_lift: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
-/2 width=1 by/ qed-.
-
-lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
-#a #I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
-#a #I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
-#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄.
-#I #L #W1 #U #i #d #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#I #L #W #U1 #i #d #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
-#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cofrees_inv_gen: ∀L,U,U0,d,i. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
- L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥.
-#L #U #U0 #d #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U -d /2 width=2 by/
-qed-.
-
-lemma cofrees_inv_lref_eq: ∀L,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃#i⦄ → ⊥.
-#L #d #i #H elim (H (#i)) -H //
-#X #H elim (lift_inv_lref2_be … H) -H //
-qed-.
-
-lemma cofrees_inv_bind: ∀a,I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃U⦄.
-/3 width=8 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
-
-lemma cofrees_inv_flat: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ~ϵ 𝐅*[d]⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-/3 width=7 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
-
-(* Basic Properties *********************************************************)
-
-lemma cofrees_lsuby_conf: ∀L1,U,d,i. L1 ⊢ i ~ϵ 𝐅*[d]⦃U⦄ →
- ∀L2. L1 ⊆[d, ∞] L2 → L2 ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-/3 width=3 by lsuby_cpys_trans/ qed-.
-
-lemma cofrees_sort: ∀L,d,i,k. L ⊢ i ~ϵ 𝐅*[d]⦃⋆k⦄.
-#L #d #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_gref: ∀L,d,i,p. L ⊢ i ~ϵ 𝐅*[d]⦃§p⦄.
-#L #d #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_bind: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d] ⦃V⦄ →
- ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ →
- ∀a. L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄.
-#L #W1 #d #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
-qed.
-
-lemma cofrees_flat: ∀L,V,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ →
- ∀I. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄.
-#L #W1 #d #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
-#W2 #U2 #HW12 #HU12 #H destruct
-elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
-qed.
-
-lemma cofrees_cpy_trans: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
- ∀i. L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄.
-/3 width=3 by cpys_strap2/ qed-.
-
-axiom cofrees_dec: ∀L,T,d,i. Decidable (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄).
-
-(* Basic negated properties *************************************************)
-
-lemma frees_cpy_div: ∀L,U1,U2,d. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 →
- ∀i. (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U1⦄ → ⊥).
-/3 width=7 by cofrees_cpy_trans/ qed-.
-
-(* Basic negated inversion lemmas *******************************************)
-
-lemma frees_inv_bind: ∀a,I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓑ{a,I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*[⫯d]⦃T⦄ → ⊥).
-#a #I #L #W #U #d #i #H elim (cofrees_dec L W d i)
-/4 width=9 by cofrees_bind, or_intror, or_introl/
-qed-.
-
-lemma frees_inv_flat: ∀I,L,V,T,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}V.T⦄ → ⊥) →
- (L ⊢ i ~ϵ 𝐅*[d]⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥).
-#I #L #W #U #d #H elim (cofrees_dec L W d)
-/4 width=8 by cofrees_flat, or_intror, or_introl/
-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_nlift.ma".
-include "basic_2/substitution/cofrees_lift.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Alternative definition of frees_ge ***************************************)
-
-lemma nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
-#L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
-qed-.
-
-lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
- (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
- (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
-#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
-#U1 #U2 #HU12 #HU2 *
-[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
- * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
-| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
- * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
- lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
- #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
- #HK12
- @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
- @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
- >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
-]
-qed-.
-
-lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
- (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
- (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
- ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
-#R #IH1 #IH2 #d #i #L #U
-generalize in match d; -d generalize in match i; -i
-@(f2_ind … rfw … L U) -L -U
-#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
--IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
-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/cofrees.ma".
-
-(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cofrees_inv_lref_be: ∀L,d,i,j. L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → d ≤ yinj j → j < i →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #i #j #Hj #Hdj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
-#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
-#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
->minus_plus <plus_minus_m_m //
-qed-.
-
-lemma cofrees_inv_be: ∀L,U,d,i. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → d ≤ yinj j → j < i → K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #d #i #_ #j #H elim (H (⋆k)) -H //
-| -IH #j #_ #d #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
- /2 width=9 by cofrees_inv_lref_be/
-| -IH #p #_ #d #i #_ #j #H elim (H (§p)) -H //
-| #a #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
- elim (cofrees_inv_bind … H1) -H1 #HW #HU
- elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=9 by/ ]
- -HW #HnU lapply (IH … HU … HnU I K V ? ? ?)
- /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/ -a -I -J -L -W -U -d
- >minus_plus_plus_l //
-| #J #W #U #Hn #d #i #H1 #j #H2 #I #K #V #HLK #Hdj #Hji destruct
- elim (cofrees_inv_flat … H1) -H1 #HW #HU
- elim (nlift_inv_flat … H2) -H2 [ /3 width=9 by/ ]
- #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cofrees_lref_skip: ∀L,d,i,j. j < i → yinj j < d → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I #K #W1 #W2 #Hdj elim (ylt_yle_false … Hdj) -i -I -L -K -W1 -W2 -X //
-]
-qed.
-
-lemma cofrees_lref_lt: ∀L,d,i,j. i < j → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
-| * #I #K #V1 #V2 #_ #_ #_ #H -I -L -K -V1 -d
- elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
-]
-qed.
-
-lemma cofrees_lref_gt: ∀I,L,K,W,d,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
- K ⊢ (i-j-1) ~ϵ 𝐅*[O]⦃W⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
- #H destruct elim (HW1 … HW12) -I -K -W1 -d
- #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
- >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
-]
-qed.
-
-lemma cofrees_lref_free: ∀L,d,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
-#L #d #i #j #Hj #Hji #X #H elim (cpys_inv_lref1_Y2 … H) -H
-[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
-| * #I #K #W1 #W2 #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
- #H elim (lt_refl_false j) -d -i -K -W1 -W2 -X /2 width=3 by lt_to_le_to_lt/
-]
-qed.
-
-(* Advanced negated inversion lemmas ****************************************)
-
-lemma frees_inv_lref_gt: ∀L,d,i,j. j < i → (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) →
- ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*[0]⦃W⦄ → ⊥) & d ≤ yinj j.
-#L #d #i #j #Hji #H elim (ylt_split j d) #Hjd
-[ elim H -H /2 width=6 by cofrees_lref_skip/
-| elim (lt_or_ge j (|L|)) #Hj
- [ elim (ldrop_O1_lt … Hj) -Hj /4 width=10 by cofrees_lref_gt, ex3_3_intro/
- | elim H -H /2 width=6 by cofrees_lref_free/
- ]
-]
-qed-.
-
-lemma frees_inv_lref_free: ∀L,d,i,j. (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
-#L #d #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
-#Hij elim H -H /2 width=6 by cofrees_lref_lt, cofrees_lref_free/
-qed-.
-
-lemma frees_inv_gen: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
- ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH #k #_ #d #i #H elim H -H //
-| #j #Hn #d #i #H elim (lt_or_eq_or_gt i j)
- [ -n #Hij elim H -H /2 width=5 by cofrees_lref_lt/
- | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
- | #Hji elim (frees_inv_lref_gt … H) // -H
- #I #K #W1 #HLK #H #Hdj elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
- #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
- #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
- #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
- >minus_plus <plus_minus_m_m //
- ]
-| -IH #p #_ #d #i #H elim H -H //
-| #a #I #W #U #Hn #d #i #H elim (frees_inv_bind … H) -H
- #H elim (IH … H) // -H -n
- /4 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn, ex2_intro/
-| #I #W #U #Hn #d #i #H elim (frees_inv_flat … H) -H
- #H elim (IH … H) // -H -n
- /4 width=9 by cpys_flat, nlift_flat_dx, nlift_flat_sn, ex2_intro/
-]
-qed-.
-
-lemma frees_ind: ∀L,d,i. ∀R:predicate term.
- (∀U1. (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1) →
- (∀U1,U2. ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (L ⊢ i ~ϵ 𝐅*[d]⦃U2⦄ → ⊥) → R U2 → R U1) →
- ∀U. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U.
-#L #d #i #R #IH1 #IH2 #U1 #H elim (frees_inv_gen … H) -H
-#U2 #H #HnU2 @(cpys_ind_dx … H) -U1 /4 width=8 by cofrees_inv_gen/
-qed-.
-
-(* Advanced negated properties **********************************************)
-
-lemma frees_be: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
- ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) →
- ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
- ∀d. d ≤ yinj j → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
-/4 width=11 by cofrees_inv_be/ qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma cofrees_lift_be: ∀d0,e0,i. d0 ≤ i → i ≤ d0 + e0 →
- ∀L,K,s. ⇩[s, d0, e0+1] L ≡ K → ∀T,U. ⇧[d0, e0+1] T ≡ U →
- ∀d. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#d0 #e0 #i #Hd0i #Hide0 #L #K #s #HLK #T1 #U1 #HTU1 #d #U2 #HU12
-elim (yle_split d0 d) #H1
-[ elim (yle_split d (d0+e0+1)) #H2
- [ letin cpys_inv ≝ cpys_inv_lift1_ge_up
- | letin cpys_inv ≝ cpys_inv_lift1_ge
- ]
-| letin cpys_inv ≝ cpys_inv_lift1_be
-]
-elim (cpys_inv … HU12 … HLK … HTU1) // #T2 #_ #HTU2 -s -L -K -U1 -T1 -d
-elim (lift_split … HTU2 i e0) /2 width=2 by ex_intro/
-qed.
-
-lemma cofrees_lift_ge: ∀d0,e0,i. d0 + e0 ≤ i →
- ∀L,K,s. ⇩[s, d0, e0] L ≡ K → ∀T,U. ⇧[d0, e0] T ≡ U →
- ∀d. K ⊢ i-e0 ~ϵ 𝐅*[d-yinj e0]⦃T⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
-#d0 #e0 #i #Hde0i #L #K #s #HLK #T1 #U1 #HTU1 #d #HT1 #U2 #HU12
-elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i
-elim (yle_split d0 d) #H1
-[ elim (yle_split d (d0+e0)) #H2
- [ elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) // >yplus_inj >yminus_Y_inj #T2 #HT12
- lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12
- | elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) // #T2
- ]
-| elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) // >yminus_Y_inj #T2 #HT12
- lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) // -HT12
-]
--s -L #HT12 #HTU2
-elim (HT1 … HT12) -T1 #V2 #HVT2
-elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L ⊢ break term 46 i ~ ϵ 𝐅 * [ break term 46 d ] ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'CoFreeStar $L $i $d $T }.
--- /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/substitution/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/multiple/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/substitution/cpy_cpy.ma".
+include "basic_2/multiple/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/substitution/cpy_lift.ma".
+include "basic_2/multiple/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_Y2: ∀G,L,T2,i,d. ⦃G, L⦄ ⊢ #i ▶*[d, ∞] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V1,V2. d ≤ i & ⇩[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #d #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/
+* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
+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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lazyeq_7.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR CLOSURES ********************************************)
+
+inductive fleq (d) (G) (L1) (T): relation3 genv lenv term ≝
+| fleq_intro: ∀L2. L1 ≡[T, d] L2 → fleq d G L1 T G L2 T
+.
+
+interpretation
+ "lazy equivalence (closure)"
+ 'LazyEq d G1 L1 T1 G2 L2 T2 = (fleq d G1 L1 T1 G2 L2 T2).
+
+(* Basic_properties *********************************************************)
+
+lemma fleq_refl: ∀d. tri_reflexive … (fleq d).
+/2 width=1 by fleq_intro/ qed.
+
+lemma fleq_sym: ∀d. tri_symmetric … (fleq d).
+#d #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fleq_inv_gen: ∀G1,G2,L1,L2,T1,T2,d. ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L1 ≡[T1, d] L2 & T1 = T2.
+#G1 #G2 #L1 #L2 #T1 #T2 #d * -G2 -L2 -T2 /2 width=1 by and3_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/multiple/lleq_lleq.ma".
+include "basic_2/multiple/fleq.ma".
+
+(* LAZY EQUIVALENCE FOR CLOSURES *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem fleq_trans: ∀d. tri_transitive … (fleq d).
+#d #G1 #G #L1 #L #T1 #T * -G -L -T
+#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/3 width=3 by lleq_trans, fleq_intro/
+qed-.
+
+theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,d.
+ ⦃G, L, T⦄ ≡[d] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[d] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ qed-.
+
+theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,d.
+ ⦃G1, L1, T1⦄ ≡[d] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[d] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ 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/suptermplus_6.ma".
+include "basic_2/substitution/fqu.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu.
+
+interpretation "plus-iterated structural successor (closure)"
+ 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fqup_ldrop: ∀G1,G2,L1,K1,K2,T1,T2,U1,e. ⇩[e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
+ ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #e #HLK1 #HTU1 #HT12 elim (eq_or_gt … e) #H destruct
+[ >(ldrop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 //
+| /3 width=5 by fqup_strap2, fqu_drop_lt/
+]
+qed-.
+
+lemma fqup_lref: ∀I,G,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
+/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_ldrop/ qed.
+
+lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄.
+/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
+
+lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊐+ ⦃G, L.ⓑ{I}V, T⦄.
+/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
+
+lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+ ⦃G, L, T⦄.
+/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
+
+lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+ ⦃G, L, V2⦄.
+/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
+
+lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I1}V1, T⦄.
+/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
+
+lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I2}V2, T⦄.
+/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 ….
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
+qed-.
+
+lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
+ (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=3 by fqu_fwd_fw, transitive_lt/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqup_wf_ind: ∀R:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
+qed-.
+
+lemma fqup_wf_ind_eq: ∀R:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
+ ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/
+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/multiple/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqup_trans: tri_transitive … fqup.
+/2 width=5 by tri_TC_transitive/ 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/suptermstar_6.ma".
+include "basic_2/substitution/fquq.ma".
+include "basic_2/multiple/fqup.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
+
+interpretation "star-iterated structural successor (closure)"
+ 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
+qed-.
+
+lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fqus_refl: tri_reflexive … fqus.
+/2 width=1 by tri_inj/ qed.
+
+lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
+
+lemma fqus_ldrop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ →
+ ∀L1,U1,e. ⇩[e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
+ ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄.
+#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
+qed-.
+
+lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
+/3 width=3 by fquq_fwd_fw, transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
+#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_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/substitution/fquq_alt.ma".
+include "basic_2/multiple/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
+#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2
+[ /3 width=5 by fqup_strap1, or_introl/
+| * #HG #HL #HT destruct /2 width=1 by or_introl/
+| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/
+| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1
+[ /2 width=5 by fqup_strap1/
+| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2
+[ /2 width=5 by fqup_strap2/
+| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
+/2 width=5 by fqus_strap1_fqu, fqup_strap1/
+qed-.
+
+lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
+/3 width=5 by fqus_strap2_fqu, fqup_strap2/
+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/multiple/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqus_trans: tri_transitive … fqus.
+/2 width=5 by tri_TC_transitive/ 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/freestar_4.ma".
+include "basic_2/substitution/lift_neg.ma".
+include "basic_2/substitution/ldrop.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+inductive frees: relation4 ynat lenv term nat ≝
+| frees_eq: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → frees d L U i
+| frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
+ (∀T. ⇧[j, 1] T ≡ U → ⊥) → ⇩[j]L ≡ K.ⓑ{I}W →
+ frees 0 K W (i-j-1) → frees d L U i.
+
+interpretation
+ "context-sensitive free variables (term)"
+ 'FreeStar L i d U = (frees d L U i).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+ (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
+ ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⇧[j, 1] T ≡ U → ⊥) &
+ ⇩[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
+#L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
+#L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
+ j = i ∨
+ ∃∃I,K,W. d ≤ yinj j & j < i & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #x #i #H elim (frees_inv … H) -H
+[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
+| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
+ >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
+#L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
+#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
+#L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
+* #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
+qed-.
+
+lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
+#L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
+ ∃∃I,K,W. d ≤ yinj j & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
+[ #H elim (lt_refl_false j) //
+| * /2 width=5 by ex3_3_intro/
+]
+qed-.
+
+lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
+#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_bind … HnX) -HnX
+ /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
+ [ /4 width=9 by frees_be, or_introl/
+ | #HnT @or_intror @(frees_be … HnT) -HnT
+ [4,5,6: /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/
+ |7: >minus_plus_plus_l //
+ |*: skip
+ ]
+ ]
+]
+qed-.
+
+lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
+#J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_flat … HnX) -HnX
+ /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
+ /4 width=9 by frees_be, or_intror, or_introl/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
+/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
+
+lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W →
+ K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
+/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
+
+lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
+qed.
+
+lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
+#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ /4 width=9 by frees_eq, nlift_bind_dx/
+| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
+ elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
+ lapply (ylt_O … H) -H #Hj
+ >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
+ /4 width=9 by frees_be, nlift_bind_dx, ldrop_inv_drop1_lt, lt_plus_to_minus/
+]
+qed.
+
+lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
+qed.
+
+lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+ L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
+qed.
+
+lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
+ ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
+#L #U #d1 #i #H elim H -L -U -d1 -i
+/3 width=9 by frees_be, frees_eq, yle_trans/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ .
+#a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
+/3 width=3 by frees_weak, or_intror, or_introl/
+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/multiple/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma frees_dec: ∀L,U,d,i. Decidable (frees d L U i).
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH /3 width=5 by frees_inv_sort, or_intror/
+| #j #Hn #d #i elim (lt_or_eq_or_gt i j) #Hji
+ [ -n @or_intror #H elim (lt_refl_false i)
+ lapply (frees_inv_lref_ge … H ?) -L -d /2 width=1 by lt_to_le/
+ | -n /2 width=1 by or_introl/
+ | elim (ylt_split j d) #Hdi
+ [ -n @or_intror #H elim (lt_refl_false i)
+ lapply (frees_inv_lref_skip … H ?) -L //
+ | elim (lt_or_ge j (|L|)) #Hj
+ [ elim (ldrop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
+ elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, ldrop_fwd_rfw, or_introl/ ] #HnW
+ @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -d
+ lapply (ldrop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
+ | -n @or_intror #H elim (lt_refl_false i)
+ lapply (frees_inv_lref_free … H ?) -d //
+ ]
+ ]
+ ]
+| -IH /3 width=5 by frees_inv_gref, or_intror/
+| #a #I #W #U #Hn #d #i destruct
+ elim (IH L W … d i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
+ elim (IH (L.ⓑ{I}W) U … (⫯d) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+ @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
+| #I #W #U #Hn #d #i destruct
+ elim (IH L W … d i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
+ elim (IH L U … d i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
+ @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
+]
+qed-.
+
+lemma frees_S: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[yinj d]⦃U⦄ → ∀I,K,W. ⇩[d] L ≡ K.ⓑ{I}W →
+ (K ⊢ i-d-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯d]⦃U⦄.
+#L #U #d #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
+* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
+lapply (yle_inv_inj … Hdj) -Hdj #Hdj
+elim (le_to_or_lt_eq … Hdj) -Hdj
+[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
+| -Hji -HnU #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct -I
+ elim HnW0 -L -U -HnW0 //
+]
+qed.
+
+(* Note: lemma 1250 *)
+lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
+ L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
+/4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
+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/rat_3.ma".
+include "basic_2/grammar/term_vector.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+inductive at: list2 nat nat → relation nat ≝
+| at_nil: ∀i. at (⟠) i i
+| at_lt : ∀des,d,e,i1,i2. i1 < d →
+ at des i1 i2 → at ({d, e} @ des) i1 i2
+| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
+ at des (i1 + e) i2 → at ({d, e} @ des) i1 i2
+.
+
+interpretation "application (generic relocation with pairs)"
+ 'RAt i1 des i2 = (at des i1 i2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2.
+#des #i1 #i2 * -des -i1 -i2
+[ //
+| #des #d #e #i1 #i2 #_ #_ #H destruct
+| #des #d #e #i1 #i2 #_ #_ #H destruct
+]
+qed-.
+
+lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2.
+/2 width=3 by at_inv_nil_aux/ qed-.
+
+fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
+ ∀d,e,des0. des = {d, e} @ des0 →
+ i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨
+ d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2.
+#des #i1 #i2 * -des -i1 -i2
+[ #i #d #e #des #H destruct
+| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
+| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_intror, conj/
+]
+qed-.
+
+lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨
+ d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2.
+/2 width=3 by at_inv_cons_aux/ qed-.
+
+lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ i1 < d → @⦃i1, des⦄ ≡ i2.
+#des #d #e #i1 #e2 #H
+elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
+lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
+elim (lt_refl_false … Hd)
+qed-.
+
+lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2.
+#des #d #e #i1 #e2 #H
+elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
+lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
+elim (lt_refl_false … Hd)
+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/multiple/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+(* Main properties **********************************************************)
+
+theorem at_mono: ∀des,i,i1. @⦃i, des⦄ ≡ i1 → ∀i2. @⦃i, des⦄ ≡ i2 → i1 = i2.
+#des #i #i1 #H elim H -des -i -i1
+[ #i #x #H <(at_inv_nil … H) -x //
+| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H
+ lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1 by/
+| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H
+ lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1 by/
+]
+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/rminus_3.ma".
+include "basic_2/multiple/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+inductive minuss: nat → relation (list2 nat nat) ≝
+| minuss_nil: ∀i. minuss i (⟠) (⟠)
+| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
+ minuss i ({d, e} @ des1) ({d - i, e} @ des2)
+| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
+ minuss i ({d, e} @ des1) des2
+.
+
+interpretation "minus (generic relocation with pairs)"
+ 'RMinus des1 i des2 = (minuss i des1 des2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠.
+#des1 #des2 #i * -des1 -des2 -i
+[ //
+| #des1 #des2 #d #e #i #_ #_ #H destruct
+| #des1 #des2 #d #e #i #_ #_ #H destruct
+]
+qed-.
+
+lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
+/2 width=4 by minuss_inv_nil1_aux/ qed-.
+
+fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
+ ∀d,e,des. des1 = {d, e} @ des →
+ d ≤ i ∧ des ▭ e + i ≡ des2 ∨
+ ∃∃des0. i < d & des ▭ i ≡ des0 &
+ des2 = {d - i, e} @ des0.
+#des1 #des2 #i * -des1 -des2 -i
+[ #i #d #e #des #H destruct
+| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
+]
+qed-.
+
+lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
+ d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
+ ∃∃des. i < d & des1 ▭ i ≡ des &
+ des2 = {d - i, e} @ des.
+/2 width=3 by minuss_inv_cons1_aux/ qed-.
+
+lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
+ d ≤ i → des1 ▭ e + i ≡ des2.
+#des1 #des2 #d #e #i #H
+elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi
+lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+elim (lt_refl_false … Hi)
+qed-.
+
+lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
+ i < d →
+ ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des.
+#des1 #des2 #d #e #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
+#Hdi #_ #Hid lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi
+#Hi elim (lt_refl_false … Hi)
+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/multiple/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with
+[ nil2 ⇒ ⟠
+| cons2 d e des ⇒ {d + i, e} @ pluss des i
+].
+
+interpretation "plus (generic relocation with pairs)"
+ 'plus x y = (pluss x y).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠.
+#i * // normalize
+#d #e #des #H destruct
+qed.
+
+lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} @ des2 →
+ ∃∃des1. des1 + i = des2 & des = {d - i, e} @ des1.
+#i #d #e #des2 * normalize
+[ #H destruct
+| #d1 #e1 #des1 #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/rdropstar_3.ma".
+include "basic_2/notation/relations/rdropstar_4.ma".
+include "basic_2/substitution/ldrop.ma".
+include "basic_2/multiple/gr2_minus.ma".
+include "basic_2/multiple/lifts.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+inductive ldrops (s:bool): list2 nat nat → relation lenv ≝
+| ldrops_nil : ∀L. ldrops s (⟠) L L
+| ldrops_cons: ∀L1,L,L2,des,d,e.
+ ldrops s des L1 L → ⇩[s, d, e] L ≡ L2 → ldrops s ({d, e} @ des) L1 L2
+.
+
+interpretation "iterated slicing (local environment) abstract"
+ 'RDropStar s des T1 T2 = (ldrops s des T1 T2).
+(*
+interpretation "iterated slicing (local environment) general"
+ 'RDropStar des T1 T2 = (ldrops true des T1 T2).
+*)
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrops_inv_nil_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → des = ⟠ → L1 = L2.
+#L1 #L2 #s #des * -L1 -L2 -des //
+#L1 #L #L2 #d #e #des #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: drop1_gen_pnil *)
+lemma ldrops_inv_nil: ∀L1,L2,s. ⇩*[s, ⟠] L1 ≡ L2 → L1 = L2.
+/2 width=4 by ldrops_inv_nil_aux/ qed-.
+
+fact ldrops_inv_cons_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 →
+ ∀d,e,tl. des = {d, e} @ tl →
+ ∃∃L. ⇩*[s, tl] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
+#L1 #L2 #s #des * -L1 -L2 -des
+[ #L #d #e #tl #H destruct
+| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+ /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop1_gen_pcons *)
+lemma ldrops_inv_cons: ∀L1,L2,s,d,e,des. ⇩*[s, {d, e} @ des] L1 ≡ L2 →
+ ∃∃L. ⇩*[s, des] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
+/2 width=3 by ldrops_inv_cons_aux/ qed-.
+
+lemma ldrops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
+ ∀L1,K2,V2. ⇩*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
+ ⇩*[s, des1] K1 ≡ K2 &
+ ⇧*[des1] V2 ≡ V1 &
+ L1 = K1. ⓑ{I} V1.
+#I #s #des #des2 #i #H elim H -des -des2 -i
+[ #i #L1 #K2 #V2 #H
+ >(ldrops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, ldrops_nil/
+| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+ elim (ldrops_inv_cons … H) -H #L #HL1 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
+ elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, ldrops_cons/ | skip ]
+ normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
+| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+ elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: drop1_skip_bind *)
+lemma ldrops_skip: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
+ ∀I. ⇩*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #s #des #H elim H -L1 -L2 -des
+[ #L #V1 #V2 #HV12 #I
+ >(lifts_inv_nil … HV12) -HV12 //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
+ elim (lifts_inv_cons … H) -H /3 width=5 by ldrop_skip, ldrops_cons/
+].
+qed.
+
+(* Basic_1: removed theorems 1: drop1_getl_trans *)
--- /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/multiple/ldrops.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[Ⓕ, des] L1 ≡ L → ∀L2,i. ⇩[i] L ≡ L2 →
+ ∃∃L0,des0,i0. ⇩[i0] L1 ≡ L0 & ⇩*[Ⓕ, des0] L0 ≡ L2 &
+ @⦃i, des⦄ ≡ i0 & des ▭ i ≡ des0.
+#L1 #L #des #H elim H -L1 -L -des
+[ /2 width=7 by ldrops_nil, minuss_nil, at_nil, ex4_3_intro/
+| #L1 #L0 #L #des #d #e #_ #HL0 #IHL0 #L2 #i #HL2
+ elim (lt_or_ge i d) #Hid
+ [ elim (ldrop_trans_le … HL0 … HL2) -L /2 width=2 by lt_to_le/
+ #L #HL0 #HL2 elim (IHL0 … HL0) -L0 /3 width=7 by ldrops_cons, minuss_lt, at_lt, ex4_3_intro/
+ | lapply (ldrop_trans_ge … HL0 … HL2 ?) -L // #HL02
+ elim (IHL0 … HL02) -L0 /3 width=7 by minuss_ge, at_ge, ex4_3_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/multiple/ldrops_ldrop.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop1_trans *)
+theorem ldrops_trans: ∀L,L2,s,des2. ⇩*[s, des2] L ≡ L2 → ∀L1,des1. ⇩*[s, des1] L1 ≡ L →
+ ⇩*[s, des2 @@ des1] L1 ≡ L2.
+#L #L2 #s #des2 #H elim H -L -L2 -des2 /3 width=3 by ldrops_cons/
+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/rliftstar_3.ma".
+include "basic_2/substitution/lift.ma".
+include "basic_2/multiple/gr2_plus.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+inductive lifts: list2 nat nat → relation term ≝
+| lifts_nil : ∀T. lifts (⟠) T T
+| lifts_cons: ∀T1,T,T2,des,d,e.
+ ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} @ des) T1 T2
+.
+
+interpretation "generic relocation (term)"
+ 'RLiftStar des T1 T2 = (lifts des T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2.
+#T1 #T2 #des * -T1 -T2 -des //
+#T1 #T #T2 #d #e #des #_ #_ #H destruct
+qed-.
+
+lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
+/2 width=3 by lifts_inv_nil_aux/ qed-.
+
+fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
+ ∀d,e,tl. des = {d, e} @ tl →
+ ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2.
+#T1 #T2 #des * -T1 -T2 -des
+[ #T #d #e #tl #H destruct
+| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+ /2 width=3 by ex2_intro/
+qed-.
+
+lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} @ des] T1 ≡ T2 →
+ ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2.
+/2 width=3 by lifts_inv_cons_aux/ qed-.
+
+(* Basic_1: was: lift1_sort *)
+lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_sort1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 →
+ ∃∃i2. @⦃i1, des⦄ ≡ i2 & T2 = #i2.
+#T2 #des elim des -des
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
+| #d #e #des #IH #i1 #H
+ elim (lifts_inv_cons … H) -H #X #H1 #H2
+ elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
+ elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
+]
+qed-.
+
+lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p.
+#T2 #p #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_gref1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+lemma lifts_inv_bind1: ∀a,I,T2,des,V1,U1. ⇧*[des] ⓑ{a,I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 &
+ T2 = ⓑ{a,I} V2. U2.
+#a #I #T2 #des elim des -des
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #d #e #des #IHdes #V1 #U1 #H
+ elim (lifts_inv_cons … H) -H #X #H #HT2
+ elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
+ elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+#I #T2 #des elim des -des
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #d #e #des #IHdes #V1 #U1 #H
+ elim (lifts_inv_cons … H) -H #X #H #HT2
+ elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
+ elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_dx/
+qed-.
+
+lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_bind: ∀a,I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+ ∀T1. ⇧*[des + 1] T1 ≡ T2 →
+ ⇧*[des] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/
+]
+qed.
+
+lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+ ∀T1. ⇧*[des] T1 ≡ T2 →
+ ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3 by lift_flat, lifts_cons/
+]
+qed.
+
+lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
+#des elim des -des /2 width=2 by lifts_nil, ex_intro/
+#d #e #des #IH #T1 elim (lift_total T1 d e)
+#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_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/substitution/lift_lift.ma".
+include "basic_2/multiple/gr2_minus.ma".
+include "basic_2/multiple/lifts.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+(* Properties concerning basic term relocation ******************************)
+
+(* Basic_1: was: lift1_xhg (right to left) *)
+lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 →
+ ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2.
+#T1 #T #des #H elim H -T1 -T -des
+[ /2 width=3/
+| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2
+ elim (IHT13 … HT2) -T #T #HT3 #HT2
+ elim (lift_trans_le … HT13 … HT3) -T3 // /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: lift1_free (right to left) *)
+lemma lifts_lift_trans: ∀des,i,i0. @⦃i, des⦄ ≡ i0 →
+ ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 →
+ ∀T1,T0. ⇧*[des0] T1 ≡ T0 →
+ ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 →
+ ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2.
+#des elim des -des normalize
+[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2
+ <(at_inv_nil … H1) -x #HT02
+ lapply (minuss_inv_nil1 … H2) -H2 #H
+ >(pluss_inv_nil2 … H) in HT10; -des0 #H
+ >(lifts_inv_nil … H) -T1 /2 width=3/
+| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
+ elim (at_inv_cons … H1) -H1 * #Hid #Hi0
+ [ elim (minuss_inv_cons1_lt … H2) -H2 [2: /2 width=1/ ] #des1 #Hdes1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
+ elim (pluss_inv_cons2 … H) -H #des2 #H1 #H2 destruct
+ elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
+ elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02
+ elim (lift_trans_le … HT1 … HT0) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
+ | >commutative_plus in Hi0; #Hi0
+ lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hdes0
+ elim (IHdes … Hi0 … Hdes0 … HT10 … HT02) -IHdes -Hi0 -Hdes0 -T0 #T0 #HT0 #HT02
+ elim (lift_split … HT0 d (i+1)) -HT0 /2 width=1/ /3 width=5/
+ ]
+]
+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/lift_lift_vector.ma".
+include "basic_2/multiple/lifts_lift.ma".
+include "basic_2/multiple/lifts_vector.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lifts1_xhg (right to left) *)
+lemma liftsv_liftv_trans_le: ∀T1s,Ts,des. ⇧*[des] T1s ≡ Ts →
+ ∀T2s:list term. ⇧[0, 1] Ts ≡ T2s →
+ ∃∃T0s. ⇧[0, 1] T1s ≡ T0s & ⇧*[des + 1] T0s ≡ T2s.
+#T1s #Ts #des #H elim H -T1s -Ts
+[ #T1s #H
+ >(liftv_inv_nil1 … H) -T1s /2 width=3/
+| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H
+ elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct
+ elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s
+ elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/
+]
+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/multiple/lifts_lift.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lift1_lift1 (left to right) *)
+theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 →
+ ⇧*[des1 @@ des2] T1 ≡ T2.
+#T1 #T #des1 #H elim H -T1 -T -des1 // /3 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/substitution/lift_vector.ma".
+include "basic_2/multiple/lifts.ma".
+
+(* GENERIC TERM VECTOR RELOCATION *******************************************)
+
+inductive liftsv (des:list2 nat nat) : relation (list term) ≝
+| liftsv_nil : liftsv des (◊) (◊)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+ ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s →
+ liftsv des (T1 @ T1s) (T2 @ T2s)
+.
+
+interpretation "generic relocation (vector)"
+ 'RLiftStar des T1s T2s = (liftsv des T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: lifts1_flat (left to right) *)
+lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 →
+ ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 &
+ T2 = Ⓐ V2s. U2.
+#V1s elim V1s -V1s normalize
+[ #T1 #T2 #des #HT12
+ @ex3_2_intro [3,4: // |1,2: skip | // ] (**) (* explicit constructor *)
+| #V1 #V1s #IHV1s #T1 #X #des #H
+ elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
+ elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
+ @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: lifts1_flat (right to left) *)
+lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 →
+ ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
+#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_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/notation/relations/lazyeq_4.ma".
+include "basic_2/multiple/llpx_sn.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2.
+
+definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
+
+interpretation
+ "lazy equivalence (local environment)"
+ 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+
+definition lleq_transitive: predicate (relation3 lenv term term) ≝
+ λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
+ ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
+ ) → (
+ ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
+ ) → (
+ ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
+ ) → (
+ ∀a,I,L1,L2,V,T,d.
+ L1 ≡[V, d]L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
+ R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
+ ) → (
+ ∀I,L1,L2,V,T,d.
+ L1 ≡[V, d]L2 → L1 ≡[T, d] L2 →
+ R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
+ ) →
+ ∀d,T,L1,L2. L1 ≡[T, d] L2 → R d T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim H -L1 -L2 -T -d /2 width=8 by/
+qed-.
+
+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] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind/ 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.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ≡[T, d] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V &
+ K1 ≡[V, yinj 0] K2 & d ≤ yinj i.
+#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref … H) /2 width=1/
+* /3 width=7 by or3_intro2, ex4_4_intro/
+qed-.
+
+lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
+ ∃K2. ⇩[i] L2 ≡ K2.
+/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-.
+
+lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
+ ∃K1. ⇩[i] L1 ≡ K1.
+/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-.
+
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
+ L1 ≡[ⓑ{a,I}V.T, d] L2 → L1 ≡[V, d] L2.
+/2 width=4 by llpx_sn_fwd_bind_sn/ 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] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
+ L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[V, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
+ L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[T, d] L2.
+/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ≡[⋆k, d] L2.
+/2 width=1 by llpx_sn_sort/ qed.
+
+lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ≡[#i, d] L2.
+/2 width=1 by llpx_sn_skip/ qed.
+
+lemma lleq_lref: ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, 0] K2 → L1 ≡[#i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, d] L2.
+/2 width=1 by llpx_sn_free/ qed.
+
+lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ≡[§p, d] L2.
+/2 width=1 by llpx_sn_gref/ qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
+ L1 ≡[V, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, d] L2.
+/2 width=1 by llpx_sn_bind/ qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,d.
+ L1 ≡[V, d] L2 → L1 ≡[T, d] L2 → L1 ≡[ⓕ{I}V.T, d] L2.
+/2 width=1 by llpx_sn_flat/ qed.
+
+lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
+/2 width=1 by llpx_sn_Y/ qed.
+
+lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
+#d #T #L1 #L2 #H @(lleq_ind … H) -d -T -L1 -L2
+/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
+qed-.
+
+lemma lleq_ge_up: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
+ ∀T,d,e. ⇧[d, e] T ≡ U →
+ dt ≤ d + e → L1 ≡[U, d] L2.
+/2 width=6 by llpx_sn_ge_up/ qed-.
+
+lemma lleq_ge: ∀L1,L2,T,d1. L1 ≡[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ≡[T, d2] L2.
+/2 width=3 by llpx_sn_ge/ qed-.
+
+lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, 0] L2.
+/2 width=1 by llpx_sn_bind_O/ qed-.
+
+(* Advancded properties on lazy pointwise exyensions ************************)
+
+lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T,d. L1 ≡[T, d] L2 → llpx_sn R d T L1 L2.
+/2 width=3 by llpx_sn_co/ 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/multiple/llpx_sn_alt.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Alternative definition (not recursive) ***********************************)
+
+theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ V1 = V2
+ ) → L1 ≡[T, d] L2.
+#L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
+qed.
+
+theorem lleq_inv_alt: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ V1 = V2.
+#L1 #L2 #T #d #H elim (llpx_sn_llpx_sn_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
+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/multiple/llpx_sn_alt_rec.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Alternative definition (recursive) ***************************************)
+
+theorem lleq_intro_alt_r: ∀L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2
+ ) → L1 ≡[T, d] L2.
+#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt_r // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed.
+
+theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. L1 ≡[T, d] L2 → S d T L1 L2.
+#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
+qed-.
+
+theorem lleq_inv_alt_r: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
+#L1 #L2 #T #d #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_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/multiple/fqus_alt.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≡[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
+ #K1 #H1 #H2 lapply (ldrop_inv_O2 … H1) -H1
+ #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+ [ elim (lleq_inv_bind … H)
+ | elim (lleq_inv_flat … H)
+ ] -H
+ /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
+ #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
+ /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
+ elim (ldrop_O1_le (Ⓕ) (e+1) L1)
+ [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+ | lapply (ldrop_fwd_length_le2 … HLK2) -K2
+ lapply (lleq_fwd_length … HL12) -T -U //
+ ]
+]
+qed-.
+
+lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≡[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
+[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≡[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
+ /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
+ #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
+ /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≡[T, 0] L2 →
+ ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
+[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT 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/multiple/llpx_sn_ldrop.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+ ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W.
+/2 width=7 by llpx_sn_bind_repl_O/ qed-.
+
+lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ≡[T, d] L2).
+/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
+
+lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
+ ∀L1,L2,T,d. L1 ≡[T, d] L2 →
+ ∀L. llpx_sn R d T L2 L → llpx_sn R d T L1 L.
+#R #HR #L1 #L2 #T #d #H @(lleq_ind … H) -L1 -L2 -T -d
+[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
+|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
+| #I #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
+ /3 width=11 by llpx_sn_lref/
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
+ /3 width=1 by llpx_sn_bind/
+| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
+ ∀L1,L2,T,d. L1 ≡[T, d] L2 →
+ ∀L. llpx_sn R d T L1 L → llpx_sn R d T L2 L.
+/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
+ ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
+ ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
+ ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_bi: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
+ ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
+elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
+qed-.
+
+lemma lleq_inv_S: ∀L1,L2,T,d. L1 ≡[T, d+1] L2 →
+ ∀I,K1,K2,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, 0] K2 → L1 ≡[T, d] L2.
+/2 width=9 by llpx_sn_inv_S/ qed-.
+
+lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
+ L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
+ ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ i < d ∨
+ ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
+ ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
+ i < d ∨
+ ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & d ≤ i.
+#L1 #L2 #d #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
+ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ≡[T, dt] K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ≡[U, dt] L2.
+/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
+
+lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ≡[T, dt] K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ≡[U, dt+e] L2.
+/2 width=9 by llpx_sn_lift_ge/ qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ≡[T, dt] K2.
+/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
+
+lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ≡[T, d] K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ≡[T, dt-e] K2.
+/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_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] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
+
+lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ≡[ⓕ{I}V.T, d] L2 → ⊥) →
+ (L1 ≡[V, d] L2 → ⊥) ∨ (L1 ≡[T, d] L2 → ⊥).
+/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
+
+lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
+ (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ 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/multiple/llpx_sn_leq.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma leq_lleq_trans: ∀L2,L,T,d. L2 ≡[T, d] L →
+ ∀L1. L1 ≃[d, ∞] L2 → L1 ≡[T, d] L.
+/2 width=3 by leq_llpx_sn_trans/ qed-.
+
+lemma lleq_leq_trans: ∀L,L1,T,d. L ≡[T, d] L1 →
+ ∀L2. L1 ≃[d, ∞] L2 → L ≡[T, d] L2.
+/2 width=3 by llpx_sn_leq_trans/ qed-.
+
+lemma lleq_leq_repl: ∀L1,L2,T,d. L1 ≡[T, d] L2 → ∀K1. K1 ≃[d, ∞] L1 →
+ ∀K2. L2 ≃[d, ∞] K2 → K1 ≡[T, d] K2.
+/2 width=5 by llpx_sn_leq_repl/ qed-.
+
+lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 →
+ ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2.
+/2 width=5 by llpx_sn_bind_repl_SO/ 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/multiple/lleq_ldrop.ma".
+
+(* Main properties **********************************************************)
+
+theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
+/2 width=3 by lleq_llpx_sn_trans/ qed-.
+
+theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ≡[d, T] L1→ L ≡[d, T] L2 → L1 ≡[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ≡[d, T] L → L2 ≡[d, T] L → L1 ≡[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1≡[T, d] L →
+ ∀L2. (L ≡[T, d] L2 → ⊥) → (L1 ≡[T, d] L2 → ⊥).
+/3 width=3 by lleq_canc_sn/ qed-.
+works with /4 width=8/ so lleq_canc_sn is more convenient
+*)
--- /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/multiple/llor.ma".
+include "basic_2/multiple/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on poinwise union for local environments **********************)
+
+lemma llpx_sn_llor_dx: ∀R,L1,L2.
+ (∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄) →
+ ∀T. llpx_sn R 0 T L1 L2 → ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
+#R #L1 #L2 #HR #T #H1 #L #H2
+elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
+elim H2 -H2 #_ #HL1 #IH2
+@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
+lapply (ldrop_fwd_length_lt2 … HLK) #HiL
+elim (ldrop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
+elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
+elim H -H /2 width=1 by/
+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/lazyor_4.ma".
+include "basic_2/multiple/frees.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition llor: relation4 term lenv lenv lenv ≝ λT,L2,L1,L.
+ ∧∧ |L1| ≤ |L2| & |L1| = |L|
+ & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → ⇩[i] L ≡ K.ⓑ{I}V →
+ (∧∧ (L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄ → ⊥) & I1 = I & V1 = V) ∨
+ (∧∧ L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄ & I1 = I & V2 = V)
+ ).
+
+interpretation
+ "lazy union (local environment)"
+ 'LazyOr L1 T L2 L = (llor T L2 L1 L).
+
+(* Basic properties *********************************************************)
+
+lemma llor_atom: ∀T,L2. ⋆ ⩖[T] L2 ≡ ⋆.
+#T #L2 @and3_intro //
+#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
+elim (ldrop_inv_atom1 … HLK1) -HLK1 #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/multiple/frees_lift.ma".
+include "basic_2/multiple/llor.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Advanced properties ******************************************************)
+
+axiom llor_total: ∀L1,L2,T. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
--- /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/substitution/ldrop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_sort: ∀L1,L2,d,k. |L1| = |L2| → llpx_sn R d (⋆k) L1 L2
+| llpx_sn_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn R d (#i) L1 L2
+| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R d (#i) L1 L2
+| llpx_sn_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R d (#i) L1 L2
+| llpx_sn_gref: ∀L1,L2,d,p. |L1| = |L2| → llpx_sn R d (§p) L1 L2
+| llpx_sn_bind: ∀a,I,L1,L2,V,T,d.
+ llpx_sn R d V L1 L2 → llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn R d (ⓑ{a,I}V.T) L1 L2
+| llpx_sn_flat: ∀I,L1,L2,V,T,d.
+ llpx_sn R d V L1 L2 → llpx_sn R d T L1 L2 → llpx_sn R d (ⓕ{I}V.T) L1 L2
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
+ ∀a,I,V,T. X = ⓑ{a,I}V.T →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #b #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #d #p #_ #b #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #d #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
+| #I #L1 #L2 #V #T #d #_ #_ #b #J #W #U #H destruct
+]
+qed-.
+
+lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
+
+fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
+ ∀I,V,T. X = ⓕ{I}V.T →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #d #i #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #d #p #_ #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #d #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #V #T #d #HV #HT #J #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
+/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_fwd_length: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d //
+#I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #HLK1 #HLK2 #_ #_ #HK12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2
+normalize //
+qed-.
+
+lemma llpx_sn_fwd_ldrop_sn: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ ∀K1,i. ⇩[i] L1 ≡ K1 → ∃K2. ⇩[i] L2 ≡ K2.
+#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
+qed-.
+
+lemma llpx_sn_fwd_ldrop_dx: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ ∀K2,i. ⇩[i] L2 ≡ K2 → ∃K1. ⇩[i] L1 ≡ K1.
+#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by ldrop_O1_le/
+qed-.
+
+fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 → ∀i. X = #i →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R (yinj 0) V1 K1 K2 &
+ R K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #X #d * -L1 -L2 -X -d
+[ #L1 #L2 #d #k #_ #j #H destruct
+| #L1 #L2 #d #i #_ #Hid #j #H destruct /2 width=1 by or3_intro1/
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
+ /3 width=9 by or3_intro2, ex5_5_intro/
+| #L1 #L2 #d #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
+| #L1 #L2 #d #p #_ #j #H destruct
+| #a #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
+| #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
+]
+qed-.
+
+lemma llpx_sn_fwd_lref: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R (yinj 0) V1 K1 K2 &
+ R K1 V1 V2 & d ≤ yinj i.
+/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
+
+lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (②{I}V.T) L1 L2 →
+ llpx_sn R d V L1 L2.
+#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
+qed-.
+
+(* Basic_properties *********************************************************)
+
+lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,d. llpx_sn R d T L L.
+#R #HR #T #L @(f2_ind … rfw … L T) -L -T
+#n #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
+#i #Hn elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
+#HiL #d elim (ylt_split i d) /2 width=1 by llpx_sn_skip/
+elim (ldrop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+qed-.
+
+lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
+#a #I #V1 #T1 #Hn #L2 #HL12
+@llpx_sn_bind /2 width=1/ (**) (* explicit constructor *)
+@IH -IH // normalize /2 width=1 by eq_f2/
+qed-.
+
+lemma llpx_sn_ge_up: ∀R,L1,L2,U,dt. llpx_sn R dt U L1 L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
+ dt ≤ d + e → llpx_sn R d U L1 L2.
+#R #L1 #L2 #U #dt #H elim H -L1 -L2 -U -dt
+[ #L1 #L2 #dt #k #HL12 #X #d #e #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
+ elim (ylt_yle_false … Hidt) -Hidt
+ @(yle_trans … Hdtde) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
+| #I #L1 #L2 #K1 #K2 #W1 #W2 #dt #i #Hdti #HLK1 #HLK2 #HW1 #HW12 #_ #X #d #e #H #_
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
+ lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
+ normalize in ⊢ (%→%→?); -I -W1 -W2 -dt /3 width=1 by llpx_sn_skip, ylt_inj/
+ | /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
+ ]
+| /2 width=1 by llpx_sn_free/
+| #L1 #L2 #dt #p #HL12 #X #d #e #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+ elim (lift_inv_bind2 … H) -H #V #T #HVW >commutative_plus #HTU #H destruct
+ @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
+ @(IHT … HTU) /2 width=1 by yle_succ/
+| #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+ elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
+ /3 width=4 by llpx_sn_flat/
+]
+qed-.
+
+(**) (* the minor premise comes first *)
+lemma llpx_sn_ge: ∀R,L1,L2,T,d1,d2. d1 ≤ d2 →
+ llpx_sn R d1 T L1 L2 → llpx_sn R d2 T L1 L2.
+#R #L1 #L2 #T #d1 #d2 * -d1 -d2 (**) (* destructed yle *)
+/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
+qed-.
+
+lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
+ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
+/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
+
+lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T,d. llpx_sn R1 d T L1 L2 → llpx_sn R2 d T L1 L2.
+#R1 #R2 #HR12 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_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/multiple/frees.ma".
+include "basic_2/multiple/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (not recursive) *)
+definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝
+ λR,d,T,L1,L2. |L1| = |L2| ∧
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2
+ ).
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IHU @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv … H) -H
+[ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
+| * #J1 #K10 #W10 #j #Hdj #Hji #HnU #HLK10 #HnW10 destruct
+ lapply (ldrop_fwd_drop2 … HLK10) #H
+ lapply (ldrop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
+ elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
+ lapply (ldrop_fwd_drop2 … HLK20) #H
+ lapply (ldrop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
+ elim (IHn K10 W10 … K20 0) -IHn -HL12 /3 width=6 by ldrop_fwd_rfw/
+ elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
+]
+qed.
+
+theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
+elim (IHU … HLK1 HLK2) /3 width=2 by frees_eq/
+#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
+lapply (ldrop_fwd_drop2 … HLK1) #H1
+lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
+@conj [ @(ldrop_fwd_length_eq1 … H1 H2) // ] -HL12
+#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
+>(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
+#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
+lapply (ldrop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
+lapply (ldrop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
+/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
+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/lift_neg.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (recursive) *)
+inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_alt_r_intro: ∀L1,L2,T,d.
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+ ) →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
+ ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
+.
+
+(* Compact definition of llpx_sn_alt_r **************************************)
+
+lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
+ ) → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
+#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+ /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+ lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+ lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+ @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
+qed-.
+
+lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < d
+ | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
+ R K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
+qed.
+
+lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
+qed.
+
+lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
+/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
+qed.
+
+lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
+ llpx_sn_alt_r R d (#i) L1 L2.
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
+lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
+/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
+qed.
+
+lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn_alt_r R 0 V1 K1 K2 → R K1 V1 V2 →
+ llpx_sn_alt_r R d (#i) L1 L2.
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
+[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
+ @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+ elim (lt_or_eq_or_gt i j) #Hij destruct
+ [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+ | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+ | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+ ]
+]
+qed.
+
+lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
+ llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
+ llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
+
+lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
+ llpx_sn_alt_r R d V L1 L2 →
+ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+ llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
+#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
+[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
+| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
+ #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
+ [ * /2 width=1 by llpx_sn_free/
+ | /2 width=1 by llpx_sn_skip/
+ | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+ ]
+| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
+ /3 width=1 by llpx_sn_bind/
+| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+(* Alternative definition of llpx_sn (recursive) ****************************)
+
+lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
+ ) → llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
+qed.
+
+lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
+#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
+qed-.
+
+lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_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/substitution/ldrop_ldrop.ma".
+include "basic_2/multiple/llpx_sn_leq.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ i < d ∨
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
+ R K1 V1 V2 & d ≤ i.
+#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono … HLK22 … HLK2) -L2 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ i < d ∨
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
+ R K1 V1 V2 & d ≤ i.
+#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #H #_ elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono … HLK11 … HLK1) -L1 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
+[ #H elim (ylt_yle_false … H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
+[ #H elim (ylt_yle_false … H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
+#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
+ ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
+/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
+[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
+ elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid
+ #H destruct /2 width=9 by llpx_sn_lref/
+| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
+ /3 width=9 by llpx_sn_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /4 width=9 by llpx_sn_bind, ldrop_drop/
+| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /3 width=9 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
+ ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
+ llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2.
+/2 width=9 by llpx_sn_inv_S_aux/ qed-.
+
+lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) →
+ ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
+/3 width=9 by ldrop_pair, conj, llpx_sn_inv_S/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
+ ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
+/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
+
+lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
+| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
+ [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/
+ #Hdi elim (lt_or_ge i (|L1|)) #HiL1
+ elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
+ elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
+ elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
+ elim (eq_bind2_dec I2 I1)
+ [ #H2 elim (HR K1 V1 V2) -HR
+ [ #H3 elim (IH K1 V1 … K2 0) destruct
+ /3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
+ ]
+ ]
+ -IH #H3 @or_intror
+ #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
+ [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+ |7,8,9: /2 width=4 by ylt_yle_false/
+ ]
+ #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
+ lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
+ #H #H0 destruct /2 width=1 by/
+ ]
+| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
+| #a #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
+| #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V … L2 d) /2 width=1 by/
+ elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+]
+-n /4 width=4 by llpx_sn_fwd_length, or_intror/
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma llpx_sn_lift_le: ∀R. l_liftable R →
+ ∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
+#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hdi #H destruct
+ [ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_skip/
+ | elim (ylt_yle_false … Hid0) -L1 -L2 -K1 -K2 -e -Hid0
+ /3 width=3 by yle_trans, yle_inj/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hdi #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_trans_lt … HLK1 … HK11) // -K1
+ elim (ldrop_trans_lt … HLK2 … HK22) // -Hdi -K2
+ /3 width=18 by llpx_sn_lref/
+ | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hdi -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
+ #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
+ #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_lift_ge: ∀R,K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
+ ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀U. ⇧[d, e] T ≡ U → d ≤ d0 → llpx_sn R (d0+e) U L1 L2.
+#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
+ [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
+ | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hid -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
+ #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
+ #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma llpx_sn_inv_lift_le: ∀R. l_deliftable_sn R →
+ ∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
+#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ [ /2 width=1 by llpx_sn_skip/
+ | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
+ elim (ldrop_conf_lt … HLK2 … HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
+ elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
+ lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
+ /3 width=10 by llpx_sn_lref/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0
+ elim (le_inv_plus_l … Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 … H) -H
+ #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d ≤ d0 → d0 ≤ yinj d + e → llpx_sn R d T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/
+ | elim (ylt_yle_false … Hid0) -L1 -L2 -Hd0 -Hid0
+ /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
+ elim (le_inv_plus_l … Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 … H) -H
+ >commutative_plus #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ d0 → llpx_sn R (d0-e) T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct [ -Hid0 | -Hded0 ]
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+ [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/
+ | elim (le_inv_plus_l … Hid) -Hid #_
+ /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0
+ /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
+ lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hded0 -Hid
+ /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 … H) -H
+ #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
+ @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_flat2 … H) -H
+ #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K1,K2,e. ⇩[e] L1 ≡ K1 → ⇩[e] L2 ≡ K2 →
+ ∀T. ⇧[0, e] T ≡ U → llpx_sn R 0 T K1 K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma llpx_sn_ldrop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K1,e. ⇩[e] L1 ≡ K1 → ∀T. ⇧[0, e] T ≡ U →
+ ∃∃K2. ⇩[e] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K1 #e #HLK1 #T #HTU elim (llpx_sn_fwd_ldrop_sn … HU … HLK1)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+lemma llpx_sn_ldrop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+ ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
+ ∃∃K1. ⇩[e] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K2 #e #HLK2 #T #HTU elim (llpx_sn_fwd_ldrop_dx … HU … HLK2)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+(* Inversion lemmas on negated lazy pointwise extension *********************)
+
+lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
+/4 width=1 by llpx_sn_bind, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
+/4 width=1 by llpx_sn_flat, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+ (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
+/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
+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_leq.ma".
+include "basic_2/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma leq_llpx_sn_trans: ∀R,L2,L,T,d. llpx_sn R d T L2 L →
+ ∀L1. L1 ≃[d, ∞] L2 → llpx_sn R d T L1 L.
+#R #L2 #L #T #d #H elim H -L2 -L -T -d
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
+[ #I #L2 #L #K2 #K #V2 #V #d #i #Hdi #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
+ elim (leq_ldrop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
+ lapply (leq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_trans_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, leq_succ/
+]
+qed-.
+
+lemma llpx_sn_leq_trans: ∀R,L,L1,T,d. llpx_sn R d T L L1 →
+ ∀L2. L1 ≃[d, ∞] L2 → llpx_sn R d T L L2.
+#R #L #L1 #T #d #H elim H -L -L1 -T -d
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
+[ #I #L #L1 #K #K1 #V #V1 #d #i #Hdi #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
+ elim (leq_ldrop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
+ lapply (leq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_conf_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, leq_succ/
+]
+qed-.
+
+lemma llpx_sn_leq_repl: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → ∀K1. K1 ≃[d, ∞] L1 →
+ ∀K2. L2 ≃[d, ∞] K2 → llpx_sn R d T K1 K2.
+/3 width=4 by llpx_sn_leq_trans, leq_llpx_sn_trans/ qed-.
+
+lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+ ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2).
+#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT
+/3 width=7 by llpx_sn_leq_repl, leq_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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpx_sn_alt.ma".
+include "basic_2/multiple/llor.ma".
+include "basic_2/multiple/lleq_alt.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Inversion lemmas on poinwise union for local environments ****************)
+
+lemma llpx_sn_llor_fwd_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T. llpx_sn R 0 T L1 L2 →
+ ∀L. L1 ⩖[T] L2 ≡ L → lpx_sn R L1 L.
+#R #HR #L1 #L2 #T #H1 #L #H2
+elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
+elim H2 -H2 #_ #HL1 #IH2
+@lpx_sn_intro_alt // #I1 #I #K1 #K #V1 #V #i #HLK1 #HLK
+lapply (ldrop_fwd_length_lt2 … HLK) #HiL
+elim (ldrop_O1_lt (Ⓕ) L2 i) // -HiL -HL1 -HL12 #I2 #K2 #V2 #HLK2
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK * [ /2 width=1 by conj/ ]
+#HnT #H1 #H2 elim (IH1 … HnT … HLK1 HLK2) -IH1 -HnT -HLK1 -HLK2 /2 width=1 by 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/substitution/lpx_sn_ldrop.ma".
+include "basic_2/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on pointwise extensions ***************************************)
+
+lemma lpx_sn_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀T,L1,L2,d. lpx_sn R L1 L2 → llpx_sn R d T L1 L2.
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
+| -HR #i elim (lt_or_ge i (|L1|))
+ [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
+ #Hi #Hn #L2 #d elim (ylt_split i d)
+ [ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
+ #Hdi #HL12 elim (ldrop_O1_lt (Ⓕ) L1 i) //
+ #I #K1 #V1 #HLK1 elim (lpx_sn_ldrop_conf … HL12 … HLK1) -HL12
+ /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
+| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
+| -HR /3 width=1 by llpx_sn_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/multiple/llpx_sn_ldrop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties about transitive closure **************************************)
+
+lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L,V1,V2,T. LTC … R L V1 V2 →
+ LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
+#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
+/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
+qed.
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/reduction/cpr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/multiple/llpx_sn_ldrop.ma".
include "basic_2/reduction/cpr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_leq.ma".
+include "basic_2/substitution/ldrop_leq.ma".
include "basic_2/reduction/cpx.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/fqus_alt.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/multiple/fqus_alt.ma".
include "basic_2/static/ssta.ma".
include "basic_2/reduction/cpx.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
include "basic_2/reduction/cpx_llpx_sn.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/multiple/llpx_sn_ldrop.ma".
include "basic_2/reduction/cpx.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
include "basic_2/notation/relations/predreducible_3.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/substitution/ldrop.ma".
(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/reduction/crr.ma".
(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/reduction/crx.ma".
(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
(**************************************************************************)
include "basic_2/notation/relations/btpred_8.ma".
-include "basic_2/relocation/fquq.ma".
-include "basic_2/substitution/lleq.ma".
+include "basic_2/substitution/fquq.ma".
+include "basic_2/multiple/lleq.ma".
include "basic_2/reduction/lpx.ma".
(* "BIG TREE" PARALLEL REDUCTION FOR CLOSURES *******************************)
(**************************************************************************)
include "basic_2/notation/relations/predsn_3.ma".
-include "basic_2/relocation/lpx_sn.ma".
+include "basic_2/substitution/lpx_sn.ma".
include "basic_2/reduction/cpr.ma".
(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lpx_sn_ldrop.ma".
-include "basic_2/relocation/fquq_alt.ma".
+include "basic_2/substitution/lpx_sn_ldrop.ma".
+include "basic_2/substitution/fquq_alt.ma".
include "basic_2/reduction/cpr_lift.ma".
include "basic_2/reduction/lpr.ma".
(* *)
(**************************************************************************)
-include "basic_2/relocation/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
+include "basic_2/substitution/lpx_sn_lpx_sn.ma".
+include "basic_2/multiple/fqup.ma".
include "basic_2/reduction/lpr_ldrop.ma".
(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/fqup.ma".
-include "basic_2/substitution/frees_lift.ma".
+include "basic_2/multiple/fqup.ma".
+include "basic_2/multiple/frees_lift.ma".
include "basic_2/reduction/lpx_ldrop.ma".
(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
(*
-lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
-
-lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
-/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
-
lemma cofrees_lsuby_conf: ∀L1,U,i. L1 ⊢ i ~ϵ 𝐅*⦃U⦄ →
∀L2. lsuby L1 L2 → L2 ⊢ i ~ϵ 𝐅*⦃U⦄.
/3 width=3 by lsuby_cpys_trans/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/relocation/lpx_sn_ldrop.ma".
+include "basic_2/substitution/lpx_sn_ldrop.ma".
include "basic_2/reduction/cpx_lift.ma".
include "basic_2/reduction/lpx.ma".
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_leq.ma".
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_leq.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
include "basic_2/reduction/cpx_leq.ma".
include "basic_2/reduction/lpx_ldrop.ma".
+++ /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/relocation/lift_neg.ma".
-include "basic_2/relocation/lift_lift.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Inversion lemmas on negated relocation ***********************************)
-
-lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 →
- ∀i. d ≤ yinj i → (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) →
- (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
- (∀V. ⇧[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥).
-#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
-[ /3 width=2 by or_introl/
-| #I #G #L #K #V #W #j #d #e #Hdj #Hjde #HLK #HVW #i #Hdi #HnW
- elim (lt_or_ge j i) #Hij
- [ @or_intror @(ex5_4_intro … HLK) // -HLK
- [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
- #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
- | -HnW /2 width=7 by lift_inv_lref2_be/
- ]
- | elim (lift_split … HVW i j) -HVW //
- #X #_ #H elim HnW -HnW //
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_bind … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_bind_sn, or_introl/
- | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
- [ /4 width=9 by nlift_bind_dx, or_introl/
- | * #J #K #W #j #Hdj #Hji #HLK #HnW
- elim (yle_inv_succ1 … Hdj) -Hdj #Hdj #Hj
- lapply (ylt_O … Hj) -Hj #Hj
- lapply (ldrop_inv_drop1_lt … HLK ?) // -HLK #HLK
- >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
- #HnU1 <minus_le_minus_minus_comm in HnW;
- /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
- ]
- ]
-| #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_flat … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_flat_sn, or_introl/
- | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
- [ /4 width=9 by nlift_flat_dx, or_introl/
- | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
- ]
-]
-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/supterm_6.ma".
-include "basic_2/grammar/cl_weight.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* activate genv *)
-inductive fqu: tri_relation genv lenv term ≝
-| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
-| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
-| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
-| fqu_drop : ∀G,L,K,T,U,e.
- ⇩[e+1] L ≡ K → ⇧[0, e+1] T ≡ U → fqu G L U G K T
-.
-
-interpretation
- "structural successor (closure)"
- 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_drop_lt: ∀G,L,K,T,U,e. 0 < e →
- ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
-#G #L #K #T #U #e #He >(plus_minus_m_m e 1) /2 width=3 by fqu_drop/
-qed.
-
-lemma fqu_lref_S_lt: ∀I,G,L,V,i. 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(i-1)⦄.
-/3 width=3 by fqu_drop, ldrop_drop, lift_lref_ge_minus/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-#G #L #K #T #U #e #HLK #HTU
-lapply (ldrop_fwd_lw_lt … HLK ?) -HLK // #HKL
-lapply (lift_fwd_tw … HTU) -e #H
-normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
-qed-.
-
-fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| < |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[1: normalize //
-|3: #a
-|5: /2 width=4 by ldrop_fwd_length_lt4/
-] #I #G #L #V #T #j #H destruct
-qed-.
-
-lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ → |L2| < |L1|.
-/2 width=7 by fqu_fwd_length_lref1_aux/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqu_wf_ind: ∀R:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
-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/suptermopt_6.ma".
-include "basic_2/relocation/fqu.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* activate genv *)
-inductive fquq: tri_relation genv lenv term ≝
-| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
-| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
-| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
-| fquq_drop : ∀G,L,K,T,U,e.
- ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → fquq G L U G K T
-.
-
-interpretation
- "optional structural successor (closure)"
- 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquq_refl: tri_reflexive … fquq.
-/2 width=3 by fquq_drop/ qed.
-
-lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
-#G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1
-lapply (ldrop_fwd_lw … HLK1) -HLK1
-lapply (lift_fwd_tw … HTU1) -HTU1
-/2 width=1 by le_plus, le_n/
-qed-.
-
-fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| ≤ |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-[ #a #I #G #L #V #T #j #H destruct
-| #G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #i #H destruct
- /2 width=3 by ldrop_fwd_length_le4/
-]
-qed-.
-
-lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
-/2 width=7 by fquq_fwd_length_lref1_aux/
-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/suptermoptalt_6.ma".
-include "basic_2/relocation/fquq.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* alternative definition of fquq *)
-definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
-
-interpretation
- "optional structural successor (closure) alternative"
- 'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquqa_refl: tri_reflexive … fquqa.
-// qed.
-
-lemma fquqa_drop: ∀G,L,K,T,U,e.
- ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
-#G #L #K #T #U #e #HLK #HTU elim (eq_or_gt e)
-/3 width=5 by fqu_drop_lt, or_introl/ #H destruct
->(ldrop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
-qed.
-
-(* Main properties **********************************************************)
-
-theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
-* #H1 #H2 #H3 destruct //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
-/2 width=1 by or_introl/
-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/rdrop_3.ma".
-include "basic_2/grammar/genv.ma".
-
-(* GLOBAL ENVIRONMENT READING ***********************************************)
-
-inductive gget (e:nat): relation genv ≝
-| gget_gt: ∀G. |G| ≤ e → gget e G (⋆)
-| gget_eq: ∀G. |G| = e + 1 → gget e G G
-| gget_lt: ∀I,G1,G2,V. e < |G1| → gget e G1 G2 → gget e (G1. ⓑ{I} V) G2
-.
-
-interpretation "global reading"
- 'RDrop e G1 G2 = (gget e G1 G2).
-
-(* basic inversion lemmas ***************************************************)
-
-lemma gget_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
-#G1 #G2 #e * -G1 -G2 //
-[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ #H2
- lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-]
-qed-.
-
-lemma gget_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
-#G1 #G2 #e * -G1 -G2 //
-[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ normalize #H2
- <(injective_plus_l … H2) in H1; -H2 #H
- elim (lt_refl_false … H)
-]
-qed-.
-
-fact gget_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
- e < |G1| → ⇩[e] G1 ≡ G2.
-#I #G #G1 #G2 #V #e * -G -G2
-[ #G #H1 #H destruct #H2
- lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
- elim (lt_refl_false … H)
-| #J #G #G2 #W #_ #HG2 #H destruct //
-]
-qed-.
-
-lemma gget_inv_lt: ∀I,G1,G2,V,e.
- ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
-/2 width=5 by gget_inv_lt_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma gget_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
-#e #G1 elim G1 -G1 /3 width=2/
-#I #V #G1 * #G2 #HG12
-elim (lt_or_eq_or_gt e (|G1|)) #He
-[ /3 width=2/
-| destruct /3 width=2/
-| @ex_intro [2: @gget_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
-]
-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/gget.ma".
-
-(* GLOBAL ENVIRONMENT READING ***********************************************)
-
-(* Main properties **********************************************************)
-
-theorem gget_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
-#G #G1 #e #H elim H -G -G1
-[ #G #He #G2 #H
- >(gget_inv_gt … H He) -H -He //
-| #G #He #G2 #H
- >(gget_inv_eq … H He) -H -He //
-| #I #G #G1 #V #He #_ #IHG1 #G2 #H
- lapply (gget_inv_lt … H He) -H -He /2 width=1/
-]
-qed-.
-
-lemma gget_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
-#G1 #G2 #e
-elim (gget_total e G1) #G #HG1
-elim (eq_genv_dec G G2) #HG2
-[ destruct /2 width=1/
-| @or_intror #HG12
- lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 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 "ground_2/lib/bool.ma".
-include "ground_2/lib/lstar.ma".
-include "basic_2/notation/relations/rdrop_5.ma".
-include "basic_2/notation/relations/rdrop_4.ma".
-include "basic_2/notation/relations/rdrop_3.ma".
-include "basic_2/grammar/lenv_length.ma".
-include "basic_2/grammar/cl_restricted_weight.ma".
-include "basic_2/relocation/lift.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Basic_1: includes: drop_skip_bind *)
-inductive ldrop (s:bool): relation4 nat nat lenv lenv ≝
-| ldrop_atom: ∀d,e. (s = Ⓕ → e = 0) → ldrop s d e (⋆) (⋆)
-| ldrop_pair: ∀I,L,V. ldrop s 0 0 (L.ⓑ{I}V) (L.ⓑ{I}V)
-| ldrop_drop: ∀I,L1,L2,V,e. ldrop s 0 e L1 L2 → ldrop s 0 (e+1) (L1.ⓑ{I}V) L2
-| ldrop_skip: ∀I,L1,L2,V1,V2,d,e.
- ldrop s d e L1 L2 → ⇧[d, e] V2 ≡ V1 →
- ldrop s (d+1) e (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
-.
-
-interpretation
- "basic slicing (local environment) abstract"
- 'RDrop s d e L1 L2 = (ldrop s d e L1 L2).
-(*
-interpretation
- "basic slicing (local environment) general"
- 'RDrop d e L1 L2 = (ldrop true d e L1 L2).
-*)
-interpretation
- "basic slicing (local environment) lget"
- 'RDrop e L1 L2 = (ldrop false O e L1 L2).
-
-definition l_liftable: predicate (lenv → relation term) ≝
- λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,d,e. ⇩[s, d, e] L ≡ K →
- ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
-
-definition l_deliftable_sn: predicate (lenv → relation term) ≝
- λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,d,e. ⇩[s, d, e] L ≡ K →
- ∀T1. ⇧[d, e] T1 ≡ U1 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
-
-definition dropable_sn: predicate (relation lenv) ≝
- λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
- ∃∃K2. R K1 K2 & ⇩[s, d, e] L2 ≡ K2.
-
-definition dropable_dx: predicate (relation lenv) ≝
- λR. ∀L1,L2. R L1 L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & R K1 K2.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ldrop_inv_atom1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆ ∧ (s = Ⓕ → e = 0).
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ /3 width=1 by conj/
-| #I #L #V #H destruct
-| #I #L1 #L2 #V #e #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was: drop_gen_sort *)
-lemma ldrop_inv_atom1: ∀L2,s,d,e. ⇩[s, d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ (s = Ⓕ → e = 0).
-/2 width=4 by ldrop_inv_atom1_aux/ qed-.
-
-fact ldrop_inv_O1_pair1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K.ⓑ{I}V →
- (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
- (0 < e ∧ ⇩[s, d, e-1] K ≡ L2).
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #K #J #W #H destruct
-| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
-| #I #L1 #L2 #V #e #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed-.
-
-lemma ldrop_inv_O1_pair1: ∀I,K,L2,V,s,e. ⇩[s, 0, e] K. ⓑ{I} V ≡ L2 →
- (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
- (0 < e ∧ ⇩[s, 0, e-1] K ≡ L2).
-/2 width=3 by ldrop_inv_O1_pair1_aux/ qed-.
-
-lemma ldrop_inv_pair1: ∀I,K,L2,V,s. ⇩[s, 0, 0] K.ⓑ{I}V ≡ L2 → L2 = K.ⓑ{I}V.
-#I #K #L2 #V #s #H
-elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
-elim (lt_refl_false … H)
-qed-.
-
-(* Basic_1: was: drop_gen_drop *)
-lemma ldrop_inv_drop1_lt: ∀I,K,L2,V,s,e.
- ⇩[s, 0, e] K.ⓑ{I}V ≡ L2 → 0 < e → ⇩[s, 0, e-1] K ≡ L2.
-#I #K #L2 #V #s #e #H #He
-elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
-elim (lt_refl_false … He)
-qed-.
-
-lemma ldrop_inv_drop1: ∀I,K,L2,V,s,e.
- ⇩[s, 0, e+1] K.ⓑ{I}V ≡ L2 → ⇩[s, 0, e] K ≡ L2.
-#I #K #L2 #V #s #e #H lapply (ldrop_inv_drop1_lt … H ?) -H //
-qed-.
-
-fact ldrop_inv_skip1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
- ⇧[d-1, e] V2 ≡ V1 &
- L2 = K2.ⓑ{I}V2.
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #J #K1 #W1 #H destruct
-| #I #L #V #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop_gen_skip_l *)
-lemma ldrop_inv_skip1: ∀I,K1,V1,L2,s,d,e. ⇩[s, d, e] K1.ⓑ{I}V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
- ⇧[d-1, e] V2 ≡ V1 &
- L2 = K2.ⓑ{I}V2.
-/2 width=3 by ldrop_inv_skip1_aux/ qed-.
-
-lemma ldrop_inv_O1_pair2: ∀I,K,V,s,e,L1. ⇩[s, 0, e] L1 ≡ K.ⓑ{I}V →
- (e = 0 ∧ L1 = K.ⓑ{I}V) ∨
- ∃∃I1,K1,V1. ⇩[s, 0, e-1] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & 0 < e.
-#I #K #V #s #e *
-[ #H elim (ldrop_inv_atom1 … H) -H #H destruct
-| #L1 #I1 #V1 #H
- elim (ldrop_inv_O1_pair1 … H) -H *
- [ #H1 #H2 destruct /3 width=1 by or_introl, conj/
- | /3 width=5 by ex3_3_intro, or_intror/
- ]
-]
-qed-.
-
-fact ldrop_inv_skip2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 &
- ⇧[d-1, e] V2 ≡ V1 &
- L1 = K1.ⓑ{I}V1.
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #J #K2 #W2 #H destruct
-| #I #L #V #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop_gen_skip_r *)
-lemma ldrop_inv_skip2: ∀I,L1,K2,V2,s,d,e. ⇩[s, d, e] L1 ≡ K2.ⓑ{I}V2 → 0 < d →
- ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 & ⇧[d-1, e] V2 ≡ V1 &
- L1 = K1.ⓑ{I}V1.
-/2 width=3 by ldrop_inv_skip2_aux/ qed-.
-
-lemma ldrop_inv_O1_gt: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → |L| < e →
- s = Ⓣ ∧ K = ⋆.
-#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize in ⊢ (?%?→?); #H1e
-[ elim (ldrop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
- #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1e)
-| elim (ldrop_inv_O1_pair1 … H) -H * #H2e #HLK destruct
- [ elim (lt_zero_false … H1e)
- | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
- ]
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ldrop_refl_atom_O2: ∀s,d. ⇩[s, d, O] ⋆ ≡ ⋆.
-/2 width=1 by ldrop_atom/ qed.
-
-(* Basic_1: was by definition: drop_refl *)
-lemma ldrop_refl: ∀L,d,s. ⇩[s, d, 0] L ≡ L.
-#L elim L -L //
-#L #I #V #IHL #d #s @(nat_ind_plus … d) -d /2 width=1 by ldrop_pair, ldrop_skip/
-qed.
-
-lemma ldrop_drop_lt: ∀I,L1,L2,V,s,e.
- ⇩[s, 0, e-1] L1 ≡ L2 → 0 < e → ⇩[s, 0, e] L1.ⓑ{I}V ≡ L2.
-#I #L1 #L2 #V #s #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by ldrop_drop/
-qed.
-
-lemma ldrop_skip_lt: ∀I,L1,L2,V1,V2,s,d,e.
- ⇩[s, d-1, e] L1 ≡ L2 → ⇧[d-1, e] V2 ≡ V1 → 0 < d →
- ⇩[s, d, e] L1. ⓑ{I} V1 ≡ L2.ⓑ{I}V2.
-#I #L1 #L2 #V1 #V2 #s #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) /2 width=1 by ldrop_skip/
-qed.
-
-lemma ldrop_O1_le: ∀s,e,L. e ≤ |L| → ∃K. ⇩[s, 0, e] L ≡ K.
-#s #e @(nat_ind_plus … e) -e /2 width=2 by ex_intro/
-#e #IHe *
-[ #H elim (le_plus_xSy_O_false … H)
-| #L #I #V normalize #H elim (IHe L) -IHe /3 width=2 by ldrop_drop, monotonic_pred, ex_intro/
-]
-qed-.
-
-lemma ldrop_O1_lt: ∀s,L,e. e < |L| → ∃∃I,K,V. ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
-#s #L elim L -L
-[ #e #H elim (lt_zero_false … H)
-| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4 by ldrop_pair, ex1_3_intro/
- #e #_ normalize #H elim (IHL e) -IHL /3 width=4 by ldrop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
-]
-qed-.
-
-lemma ldrop_O1_pair: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → e ≤ |L| → ∀I,V.
- ∃∃J,W. ⇩[s, 0, e] L.ⓑ{I}V ≡ K.ⓑ{J}W.
-#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize #He #I #V
-[ elim (ldrop_inv_atom1 … H) -H #H <(le_n_O_to_eq … He) -e
- #Hs destruct /2 width=3 by ex1_2_intro/
-| elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK destruct /2 width=3 by ex1_2_intro/
- elim (IHL … HLK … Z X) -IHL -HLK
- /3 width=3 by ldrop_drop_lt, le_plus_to_minus, ex1_2_intro/
-]
-qed-.
-
-lemma ldrop_O1_ge: ∀L,e. |L| ≤ e → ⇩[Ⓣ, 0, e] L ≡ ⋆.
-#L elim L -L [ #e #_ @ldrop_atom #H destruct ]
-#L #I #V #IHL #e @(nat_ind_plus … e) -e [ #H elim (le_plus_xSy_O_false … H) ]
-normalize /4 width=1 by ldrop_drop, monotonic_pred/
-qed.
-
-lemma ldrop_split: ∀L1,L2,d,e2,s. ⇩[s, d, e2] L1 ≡ L2 → ∀e1. e1 ≤ e2 →
- ∃∃L. ⇩[s, d, e2 - e1] L1 ≡ L & ⇩[s, d, e1] L ≡ L2.
-#L1 #L2 #d #e2 #s #H elim H -L1 -L2 -d -e2
-[ #d #e2 #Hs #e1 #He12 @(ex2_intro … (⋆))
- @ldrop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
-| #I #L1 #V #e1 #He1 lapply (le_n_O_to_eq … He1) -He1
- #H destruct /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #e2 #HL12 #IHL12 #e1 @(nat_ind_plus … e1) -e1
- [ /3 width=3 by ldrop_drop, ex2_intro/
- | -HL12 #e1 #_ #He12 lapply (le_plus_to_le_r … He12) -He12
- #He12 elim (IHL12 … He12) -IHL12 >minus_plus_plus_l
- #L #HL1 #HL2 elim (lt_or_ge (|L1|) (e2-e1)) #H0
- [ elim (ldrop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
- elim (ldrop_inv_atom1 … HL2) -HL2 #H #_ destruct
- @(ex2_intro … (⋆)) [ @ldrop_O1_ge normalize // ]
- @ldrop_atom #H destruct
- | elim (ldrop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by ldrop_drop, ex2_intro/
- ]
- ]
-| #I #L1 #L2 #V1 #V2 #d #e2 #_ #HV21 #IHL12 #e1 #He12 elim (IHL12 … He12) -IHL12
- #L #HL1 #HL2 elim (lift_split … HV21 d e1) -HV21 /3 width=5 by ldrop_skip, ex2_intro/
-]
-qed-.
-
-lemma ldrop_FT: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-/3 width=1 by ldrop_atom, ldrop_drop, ldrop_skip/
-qed.
-
-lemma ldrop_gen: ∀L1,L2,s,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[s, d, e] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by ldrop_FT/
-qed-.
-
-lemma ldrop_T: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by ldrop_FT/
-qed-.
-
-lemma l_liftable_LTC: ∀R. l_liftable R → l_liftable (LTC … R).
-#R #HR #K #T1 #T2 #H elim H -T2
-[ /3 width=10 by inj/
-| #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T d e) /4 width=12 by step/
-]
-qed-.
-
-lemma l_deliftable_sn_LTC: ∀R. l_deliftable_sn R → l_deliftable_sn (LTC … R).
-#R #HR #L #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #K #s #d #e #HLK #T1 #HTU1
- elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
-| #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
-#R #HR #L1 #K1 #s #d #e #HLK1 #L2 #H elim H -L2
-[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
- /3 width=3 by step, ex2_intro/
-]
-qed-.
-
-lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
-#R #HR #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #K2 #s #e #HLK2 elim (HR … HL12 … HLK2) -HR -L2
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IHL1 #K2 #s #e #HLK2 elim (HR … HL2 … HLK2) -HR -L2
- #K #HLK #HK2 elim (IHL1 … HLK) -L
- /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma l_deliftable_sn_llstar: ∀R. l_deliftable_sn R →
- ∀l. l_deliftable_sn (llstar … R l).
-#R #HR #l #L #U1 #U2 #H @(lstar_ind_r … l U2 H) -l -U2
-[ /2 width=3 by lstar_O, ex2_intro/
-| #l #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
-]
-qed-.
-
-(* Basic forvard lemmas *****************************************************)
-
-(* Basic_1: was: drop_S *)
-lemma ldrop_fwd_drop2: ∀L1,I2,K2,V2,s,e. ⇩[s, O, e] L1 ≡ K2. ⓑ{I2} V2 →
- ⇩[s, O, e + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #s #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #H
- [ -IHL1 destruct /2 width=1 by ldrop_drop/
- | @ldrop_drop >(plus_minus_m_m e 1) /2 width=3 by/
- ]
-]
-qed-.
-
-lemma ldrop_fwd_length_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → |L1| ≤ d → |L2| = |L1|.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
-[ #I #L1 #L2 #V #e #_ #_ #H elim (le_plus_xSy_O_false … H)
-| /4 width=2 by le_plus_to_le_r, eq_f/
-]
-qed-.
-
-lemma ldrop_fwd_length_le_le: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → e ≤ |L1| - d → |L2| = |L1| - e.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
-[ /3 width=2 by le_plus_to_le_r/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 >minus_plus_plus_l
- #Hd #He lapply (le_plus_to_le_r … Hd) -Hd
- #Hd >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
-]
-qed-.
-
-lemma ldrop_fwd_length_le_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → |L1| - d ≤ e → |L2| = d.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e normalize
-[ /2 width=1 by le_n_O_to_eq/
-| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false … H)
-| /3 width=2 by le_plus_to_le_r/
-| /4 width=2 by le_plus_to_le_r, eq_f/
-]
-qed-.
-
-lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L1| = |L2| + e.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1 by/
-qed-.
-
-lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by plus_minus, le_n/
-qed-.
-
-lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e = |L1| - |L2|.
-#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
-qed-.
-
-lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e ≤ |L1|.
-#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
-qed-.
-
-lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
-qed-.
-
-lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e.
- ⇩[Ⓕ, d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
-#L1 #I2 #K2 #V2 #d #e #H
-lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
-qed-.
-
-lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|.
-#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by lt_minus_to_plus_r/
-qed-.
-
-lemma ldrop_fwd_length_eq1: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- |L1| = |L2| → |K1| = |K2|.
-#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
-lapply (ldrop_fwd_length … HLK1) -HLK1
-lapply (ldrop_fwd_length … HLK2) -HLK2
-/2 width=2 by injective_plus_r/
-qed-.
-
-lemma ldrop_fwd_length_eq2: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- |K1| = |K2| → |L1| = |L2|.
-#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
-lapply (ldrop_fwd_length … HLK1) -HLK1
-lapply (ldrop_fwd_length … HLK2) -HLK2 //
-qed-.
-
-lemma ldrop_fwd_lw: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e // normalize
-[ /2 width=3 by transitive_le/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12
- >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
-]
-qed-.
-
-lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #H >H -H //
-| #I #L #V #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V #e #HL12 #_ #_
- lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
- @(le_to_lt_to_lt … HL12) -HL12 //
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
- >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
-]
-qed-.
-
-lemma ldrop_fwd_rfw: ∀I,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
-#I #L #K #V #i #HLK lapply (ldrop_fwd_lw … HLK) -HLK
-normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact ldrop_inv_O2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → e = 0 → L1 = L2.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
-[ //
-| //
-| #I #L1 #L2 #V #e #_ #_ >commutative_plus normalize #H destruct
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H
- >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e //
-]
-qed-.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma ldrop_inv_O2: ∀L1,L2,s,d. ⇩[s, d, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5 by ldrop_inv_O2_aux/ qed-.
-
-lemma ldrop_inv_length_eq: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L1| = |L2| → e = 0.
-#L1 #L2 #d #e #H #HL12 lapply (ldrop_fwd_length_minus4 … H) //
-qed-.
-
-lemma ldrop_inv_refl: ∀L,d,e. ⇩[Ⓕ, d, e] L ≡ L → e = 0.
-/2 width=5 by ldrop_inv_length_eq/ qed-.
-
-fact ldrop_inv_FT_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 →
- ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → d = 0 →
- ⇩[Ⓕ, d, e] L1 ≡ K.ⓑ{I}V.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #_ #J #K #W #H destruct
-| #I #L #V #J #K #W #H destruct //
-| #I #L1 #L2 #V #e #_ #IHL12 #J #K #W #H1 #H2 destruct
- /3 width=1 by ldrop_drop/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #_ #J #K #W #_ #_
- <plus_n_Sm #H destruct
-]
-qed-.
-
-lemma ldrop_inv_FT: ∀I,L,K,V,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
-/2 width=5 by ldrop_inv_FT_aux/ qed.
-
-lemma ldrop_inv_gen: ∀I,L,K,V,s,e. ⇩[s, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
-#I #L #K #V * /2 width=1 by ldrop_inv_FT/
-qed-.
-
-lemma ldrop_inv_T: ∀I,L,K,V,s,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
-#I #L #K #V * /2 width=1 by ldrop_inv_FT/
-qed-.
-
-(* Basic_1: removed theorems 50:
- drop_ctail drop_skip_flat
- cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- drop_clear drop_clear_O drop_clear_S
- clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
- clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
- getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
- getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
- drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
- getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
- getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
- getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
-*)
+++ /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/lift_lift.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop_mono *)
-theorem ldrop_mono: ∀L,L1,s1,d,e. ⇩[s1, d, e] L ≡ L1 →
- ∀L2,s2. ⇩[s2, d, e] L ≡ L2 → L1 = L2.
-#L #L1 #s1 #d #e #H elim H -L -L1 -d -e
-[ #d #e #He #L2 #s2 #H elim (ldrop_inv_atom1 … H) -H //
-| #I #K #V #L2 #s2 #HL12 <(ldrop_inv_O2 … HL12) -L2 //
-| #I #L #K #V #e #_ #IHLK #L2 #s2 #H
- lapply (ldrop_inv_drop1 … H) -H /2 width=2 by/
-| #I #L #K1 #T #V1 #d #e #_ #HVT1 #IHLK1 #X #s2 #H
- elim (ldrop_inv_skip1 … H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
- >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
- >(IHLK1 … HLK2) -IHLK1 -HLK2 //
-]
-qed-.
-
-(* Basic_1: was: drop_conf_ge *)
-theorem ldrop_conf_ge: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
- ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
- ⇩[s2, 0, e2 - e1] L1 ≡ L2.
-#L #L1 #s1 #d1 #e1 #H elim H -L -L1 -d1 -e1 //
-[ #d #e #_ #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
- #H #He destruct
- @ldrop_atom #H >He // (**) (* explicit constructor *)
-| #I #L #K #V #e #_ #IHLK #L2 #s2 #e2 #H #He2
- lapply (ldrop_inv_drop1_lt … H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
- <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
-| #I #L #K #V1 #V2 #d #e #_ #_ #IHLK #L2 #s2 #e2 #H #Hdee2
- lapply (transitive_le 1 … Hdee2) // #He2
- lapply (ldrop_inv_drop1_lt … H ?) -H // -He2 #HL2
- lapply (transitive_le (1+e) … Hdee2) // #Hee2
- @ldrop_drop_lt >minus_minus_comm /3 width=1 by lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
-]
-qed.
-
-(* Note: apparently this was missing in basic_1 *)
-theorem ldrop_conf_be: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. ⇩[s1, 0, d1 + e1 - e2] L2 ≡ L & ⇩[d1] L1 ≡ L.
-#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #He1 #L2 #e2 #H #Hd1 #_ elim (ldrop_inv_atom1 … H) -H #H #He2 destruct
- >(He2 ?) in Hd1; // -He2 #Hd1 <(le_n_O_to_eq … Hd1) -d1
- /4 width=3 by ldrop_atom, ex2_intro/
-| normalize #I #L #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_O2 … HL2) -HL2 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| normalize #I #L0 #K0 #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1_pair1 … H) -H * * #He2 #HL20
- [ -IHLK0 -He21 destruct <minus_n_O /3 width=3 by ldrop_drop, ex2_intro/
- | -HLK0 <minus_le_minus_minus_comm //
- elim (IHLK0 … HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
- ]
-| #I #L0 #K0 #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_drop1_lt … H ?) -H // #HL02
- elim (IHLK0 … HL02) -L0 /3 width=3 by ldrop_drop, monotonic_pred, ex2_intro/
-]
-qed-.
-
-(* Note: apparently this was missing in basic_1 *)
-theorem ldrop_conf_le: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
- ∀L2,s2,e2. ⇩[s2, 0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ⇩[s2, 0, e2] L1 ≡ L & ⇩[s1, d1 - e2, e1] L2 ≡ L.
-#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #He1 #L2 #s2 #e2 #H elim (ldrop_inv_atom1 … H) -H
- #H #He2 #_ destruct /4 width=3 by ldrop_atom, ex2_intro/
-| #I #K0 #V0 #L2 #s2 #e2 #H1 #H2
- lapply (le_n_O_to_eq … H2) -H2 #H destruct
- lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| #I #K0 #K1 #V0 #e1 #HK01 #_ #L2 #s2 #e2 #H1 #H2
- lapply (le_n_O_to_eq … H2) -H2 #H destruct
- lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by ldrop_drop, ex2_intro/
-| #I #K0 #K1 #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #s2 #e2 #H #He2d1
- elim (ldrop_inv_O1_pair1 … H) -H *
- [ -IHK01 -He2d1 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
- | -HK01 -HV10 #He2 #HK0L2
- elim (IHK01 … HK0L2) -IHK01 -HK0L2 /2 width=1 by monotonic_pred/
- >minus_le_minus_minus_comm /3 width=3 by ldrop_drop_lt, ex2_intro/
- ]
-]
-qed-.
-
-(* Note: with "s2", the conclusion parameter is "s1 ∨ s2" *)
-(* Basic_1: was: drop_trans_ge *)
-theorem ldrop_trans_ge: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
- ∀L2,e2. ⇩[e2] L ≡ L2 → d1 ≤ e2 → ⇩[s1, 0, e1 + e2] L1 ≡ L2.
-#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
-[ #d1 #e1 #He1 #L2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
- #H #He2 destruct /4 width=1 by ldrop_atom, eq_f2/
-| /2 width=1 by ldrop_gen/
-| /3 width=1 by ldrop_drop/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 #L #e2 #H #Hde2
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
- lapply (ldrop_inv_drop1_lt … H ?) -H // #HL2
- @ldrop_drop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
-]
-qed.
-
-(* Basic_1: was: drop_trans_le *)
-theorem ldrop_trans_le: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
- ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → e2 ≤ d1 →
- ∃∃L0. ⇩[s2, 0, e2] L1 ≡ L0 & ⇩[s1, d1 - e2, e1] L0 ≡ L2.
-#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
-[ #d1 #e1 #He1 #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
- #H #He2 destruct /4 width=3 by ldrop_atom, ex2_intro/
-| #I #K #V #L2 #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
- #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| #I #L1 #L2 #V #e #_ #IHL12 #L #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
- #H destruct elim (IHL12 … HL2) -IHL12 -HL2 //
- #L0 #H #HL0 lapply (ldrop_inv_O2 … H) -H #H destruct
- /3 width=5 by ldrop_pair, ldrop_drop, ex2_intro/
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV12 #IHL12 #L #s2 #e2 #H #He2d
- elim (ldrop_inv_O1_pair1 … H) -H *
- [ -He2d -IHL12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
- | -HL12 -HV12 #He2 #HL2
- elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by ldrop_drop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma l_liftable_llstar: ∀R. l_liftable R → ∀l. l_liftable (llstar … R l).
-#R #HR #l #K #T1 #T2 #H @(lstar_ind_r … l T2 H) -l -T2
-[ #L #s #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
- >(lift_mono … HTU2 … HTU1) -T1 -U2 -d -e //
-| #l #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T d e) /3 width=12 by lstar_dx/
-]
-qed-.
-
-(* Basic_1: was: drop_conf_lt *)
-lemma ldrop_conf_lt: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
- ∀I,K2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ K2.ⓑ{I}V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ⇩[s2, 0, e2] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[s1, d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
-#L #L1 #s1 #d1 #e1 #H1 #I #K2 #V2 #s2 #e2 #H2 #He2d1
-elim (ldrop_conf_le … H1 … H2) -L /2 width=2 by lt_to_le/ #K #HL1K #HK2
-elim (ldrop_inv_skip1 … HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/
-#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Note: apparently this was missing in basic_1 *)
-lemma ldrop_trans_lt: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
- ∀I,L2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ L2.ⓑ{I}V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃L0,V0. ⇩[s2, 0, e2] L1 ≡ L0.ⓑ{I}V0 &
- ⇩[s1, d, e1] L0 ≡ L2 & ⇧[d, e1] V2 ≡ V0.
-#L1 #L #s1 #d1 #e1 #HL1 #I #L2 #V2 #s2 #e2 #HL2 #Hd21
-elim (ldrop_trans_le … HL1 … HL2) -L /2 width=1 by lt_to_le/ #L0 #HL10 #HL02
-elim (ldrop_inv_skip2 … HL02) -HL02 /2 width=1 by lt_plus_to_minus_r/ #L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma ldrop_trans_ge_comm: ∀L1,L,L2,s1,d1,e1,e2.
- ⇩[s1, d1, e1] L1 ≡ L → ⇩[e2] L ≡ L2 → d1 ≤ e2 →
- ⇩[s1, 0, e2 + e1] L1 ≡ L2.
-#L1 #L #L2 #s1 #d1 #e1 #e2
->commutative_plus /2 width=5 by ldrop_trans_ge/
-qed.
-
-lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[e1] L ≡ K.ⓑ{I1}V1 →
- ∀I2,V2,e2. ⇩[e2] L ≡ K.ⓑ{I2}V2 →
- ∧∧ e1 = e2 & I1 = I2 & V1 = V2.
-#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2
-elim (le_or_ge e1 e2) #He
-[ lapply (ldrop_conf_ge … HLK1 … HLK2 ?)
-| lapply (ldrop_conf_ge … HLK2 … HLK1 ?)
-] -HLK1 -HLK2 // #HK
-lapply (ldrop_fwd_length_minus2 … HK) #H
-elim (discr_minus_x_xy … H) -H
-[1,3: normalize <plus_n_Sm #H destruct ]
-#H >H in HK; #HK
-lapply (ldrop_inv_O2 … HK) -HK #H destruct
-lapply (inv_eq_minus_O … H) -H /3 width=1 by le_to_le_to_eq, and3_intro/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma ldrop_fwd_be: ∀L,K,s,d,e,i. ⇩[s, d, e] L ≡ K → |K| ≤ i → i < d → |L| ≤ i.
-#L #K #s #d #e #i #HLK #HK #Hd elim (lt_or_ge i (|L|)) //
-#HL elim (ldrop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
-elim (ldrop_conf_lt … HLK … HLK0) // -HLK -HLK0 -Hd
-#K1 #V1 #HK1 #_ #_ lapply (ldrop_fwd_length_lt2 … HK1) -I -K1 -V1
-#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/grammar/leq_leq.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-definition dedropable_sn: predicate (relation lenv) ≝
- λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
- ∃∃L2. R L1 L2 & ⇩[s, d, e] L2 ≡ K2 & L1 ≃[d, e] L2.
-
-(* Properties on equivalence ************************************************)
-
-lemma leq_ldrop_trans_be: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
- ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W →
- d ≤ i → i < d + e →
- ∃∃K1. K1 ≃[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #J #K2 #W #s #i #H
- elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #e #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct >ypred_succ
- /2 width=3 by ldrop_pair, ex2_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=3 by ldrop_drop_lt, ex2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J #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=3 by ylt_O, ldrop_drop_lt, ex2_intro/
-]
-qed-.
-
-lemma leq_ldrop_conf_be: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
- ∀I,K1,W,s,i. ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W →
- d ≤ i → i < d + e →
- ∃∃K2. K1 ≃[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W.
-#L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hdi #Hide
-elim (leq_ldrop_trans_be … (leq_sym … HL12) … HLK1) // -L1 -Hdi -Hide
-/3 width=3 by leq_sym, ex2_intro/
-qed-.
-
-lemma ldrop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
- ∃∃L2. L1 ≃[0, i] L2 & ⇩[i] L2 ≡ K2.
-#K2 #i @(nat_ind_plus … i) -i
-[ /3 width=3 by leq_O2, ex2_intro/
-| #i #IHi #Y #Hi elim (ldrop_O1_lt (Ⓕ) Y 0) //
- #I #L1 #V #H lapply (ldrop_inv_O2 … H) -H #H destruct
- normalize in Hi; elim (IHi L1) -IHi
- /3 width=5 by ldrop_drop, leq_pair, injective_plus_l, ex2_intro/
-]
-qed-.
-
-lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
-#R #HR #L1 #K1 #s #d #e #HLK1 #K2 #H elim H -K2
-[ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
- /3 width=4 by inj, ex3_intro/
-| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
- /3 width=6 by leq_trans, step, ex3_intro/
-]
-qed-.
-
-(* Inversion lemmas on equivalence ******************************************)
-
-lemma ldrop_O1_inj: ∀i,L1,L2,K. ⇩[i] L1 ≡ K → ⇩[i] L2 ≡ K → L1 ≃[i, ∞] L2.
-#i @(nat_ind_plus … i) -i
-[ #L1 #L2 #K #H <(ldrop_inv_O2 … H) -K #H <(ldrop_inv_O2 … H) -L1 //
-| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
- lapply (ldrop_fwd_length … HLK1)
- <(ldrop_fwd_length … HLK2) [ /4 width=5 by ldrop_inv_drop1, leq_succ/ ]
- normalize <plus_n_Sm #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/notation/relations/rlift_4.ma".
-include "basic_2/grammar/term_weight.ma".
-include "basic_2/grammar/term_simple.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Basic_1: includes:
- lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
-*)
-inductive lift: relation4 nat nat term term ≝
-| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k)
-| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
-| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
-| lift_gref : ∀p,d,e. lift d e (§p) (§p)
-| lift_bind : ∀a,I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 → lift (d + 1) e T1 T2 →
- lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| lift_flat : ∀I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 → lift d e T1 T2 →
- lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lift_inv_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
-qed-.
-
-lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
-/2 width=4 by lift_inv_O2_aux/ qed-.
-
-fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5 by lift_inv_sort1_aux/ qed-.
-
-fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
- (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1/
-| #j #d #e #Hj #i #Hi destruct /3 width=1/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed-.
-
-lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
- (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2 width=3 by lift_inv_lref1_aux/ qed-.
-
-lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i.
-#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
-#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5 by lift_inv_gref1_aux/ qed-.
-
-fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T2 = ⓑ{a,I} V2. U2.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #p #d #e #a #I #V1 #U1 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
-]
-qed-.
-
-lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T2 = ⓑ{a,I} V2. U2.
-/2 width=3 by lift_inv_bind1_aux/ qed-.
-
-fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T2 = ⓕ{I} V2. U2.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #p #d #e #I #V1 #U1 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
-]
-qed-.
-
-lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T2 = ⓕ{I} V2. U2.
-/2 width=3 by lift_inv_flat1_aux/ qed-.
-
-fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5 by lift_inv_sort2_aux/ qed-.
-
-fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
- (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
- (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-/2 width=3 by lift_inv_lref2_aux/ qed-.
-
-(* Basic_1: was: lift_gen_lref_lt *)
-lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-(* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
- d ≤ i → i < d + e → ⊥.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
-[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
-lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
-elim (lt_refl_false … H)
-qed-.
-
-(* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5 by lift_inv_gref2_aux/ qed-.
-
-fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T1 = ⓑ{a,I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #p #d #e #a #I #V2 #U2 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: ∀d,e,T1,a,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{a,I} V2. U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T1 = ⓑ{a,I} V1. U1.
-/2 width=3 by lift_inv_bind2_aux/ qed-.
-
-fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T1 = ⓕ{I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #p #d #e #I #V2 #U2 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
-]
-qed-.
-
-(* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓕ{I} V2. U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T1 = ⓕ{I} V1. U1.
-/2 width=3 by lift_inv_flat2_aux/ qed-.
-
-lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → ⊥.
-#d #e #J #V elim V -V
-[ * #i #T #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
- ]
-]
-qed-.
-
-(* Basic_1: was: thead_x_lift_y_y *)
-lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → ⊥.
-#J #T elim T -T
-[ * #i #V #d #e #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #d #e #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
- ]
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lift_fwd_pair1: ∀I,T2,V1,U1,d,e. ⇧[d,e] ②{I}V1.U1 ≡ T2 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & T2 = ②{I}V2.U2.
-* [ #a ] #I #T2 #V1 #U1 #d #e #H
-[ elim (lift_inv_bind1 … H) -H /2 width=4/
-| elim (lift_inv_flat1 … H) -H /2 width=4/
-]
-qed-.
-
-lemma lift_fwd_pair2: ∀I,T1,V2,U2,d,e. ⇧[d,e] T1 ≡ ②{I}V2.U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & T1 = ②{I}V1.U1.
-* [ #a ] #I #T1 #V2 #U2 #d #e #H
-[ elim (lift_inv_bind2 … H) -H /2 width=4/
-| elim (lift_inv_flat2 … H) -H /2 width=4/
-]
-qed-.
-
-lemma lift_fwd_tw: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
-qed-.
-
-lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: lift_lref_gt *)
-lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ⇧[d, e] #(i - e) ≡ #i.
-#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
-qed.
-
-lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
-/2 width=1/ qed-.
-
-(* Basic_1: was: lift_r *)
-lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T.
-#T elim T -T
-[ * #i // #d elim (lt_or_ge i d) /2 width=1/
-| * /2 width=1/
-]
-qed.
-
-lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
-#T1 elim T1 -T1
-[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
-| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e
- elim (IHV1 d e) -IHV1 #V2 #HV12
- [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
- | elim (IHT1 d e) -IHT1 /3 width=2/
- ]
-]
-qed.
-
-(* Basic_1: was: lift_free (right to left) *)
-lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 →
- ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
- ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2.
-#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2
-[ /3 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
- >(plus_minus_m_m e2 e1 ?) // /3 width=3/
-| /3 width=3/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT d2 … ? ? He12) // /3 width=5/
-]
-qed.
-
-(* Basic_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
-#T1 elim T1 -T1
-[ * [1,3: /3 width=2/ ] #i #d #e
- elim (lt_dec i d) #Hid
- [ /4 width=2/
- | lapply (false_lt_to_le … Hid) -Hid #Hid
- elim (lt_dec i (d + e)) #Hide
- [ @or_intror * #T1 #H
- elim (lift_inv_lref2_be … H Hid Hide)
- | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
- ]
- ]
-| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e
- [ elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
- [ * #T1 #HT12 @or_introl /3 width=2/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2/
- ]
- | elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 d e) -IHT2
- [ * #T1 #HT12 /4 width=2/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2/
- ]
- ]
-]
-qed.
-
-(* Basic_1: removed theorems 7:
- lift_head lift_gen_head
- lift_weight_map lift_weight lift_weight_add lift_weight_add_O
- lift_tlt_dx
-*)
+++ /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/lift.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Main properies ***********************************************************)
-
-(* Basic_1: was: lift_inj *)
-theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
-#d #e #T1 #U #H elim H -d -e -T1 -U
-[ #k #d #e #X #HX
- lapply (lift_inv_sort2 … HX) -HX //
-| #i #d #e #Hid #X #HX
- lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/
-| #p #d #e #X #HX
- lapply (lift_inv_gref2 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-]
-qed-.
-
-(* Basic_1: was: lift_gen_lift *)
-theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T →
- d1 ≤ d2 →
- ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
- [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
- | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
- elim (le_inv_plus_l … H) -H #Hide2 #He2i
- lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
- >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
- /4 width=3/
- ]
-| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/
-| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
- elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2) /2 width=1/ /3 width=5/
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
- elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
- elim (IHU … HU2) // /3 width=5/
-]
-qed.
-
-(* Note: apparently this was missing in basic_1 *)
-theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
- e ≤ e1 → e1 ≤ e + e2 →
- ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
- [ elim (lift_inv_lref2_be … H) -H // /2 width=1/
- | >(lift_inv_lref2_ge … H ?) -H //
- lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
- elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
- @ex2_intro [2: /2 width=1/ | skip ] -Hd1e12
- @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1/ ]
- ]
-| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
-| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
- elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- elim (IHV1 … HV2) -V // >plus_plus_comm_23 in HT2; #HT2
- elim (IHT1 … HT2) -T // -He1 -He1e2 /3 width=5/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
- elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- elim (IHV1 … HV2) -V //
- elim (IHT1 … HT2) -T // -He1 -He1e2 /3 width=5/
-]
-qed.
-
-theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2.
-#d #e #T #U1 #H elim H -d -e -T -U1
-[ #k #d #e #X #HX
- lapply (lift_inv_sort1 … HX) -HX //
-| #i #d #e #Hid #X #HX
- lapply (lift_inv_lref1_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref1_ge … HX ?) -HX //
-| #p #d #e #X #HX
- lapply (lift_inv_gref1 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-]
-qed-.
-
-(* Basic_1: was: lift_free (left to right) *)
-theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 →
- d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
- >(lift_inv_sort1 … HT2) -HT2 //
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
- | -Hd21 /2 width=1/
- ]
-| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
- >(lift_inv_gref1 … HT2) -HT2 //
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) // /2 width=1/
-]
-qed.
-
-(* Basic_1: was: lift_d (right to left) *)
-theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 →
- ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
- elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
- lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
- >plus_plus_comm_23 /4 width=3/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1/ /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: lift_d (left to right) *)
-theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
- ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
- lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
- lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e
- lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1/ #T
- <plus_minus /2 width=2/ /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 // /3 width=5/
-]
-qed.
-
-(* Advanced properties ******************************************************)
-
-lemma lift_conf_O1: ∀T,T1,d1,e1. ⇧[d1, e1] T ≡ T1 → ∀T2,e2. ⇧[0, e2] T ≡ T2 →
- ∃∃T0. ⇧[0, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T2 ≡ T0.
-#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
-elim (lift_total T1 0 e2) #T0 #HT10
-elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
-lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3/
-qed.
-
-lemma lift_conf_be: ∀T,T1,d,e1. ⇧[d, e1] T ≡ T1 → ∀T2,e2. ⇧[d, e2] T ≡ T2 →
- e1 ≤ e2 → ⇧[d + e1, e2 - e1] T1 ≡ T2.
-#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
-elim (lift_split … HT2 (d+e1) e1) -HT2 // #X #H
->(lift_mono … H … HT1) -T //
-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/lift_lift.ma".
-include "basic_2/relocation/lift_vector.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-(* Main properies ***********************************************************)
-
-theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s →
- ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s.
-#Ts #U1s #d #e #H elim H -Ts -U1s
-[ #U2s #H >(liftv_inv_nil1 … H) -H //
-| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
- elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
- >(lift_mono … HTU1 … HTU2) -T /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/relocation/lift.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Properties on negated basic relocation ***********************************)
-
-lemma nlift_lref_be_SO: ∀X,i. ⇧[i, 1] X ≡ #i → ⊥.
-/2 width=7 by lift_inv_lref2_be/ qed-.
-
-lemma nlift_bind_sn: ∀W,d,e. (∀V. ⇧[d, e] V ≡ W → ⊥) →
- ∀a,I,U. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥).
-#W #d #e #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_bind_dx: ∀U,d,e. (∀T. ⇧[d+1, e] T ≡ U → ⊥) →
- ∀a,I,W. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥).
-#U #d #e #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_flat_sn: ∀W,d,e. (∀V. ⇧[d, e] V ≡ W → ⊥) →
- ∀I,U. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥).
-#W #d #e #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_flat_dx: ∀U,d,e. (∀T. ⇧[d, e] T ≡ U → ⊥) →
- ∀I,W. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥).
-#U #d #e #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
-qed-.
-
-(* Inversion lemmas on negated basic relocation *****************************)
-
-lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⇧[i, 1] X ≡ #j → ⊥) → j = i.
-#i #j elim (lt_or_eq_or_gt i j) // #Hij #H
-[ elim (H (#(j-1))) -H /2 width=1 by lift_lref_ge_minus/
-| elim (H (#j)) -H /2 width=1 by lift_lref_lt/
-]
-qed-.
-
-lemma nlift_inv_bind: ∀a,I,W,U,d,e. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥) →
- (∀V. ⇧[d, e] V ≡ W → ⊥) ∨ (∀T. ⇧[d+1, e] T ≡ U → ⊥).
-#a #I #W #U #d #e #H elim (is_lift_dec W d e)
-[ * /4 width=2 by lift_bind, or_intror/
-| /4 width=2 by ex_intro, or_introl/
-]
-qed-.
-
-lemma nlift_inv_flat: ∀I,W,U,d,e. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥) →
- (∀V. ⇧[d, e] V ≡ W → ⊥) ∨ (∀T. ⇧[d, e] T ≡ U → ⊥).
-#I #W #U #d #e #H elim (is_lift_dec W d e)
-[ * /4 width=2 by lift_flat, or_intror/
-| /4 width=2 by ex_intro, or_introl/
-]
-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/term_vector.ma".
-include "basic_2/relocation/lift.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-inductive liftv (d,e:nat) : relation (list term) ≝
-| liftv_nil : liftv d e (◊) (◊)
-| liftv_cons: ∀T1s,T2s,T1,T2.
- ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s →
- liftv d e (T1 @ T1s) (T2 @ T2s)
-.
-
-interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
-#T1s #T2s #d #e * -T1s -T2s //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
-qed-.
-
-lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
-/2 width=5 by liftv_inv_nil1_aux/ qed-.
-
-fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
- ∀U1,U1s. T1s = U1 @ U1s →
- ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
- T2s = U2 @ U2s.
-#T1s #T2s #d #e * -T1s -T2s
-[ #U1 #U1s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 @ U1s ≡ T2s →
- ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
- T2s = U2 @ U2s.
-/2 width=3 by liftv_inv_cons1_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s.
-#d #e #T1s elim T1s -T1s
-[ /2 width=2 by liftv_nil, ex_intro/
-| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 d e) /3 width=2 by liftv_cons, ex_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/grammar/lenv_length.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-inductive lpx_sn (R:relation3 lenv term term): relation lenv ≝
-| lpx_sn_atom: lpx_sn R (⋆) (⋆)
-| lpx_sn_pair: ∀I,K1,K2,V1,V2.
- lpx_sn R K1 K2 → R K1 V1 V2 →
- lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
-#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #L1 #L2 * -L1 -L2
-[ #J #K1 #V1 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #L1 #L2 * -L1 -L2
-[ #J #K2 #V2 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
- ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
- lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
- ∧∧ lpx_sn R L1 L2 & R L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
-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.ma".
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* alternative definition of lpx_sn *)
-definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
- λR,L1,L2. |L1| = |L2| ∧
- (∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2
- ).
-
-(* Basic forward lemmas ******************************************************)
-
-lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
-#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_sn/
-qed-.
-
-lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #I1 #L2 #K1 #V1 #H elim H -H
-#H #IH elim (length_inv_pos_sn … H) -H
-#I2 #K2 #V2 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
-qed-.
-
-lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
-#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_dx/
-qed-.
-
-lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #I2 #L1 #K2 #V2 #H elim H -H
-#H #IH elim (length_inv_pos_dx … H) -H
-#I1 #K1 #V1 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
-#R @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
-#H destruct
-qed.
-
-lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
- lpx_sn_alt R L1 L2 → R L1 V1 V2 →
- lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
-#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
-#HL12 #IH @conj normalize //
-#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
-[ #HLK1 #HLK2
- lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
- lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
- /2 width=1 by conj/
-| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
-#R #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
-#R #L1 elim L1 -L1
-[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
-| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
-]
-qed-.
-
-(* alternative definition of lpx_sn *****************************************)
-
-lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2
- ) → lpx_sn R L1 L2.
-/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
-
-lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2.
-#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H elim H -H /3 width=4 by 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/relocation/ldrop_leq.ma".
-include "basic_2/relocation/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properies on dropping ****************************************************)
-
-lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by ldrop_drop_lt, ex3_2_intro/
- ]
-]
-qed-.
-
-lemma lpx_sn_ldrop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K2,V2,i. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by ldrop_drop_lt, ex3_2_intro/
- ]
-]
-qed-.
-
-lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_sn R).
-#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (HR … HV12 … HLK1 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
-]
-qed-.
-
-lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
- l_liftable R → dedropable_sn (lpx_sn R).
-#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #K2 #V2 #HK12 #HV12 #H destruct
- lapply (lpx_sn_fwd_length … HK12)
- #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
- /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
- @leq_O2 normalize //
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
- /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1
- /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
-]
-qed-.
-
-fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
- d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
- #K1 #V1 #HK12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
- #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
- <plus_n_Sm #H destruct
-]
-qed-.
-
-lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
-/2 width=5 by lpx_sn_dropable_aux/ 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/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
-#R #HR #L1 #L #H elim H -L1 -L //
-#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
-elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
-qed-.
-
-theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
- confluent2 … (lpx_sn R1) (lpx_sn R2).
-#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
-[ #_ #X1 #H1 #X2 #H2 -n
- >(lpx_sn_inv_atom1 … H1) -X1
- >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
-| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
- elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
- elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, 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/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properties on transitive_closure *****************************************)
-
-lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
-#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
-[ /2 width=1 by lpx_sn_refl/
-| /3 width=1 by TC_reflexive, lpx_sn_refl/
-| /3 width=5 by lpx_sn_pair, step/
-]
-qed-.
-
-lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
- TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
-(* Inversion lemmas on transitive closure ***********************************)
-
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-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 elim (le_plus_xSy_O_false … H)
-| #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/atomicarity_4.ma".
include "basic_2/grammar/aarity.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/substitution/ldrop.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/static/aaa.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/fqus_alt.ma".
+include "basic_2/multiple/fqus_alt.ma".
include "basic_2/static/aaa_lift.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/static/aaa.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/ldrops.ma".
+include "basic_2/multiple/ldrops.ma".
include "basic_2/static/aaa_lift.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_ldrop.ma".
include "basic_2/static/aaa.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
include "basic_2/notation/relations/degree_6.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/substitution/ldrop.ma".
include "basic_2/static/sd.ma".
(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/static/da.ma".
(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
(**************************************************************************)
include "basic_2/notation/relations/lrsubeqc_2.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/substitution/ldrop.ma".
(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
(* *)
(**************************************************************************)
-include "basic_2/substitution/llpx_sn_ldrop.ma".
+include "basic_2/multiple/llpx_sn_ldrop.ma".
include "basic_2/static/ssta.ma".
(* STRATIFIED STATIC TYPE ASSIGNMENT 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 "ground_2/ynat/ynat_max.ma".
+include "basic_2/notation/relations/psubst_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/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/substitution/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/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/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_plus1_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/substitution/lift_neg.ma".
+include "basic_2/substitution/lift_lift.ma".
+include "basic_2/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on negated relocation ***********************************)
+
+lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 →
+ ∀i. d ≤ yinj i → (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) →
+ (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) ∨
+ ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
+ (∀V. ⇧[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
+[ /3 width=2 by or_introl/
+| #I #G #L #K #V #W #j #d #e #Hdj #Hjde #HLK #HVW #i #Hdi #HnW
+ elim (lt_or_ge j i) #Hij
+ [ @or_intror @(ex5_4_intro … HLK) // -HLK
+ [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
+ #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
+ | -HnW /2 width=7 by lift_inv_lref2_be/
+ ]
+ | elim (lift_split … HVW i j) -HVW //
+ #X #_ #H elim HnW -HnW //
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_bind … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_bind_sn, or_introl/
+ | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
+ [ /4 width=9 by nlift_bind_dx, or_introl/
+ | * #J #K #W #j #Hdj #Hji #HLK #HnW
+ elim (yle_inv_succ1 … Hdj) -Hdj #Hdj #Hj
+ lapply (ylt_O … Hj) -Hj #Hj
+ lapply (ldrop_inv_drop1_lt … HLK ?) // -HLK #HLK
+ >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
+ #HnU1 <minus_le_minus_minus_comm in HnW;
+ /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
+ ]
+ ]
+| #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_flat … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_flat_sn, or_introl/
+ | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
+ [ /4 width=9 by nlift_flat_dx, or_introl/
+ | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
+ ]
+]
+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_Y2: ∀G,L,T2,i,d. ⦃G, L⦄ ⊢ #i ▶*[d, ∞] T2 →
- T2 = #i ∨
- ∃∃I,K,V1,V2. d ≤ i & ⇩[i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⇧[O, i+1] V2 ≡ T2.
-#G #L #T2 #i #d #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/
-* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
-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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lazyeq_7.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR CLOSURES ********************************************)
-
-inductive fleq (d) (G) (L1) (T): relation3 genv lenv term ≝
-| fleq_intro: ∀L2. L1 ≡[T, d] L2 → fleq d G L1 T G L2 T
-.
-
-interpretation
- "lazy equivalence (closure)"
- 'LazyEq d G1 L1 T1 G2 L2 T2 = (fleq d G1 L1 T1 G2 L2 T2).
-
-(* Basic_properties *********************************************************)
-
-lemma fleq_refl: ∀d. tri_reflexive … (fleq d).
-/2 width=1 by fleq_intro/ qed.
-
-lemma fleq_sym: ∀d. tri_symmetric … (fleq d).
-#d #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fleq_inv_gen: ∀G1,G2,L1,L2,T1,T2,d. ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L1 ≡[T1, d] L2 & T1 = T2.
-#G1 #G2 #L1 #L2 #T1 #T2 #d * -G2 -L2 -T2 /2 width=1 by and3_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/substitution/lleq_lleq.ma".
-include "basic_2/substitution/fleq.ma".
-
-(* LAZY EQUIVALENCE FOR CLOSURES *******************************************)
-
-(* Main properties **********************************************************)
-
-theorem fleq_trans: ∀d. tri_transitive … (fleq d).
-#d #G1 #G #L1 #L #T1 #T * -G -L -T
-#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
-/3 width=3 by lleq_trans, fleq_intro/
-qed-.
-
-theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,d.
- ⦃G, L, T⦄ ≡[d] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[d] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄.
-/3 width=5 by fleq_trans, fleq_sym/ qed-.
-
-theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,d.
- ⦃G1, L1, T1⦄ ≡[d] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[d] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[d] ⦃G2, L2, T2⦄.
-/3 width=5 by fleq_trans, fleq_sym/ 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/supterm_6.ma".
+include "basic_2/grammar/cl_weight.ma".
+include "basic_2/substitution/ldrop.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* activate genv *)
+inductive fqu: tri_relation genv lenv term ≝
+| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
+| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
+| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
+| fqu_drop : ∀G,L,K,T,U,e.
+ ⇩[e+1] L ≡ K → ⇧[0, e+1] T ≡ U → fqu G L U G K T
+.
+
+interpretation
+ "structural successor (closure)"
+ 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_drop_lt: ∀G,L,K,T,U,e. 0 < e →
+ ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
+#G #L #K #T #U #e #He >(plus_minus_m_m e 1) /2 width=3 by fqu_drop/
+qed.
+
+lemma fqu_lref_S_lt: ∀I,G,L,V,i. 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(i-1)⦄.
+/3 width=3 by fqu_drop, ldrop_drop, lift_lref_ge_minus/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+#G #L #K #T #U #e #HLK #HTU
+lapply (ldrop_fwd_lw_lt … HLK ?) -HLK // #HKL
+lapply (lift_fwd_tw … HTU) -e #H
+normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
+qed-.
+
+fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| < |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[1: normalize //
+|3: #a
+|5: /2 width=4 by ldrop_fwd_length_lt4/
+] #I #G #L #V #T #j #H destruct
+qed-.
+
+lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ → |L2| < |L1|.
+/2 width=7 by fqu_fwd_length_lref1_aux/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqu_wf_ind: ∀R:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
+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/suptermplus_6.ma".
-include "basic_2/relocation/fqu.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu.
-
-interpretation "plus-iterated structural successor (closure)"
- 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed.
-
-lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed.
-
-lemma fqup_ldrop: ∀G1,G2,L1,K1,K2,T1,T2,U1,e. ⇩[e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
- ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #e #HLK1 #HTU1 #HT12 elim (eq_or_gt … e) #H destruct
-[ >(ldrop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 //
-| /3 width=5 by fqup_strap2, fqu_drop_lt/
-]
-qed-.
-
-lemma fqup_lref: ∀I,G,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
-/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_ldrop/ qed.
-
-lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄.
-/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-
-lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊐+ ⦃G, L.ⓑ{I}V, T⦄.
-/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-
-lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+ ⦃G, L, T⦄.
-/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-
-lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+ ⦃G, L, V2⦄.
-/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-
-lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I1}V1, T⦄.
-/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-
-lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I2}V2, T⦄.
-/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
-
-(* Basic eliminators ********************************************************)
-
-lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
-qed-.
-
-lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=3 by fqu_fwd_fw, transitive_lt/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqup_wf_ind: ∀R:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
-qed-.
-
-lemma fqup_wf_ind_eq: ∀R:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
- ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/
-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/fqup.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fqup_trans: tri_transitive … fqup.
-/2 width=5 by tri_TC_transitive/ 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/suptermopt_6.ma".
+include "basic_2/substitution/fqu.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* activate genv *)
+inductive fquq: tri_relation genv lenv term ≝
+| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
+| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
+| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
+| fquq_drop : ∀G,L,K,T,U,e.
+ ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → fquq G L U G K T
+.
+
+interpretation
+ "optional structural successor (closure)"
+ 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquq_refl: tri_reflexive … fquq.
+/2 width=3 by fquq_drop/ qed.
+
+lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
+#G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1
+lapply (ldrop_fwd_lw … HLK1) -HLK1
+lapply (lift_fwd_tw … HTU1) -HTU1
+/2 width=1 by le_plus, le_n/
+qed-.
+
+fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| ≤ |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+[ #a #I #G #L #V #T #j #H destruct
+| #G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #i #H destruct
+ /2 width=3 by ldrop_fwd_length_le4/
+]
+qed-.
+
+lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
+/2 width=7 by fquq_fwd_length_lref1_aux/
+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/suptermoptalt_6.ma".
+include "basic_2/substitution/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* alternative definition of fquq *)
+definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
+
+interpretation
+ "optional structural successor (closure) alternative"
+ 'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquqa_refl: tri_reflexive … fquqa.
+// qed.
+
+lemma fquqa_drop: ∀G,L,K,T,U,e.
+ ⇩[e] L ≡ K → ⇧[0, e] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
+#G #L #K #T #U #e #HLK #HTU elim (eq_or_gt e)
+/3 width=5 by fqu_drop_lt, or_introl/ #H destruct
+>(ldrop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
+qed.
+
+(* Main properties **********************************************************)
+
+theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
+* #H1 #H2 #H3 destruct //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
+/2 width=1 by or_introl/
+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/suptermstar_6.ma".
-include "basic_2/relocation/fquq.ma".
-include "basic_2/substitution/fqup.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
-
-interpretation "star-iterated structural successor (closure)"
- 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
-qed-.
-
-lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fqus_refl: tri_reflexive … fqus.
-/2 width=1 by tri_inj/ qed.
-
-lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed-.
-
-lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed-.
-
-lemma fqus_ldrop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ →
- ∀L1,U1,e. ⇩[e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
- ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄.
-#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
-qed-.
-
-lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
-/3 width=3 by fquq_fwd_fw, transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
-#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_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/fquq_alt.ma".
-include "basic_2/substitution/fqus.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
-#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2
-[ /3 width=5 by fqup_strap1, or_introl/
-| * #HG #HL #HT destruct /2 width=1 by or_introl/
-| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/
-| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1
-[ /2 width=5 by fqup_strap1/
-| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
-]
-qed-.
-
-lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2
-[ /2 width=5 by fqup_strap2/
-| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
-]
-qed-.
-
-lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
-/2 width=5 by fqus_strap1_fqu, fqup_strap1/
-qed-.
-
-lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
-/3 width=5 by fqus_strap2_fqu, fqup_strap2/
-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/fqus.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fqus_trans: tri_transitive … fqus.
-/2 width=5 by tri_TC_transitive/ 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/freestar_4.ma".
-include "basic_2/relocation/lift_neg.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-inductive frees: relation4 ynat lenv term nat ≝
-| frees_eq: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → frees d L U i
-| frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
- (∀T. ⇧[j, 1] T ≡ U → ⊥) → ⇩[j]L ≡ K.ⓑ{I}W →
- frees 0 K W (i-j-1) → frees d L U i.
-
-interpretation
- "context-sensitive free variables (term)"
- 'FreeStar L i d U = (frees d L U i).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⇧[j, 1] T ≡ U → ⊥) &
- ⇩[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
-qed-.
-
-lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
-#L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
-qed-.
-
-lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
-#L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
-qed-.
-
-lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
- j = i ∨
- ∃∃I,K,W. d ≤ yinj j & j < i & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #x #i #H elim (frees_inv … H) -H
-[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
-| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
- >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
-]
-qed-.
-
-lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
-#L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
-* #I #K #W #_ #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
-#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
-qed-.
-
-lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
-#L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
-* #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
-qed-.
-
-lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
-#L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
-* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
-qed-.
-
-lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
- ∃∃I,K,W. d ≤ yinj j & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
-[ #H elim (lt_refl_false j) //
-| * /2 width=5 by ex3_3_intro/
-]
-qed-.
-
-lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
-[ #HnX elim (nlift_inv_bind … HnX) -HnX
- /4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
- [ /4 width=9 by frees_be, or_introl/
- | #HnT @or_intror @(frees_be … HnT) -HnT
- [4,5,6: /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/
- |7: >minus_plus_plus_l //
- |*: skip
- ]
- ]
-]
-qed-.
-
-lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
-#J #L #V #U #d #i #H elim (frees_inv … H) -H
-[ #HnX elim (nlift_inv_flat … HnX) -HnX
- /4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
- /4 width=9 by frees_be, or_intror, or_introl/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
-/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
-
-lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W →
- K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
-/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
-
-lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
-/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
-qed.
-
-lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
-[ /4 width=9 by frees_eq, nlift_bind_dx/
-| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
- elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
- lapply (ylt_O … H) -H #Hj
- >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
- /4 width=9 by frees_be, nlift_bind_dx, ldrop_inv_drop1_lt, lt_plus_to_minus/
-]
-qed.
-
-lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
-/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
-qed.
-
-lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
-/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
-qed.
-
-lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
- ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
-#L #U #d1 #i #H elim H -L -U -d1 -i
-/3 width=9 by frees_be, frees_eq, yle_trans/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ .
-#a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
-/3 width=3 by frees_weak, or_intror, or_introl/
-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/substitution/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma frees_dec: ∀L,U,d,i. Decidable (frees d L U i).
-#L #U @(f2_ind … rfw … L U) -L -U
-#n #IH #L * *
-[ -IH /3 width=5 by frees_inv_sort, or_intror/
-| #j #Hn #d #i elim (lt_or_eq_or_gt i j) #Hji
- [ -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_ge … H ?) -L -d /2 width=1 by lt_to_le/
- | -n /2 width=1 by or_introl/
- | elim (ylt_split j d) #Hdi
- [ -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_skip … H ?) -L //
- | elim (lt_or_ge j (|L|)) #Hj
- [ elim (ldrop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
- elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, ldrop_fwd_rfw, or_introl/ ] #HnW
- @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -d
- lapply (ldrop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
- | -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_free … H ?) -d //
- ]
- ]
- ]
-| -IH /3 width=5 by frees_inv_gref, or_intror/
-| #a #I #W #U #Hn #d #i destruct
- elim (IH L W … d i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
- elim (IH (L.ⓑ{I}W) U … (⫯d) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
- @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
-| #I #W #U #Hn #d #i destruct
- elim (IH L W … d i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
- elim (IH L U … d i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
- @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
-]
-qed-.
-
-lemma frees_S: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[yinj d]⦃U⦄ → ∀I,K,W. ⇩[d] L ≡ K.ⓑ{I}W →
- (K ⊢ i-d-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯d]⦃U⦄.
-#L #U #d #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
-* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
-lapply (yle_inv_inj … Hdj) -Hdj #Hdj
-elim (le_to_or_lt_eq … Hdj) -Hdj
-[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
-| -Hji -HnU #H destruct
- lapply (ldrop_mono … HLK0 … HLK) #H destruct -I
- elim HnW0 -L -U -HnW0 //
-]
-qed.
-
-(* Note: lemma 1250 *)
-lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
- L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
-#a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
-/4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
-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/rdrop_3.ma".
+include "basic_2/grammar/genv.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+inductive gget (e:nat): relation genv ≝
+| gget_gt: ∀G. |G| ≤ e → gget e G (⋆)
+| gget_eq: ∀G. |G| = e + 1 → gget e G G
+| gget_lt: ∀I,G1,G2,V. e < |G1| → gget e G1 G2 → gget e (G1. ⓑ{I} V) G2
+.
+
+interpretation "global reading"
+ 'RDrop e G1 G2 = (gget e G1 G2).
+
+(* basic inversion lemmas ***************************************************)
+
+lemma gget_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ #H2
+ lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+]
+qed-.
+
+lemma gget_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ normalize #H2
+ <(injective_plus_l … H2) in H1; -H2 #H
+ elim (lt_refl_false … H)
+]
+qed-.
+
+fact gget_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
+ e < |G1| → ⇩[e] G1 ≡ G2.
+#I #G #G1 #G2 #V #e * -G -G2
+[ #G #H1 #H destruct #H2
+ lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
+ elim (lt_refl_false … H)
+| #J #G #G2 #W #_ #HG2 #H destruct //
+]
+qed-.
+
+lemma gget_inv_lt: ∀I,G1,G2,V,e.
+ ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
+/2 width=5 by gget_inv_lt_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma gget_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
+#e #G1 elim G1 -G1 /3 width=2/
+#I #V #G1 * #G2 #HG12
+elim (lt_or_eq_or_gt e (|G1|)) #He
+[ /3 width=2/
+| destruct /3 width=2/
+| @ex_intro [2: @gget_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
+]
+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/gget.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+(* Main properties **********************************************************)
+
+theorem gget_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
+#G #G1 #e #H elim H -G -G1
+[ #G #He #G2 #H
+ >(gget_inv_gt … H He) -H -He //
+| #G #He #G2 #H
+ >(gget_inv_eq … H He) -H -He //
+| #I #G #G1 #V #He #_ #IHG1 #G2 #H
+ lapply (gget_inv_lt … H He) -H -He /2 width=1/
+]
+qed-.
+
+lemma gget_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
+#G1 #G2 #e
+elim (gget_total e G1) #G #HG1
+elim (eq_genv_dec G G2) #HG2
+[ destruct /2 width=1/
+| @or_intror #HG12
+ lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 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/notation/relations/rat_3.ma".
-include "basic_2/grammar/term_vector.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-inductive at: list2 nat nat → relation nat ≝
-| at_nil: ∀i. at (⟠) i i
-| at_lt : ∀des,d,e,i1,i2. i1 < d →
- at des i1 i2 → at ({d, e} @ des) i1 i2
-| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
- at des (i1 + e) i2 → at ({d, e} @ des) i1 i2
-.
-
-interpretation "application (generic relocation with pairs)"
- 'RAt i1 des i2 = (at des i1 i2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2.
-#des #i1 #i2 * -des -i1 -i2
-[ //
-| #des #d #e #i1 #i2 #_ #_ #H destruct
-| #des #d #e #i1 #i2 #_ #_ #H destruct
-]
-qed-.
-
-lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2.
-/2 width=3 by at_inv_nil_aux/ qed-.
-
-fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
- ∀d,e,des0. des = {d, e} @ des0 →
- i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨
- d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2.
-#des #i1 #i2 * -des -i1 -i2
-[ #i #d #e #des #H destruct
-| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
-| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_intror, conj/
-]
-qed-.
-
-lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
- i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨
- d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2.
-/2 width=3 by at_inv_cons_aux/ qed-.
-
-lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
- i1 < d → @⦃i1, des⦄ ≡ i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
-lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false … Hd)
-qed-.
-
-lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
- d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
-lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false … Hd)
-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/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-(* Main properties **********************************************************)
-
-theorem at_mono: ∀des,i,i1. @⦃i, des⦄ ≡ i1 → ∀i2. @⦃i, des⦄ ≡ i2 → i1 = i2.
-#des #i #i1 #H elim H -des -i -i1
-[ #i #x #H <(at_inv_nil … H) -x //
-| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H
- lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1 by/
-| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H
- lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1 by/
-]
-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/rminus_3.ma".
-include "basic_2/substitution/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-inductive minuss: nat → relation (list2 nat nat) ≝
-| minuss_nil: ∀i. minuss i (⟠) (⟠)
-| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
- minuss i ({d, e} @ des1) ({d - i, e} @ des2)
-| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
- minuss i ({d, e} @ des1) des2
-.
-
-interpretation "minus (generic relocation with pairs)"
- 'RMinus des1 i des2 = (minuss i des1 des2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠.
-#des1 #des2 #i * -des1 -des2 -i
-[ //
-| #des1 #des2 #d #e #i #_ #_ #H destruct
-| #des1 #des2 #d #e #i #_ #_ #H destruct
-]
-qed-.
-
-lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
-/2 width=4 by minuss_inv_nil1_aux/ qed-.
-
-fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
- ∀d,e,des. des1 = {d, e} @ des →
- d ≤ i ∧ des ▭ e + i ≡ des2 ∨
- ∃∃des0. i < d & des ▭ i ≡ des0 &
- des2 = {d - i, e} @ des0.
-#des1 #des2 #i * -des1 -des2 -i
-[ #i #d #e #des #H destruct
-| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
-| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
-]
-qed-.
-
-lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
- d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
- ∃∃des. i < d & des1 ▭ i ≡ des &
- des2 = {d - i, e} @ des.
-/2 width=3 by minuss_inv_cons1_aux/ qed-.
-
-lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
- d ≤ i → des1 ▭ e + i ≡ des2.
-#des1 #des2 #d #e #i #H
-elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi
-lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
-elim (lt_refl_false … Hi)
-qed-.
-
-lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
- i < d →
- ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des.
-#des1 #des2 #d #e #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
-#Hdi #_ #Hid lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi
-#Hi elim (lt_refl_false … Hi)
-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/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with
-[ nil2 ⇒ ⟠
-| cons2 d e des ⇒ {d + i, e} @ pluss des i
-].
-
-interpretation "plus (generic relocation with pairs)"
- 'plus x y = (pluss x y).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠.
-#i * // normalize
-#d #e #des #H destruct
-qed.
-
-lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} @ des2 →
- ∃∃des1. des1 + i = des2 & des = {d - i, e} @ des1.
-#i #d #e #des2 * normalize
-[ #H destruct
-| #d1 #e1 #des1 #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 "ground_2/lib/bool.ma".
+include "ground_2/lib/lstar.ma".
+include "basic_2/notation/relations/rdrop_5.ma".
+include "basic_2/notation/relations/rdrop_4.ma".
+include "basic_2/notation/relations/rdrop_3.ma".
+include "basic_2/grammar/lenv_length.ma".
+include "basic_2/grammar/cl_restricted_weight.ma".
+include "basic_2/substitution/lift.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Basic_1: includes: drop_skip_bind *)
+inductive ldrop (s:bool): relation4 nat nat lenv lenv ≝
+| ldrop_atom: ∀d,e. (s = Ⓕ → e = 0) → ldrop s d e (⋆) (⋆)
+| ldrop_pair: ∀I,L,V. ldrop s 0 0 (L.ⓑ{I}V) (L.ⓑ{I}V)
+| ldrop_drop: ∀I,L1,L2,V,e. ldrop s 0 e L1 L2 → ldrop s 0 (e+1) (L1.ⓑ{I}V) L2
+| ldrop_skip: ∀I,L1,L2,V1,V2,d,e.
+ ldrop s d e L1 L2 → ⇧[d, e] V2 ≡ V1 →
+ ldrop s (d+1) e (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+.
+
+interpretation
+ "basic slicing (local environment) abstract"
+ 'RDrop s d e L1 L2 = (ldrop s d e L1 L2).
+(*
+interpretation
+ "basic slicing (local environment) general"
+ 'RDrop d e L1 L2 = (ldrop true d e L1 L2).
+*)
+interpretation
+ "basic slicing (local environment) lget"
+ 'RDrop e L1 L2 = (ldrop false O e L1 L2).
+
+definition l_liftable: predicate (lenv → relation term) ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
+
+definition l_deliftable_sn: predicate (lenv → relation term) ≝
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T1. ⇧[d, e] T1 ≡ U1 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: predicate (relation lenv) ≝
+ λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
+ ∃∃K2. R K1 K2 & ⇩[s, d, e] L2 ≡ K2.
+
+definition dropable_dx: predicate (relation lenv) ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & R K1 K2.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrop_inv_atom1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆ ∧ (s = Ⓕ → e = 0).
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ /3 width=1 by conj/
+| #I #L #V #H destruct
+| #I #L1 #L2 #V #e #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was: drop_gen_sort *)
+lemma ldrop_inv_atom1: ∀L2,s,d,e. ⇩[s, d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ (s = Ⓕ → e = 0).
+/2 width=4 by ldrop_inv_atom1_aux/ qed-.
+
+fact ldrop_inv_O1_pair1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K.ⓑ{I}V →
+ (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
+ (0 < e ∧ ⇩[s, d, e-1] K ≡ L2).
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #K #J #W #H destruct
+| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
+| #I #L1 #L2 #V #e #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed-.
+
+lemma ldrop_inv_O1_pair1: ∀I,K,L2,V,s,e. ⇩[s, 0, e] K. ⓑ{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K.ⓑ{I}V) ∨
+ (0 < e ∧ ⇩[s, 0, e-1] K ≡ L2).
+/2 width=3 by ldrop_inv_O1_pair1_aux/ qed-.
+
+lemma ldrop_inv_pair1: ∀I,K,L2,V,s. ⇩[s, 0, 0] K.ⓑ{I}V ≡ L2 → L2 = K.ⓑ{I}V.
+#I #K #L2 #V #s #H
+elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
+elim (lt_refl_false … H)
+qed-.
+
+(* Basic_1: was: drop_gen_drop *)
+lemma ldrop_inv_drop1_lt: ∀I,K,L2,V,s,e.
+ ⇩[s, 0, e] K.ⓑ{I}V ≡ L2 → 0 < e → ⇩[s, 0, e-1] K ≡ L2.
+#I #K #L2 #V #s #e #H #He
+elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
+elim (lt_refl_false … He)
+qed-.
+
+lemma ldrop_inv_drop1: ∀I,K,L2,V,s,e.
+ ⇩[s, 0, e+1] K.ⓑ{I}V ≡ L2 → ⇩[s, 0, e] K ≡ L2.
+#I #K #L2 #V #s #e #H lapply (ldrop_inv_drop1_lt … H ?) -H //
+qed-.
+
+fact ldrop_inv_skip1_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L2 = K2.ⓑ{I}V2.
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #J #K1 #W1 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop_gen_skip_l *)
+lemma ldrop_inv_skip1: ∀I,K1,V1,L2,s,d,e. ⇩[s, d, e] K1.ⓑ{I}V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L2 = K2.ⓑ{I}V2.
+/2 width=3 by ldrop_inv_skip1_aux/ qed-.
+
+lemma ldrop_inv_O1_pair2: ∀I,K,V,s,e,L1. ⇩[s, 0, e] L1 ≡ K.ⓑ{I}V →
+ (e = 0 ∧ L1 = K.ⓑ{I}V) ∨
+ ∃∃I1,K1,V1. ⇩[s, 0, e-1] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & 0 < e.
+#I #K #V #s #e *
+[ #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #H
+ elim (ldrop_inv_O1_pair1 … H) -H *
+ [ #H1 #H2 destruct /3 width=1 by or_introl, conj/
+ | /3 width=5 by ex3_3_intro, or_intror/
+ ]
+]
+qed-.
+
+fact ldrop_inv_skip2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 &
+ ⇧[d-1, e] V2 ≡ V1 &
+ L1 = K1.ⓑ{I}V1.
+#L1 #L2 #s #d #e * -L1 -L2 -d -e
+[ #d #e #_ #_ #J #K2 #W2 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop_gen_skip_r *)
+lemma ldrop_inv_skip2: ∀I,L1,K2,V2,s,d,e. ⇩[s, d, e] L1 ≡ K2.ⓑ{I}V2 → 0 < d →
+ ∃∃K1,V1. ⇩[s, d-1, e] K1 ≡ K2 & ⇧[d-1, e] V2 ≡ V1 &
+ L1 = K1.ⓑ{I}V1.
+/2 width=3 by ldrop_inv_skip2_aux/ qed-.
+
+lemma ldrop_inv_O1_gt: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → |L| < e →
+ s = Ⓣ ∧ K = ⋆.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize in ⊢ (?%?→?); #H1e
+[ elim (ldrop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
+ #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1e)
+| elim (ldrop_inv_O1_pair1 … H) -H * #H2e #HLK destruct
+ [ elim (lt_zero_false … H1e)
+ | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
+ ]
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ldrop_refl_atom_O2: ∀s,d. ⇩[s, d, O] ⋆ ≡ ⋆.
+/2 width=1 by ldrop_atom/ qed.
+
+(* Basic_1: was by definition: drop_refl *)
+lemma ldrop_refl: ∀L,d,s. ⇩[s, d, 0] L ≡ L.
+#L elim L -L //
+#L #I #V #IHL #d #s @(nat_ind_plus … d) -d /2 width=1 by ldrop_pair, ldrop_skip/
+qed.
+
+lemma ldrop_drop_lt: ∀I,L1,L2,V,s,e.
+ ⇩[s, 0, e-1] L1 ≡ L2 → 0 < e → ⇩[s, 0, e] L1.ⓑ{I}V ≡ L2.
+#I #L1 #L2 #V #s #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by ldrop_drop/
+qed.
+
+lemma ldrop_skip_lt: ∀I,L1,L2,V1,V2,s,d,e.
+ ⇩[s, d-1, e] L1 ≡ L2 → ⇧[d-1, e] V2 ≡ V1 → 0 < d →
+ ⇩[s, d, e] L1. ⓑ{I} V1 ≡ L2.ⓑ{I}V2.
+#I #L1 #L2 #V1 #V2 #s #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) /2 width=1 by ldrop_skip/
+qed.
+
+lemma ldrop_O1_le: ∀s,e,L. e ≤ |L| → ∃K. ⇩[s, 0, e] L ≡ K.
+#s #e @(nat_ind_plus … e) -e /2 width=2 by ex_intro/
+#e #IHe *
+[ #H elim (le_plus_xSy_O_false … H)
+| #L #I #V normalize #H elim (IHe L) -IHe /3 width=2 by ldrop_drop, monotonic_pred, ex_intro/
+]
+qed-.
+
+lemma ldrop_O1_lt: ∀s,L,e. e < |L| → ∃∃I,K,V. ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
+#s #L elim L -L
+[ #e #H elim (lt_zero_false … H)
+| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4 by ldrop_pair, ex1_3_intro/
+ #e #_ normalize #H elim (IHL e) -IHL /3 width=4 by ldrop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
+]
+qed-.
+
+lemma ldrop_O1_pair: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → e ≤ |L| → ∀I,V.
+ ∃∃J,W. ⇩[s, 0, e] L.ⓑ{I}V ≡ K.ⓑ{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize #He #I #V
+[ elim (ldrop_inv_atom1 … H) -H #H <(le_n_O_to_eq … He) -e
+ #Hs destruct /2 width=3 by ex1_2_intro/
+| elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK destruct /2 width=3 by ex1_2_intro/
+ elim (IHL … HLK … Z X) -IHL -HLK
+ /3 width=3 by ldrop_drop_lt, le_plus_to_minus, ex1_2_intro/
+]
+qed-.
+
+lemma ldrop_O1_ge: ∀L,e. |L| ≤ e → ⇩[Ⓣ, 0, e] L ≡ ⋆.
+#L elim L -L [ #e #_ @ldrop_atom #H destruct ]
+#L #I #V #IHL #e @(nat_ind_plus … e) -e [ #H elim (le_plus_xSy_O_false … H) ]
+normalize /4 width=1 by ldrop_drop, monotonic_pred/
+qed.
+
+lemma ldrop_split: ∀L1,L2,d,e2,s. ⇩[s, d, e2] L1 ≡ L2 → ∀e1. e1 ≤ e2 →
+ ∃∃L. ⇩[s, d, e2 - e1] L1 ≡ L & ⇩[s, d, e1] L ≡ L2.
+#L1 #L2 #d #e2 #s #H elim H -L1 -L2 -d -e2
+[ #d #e2 #Hs #e1 #He12 @(ex2_intro … (⋆))
+ @ldrop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
+| #I #L1 #V #e1 #He1 lapply (le_n_O_to_eq … He1) -He1
+ #H destruct /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #e2 #HL12 #IHL12 #e1 @(nat_ind_plus … e1) -e1
+ [ /3 width=3 by ldrop_drop, ex2_intro/
+ | -HL12 #e1 #_ #He12 lapply (le_plus_to_le_r … He12) -He12
+ #He12 elim (IHL12 … He12) -IHL12 >minus_plus_plus_l
+ #L #HL1 #HL2 elim (lt_or_ge (|L1|) (e2-e1)) #H0
+ [ elim (ldrop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
+ elim (ldrop_inv_atom1 … HL2) -HL2 #H #_ destruct
+ @(ex2_intro … (⋆)) [ @ldrop_O1_ge normalize // ]
+ @ldrop_atom #H destruct
+ | elim (ldrop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by ldrop_drop, ex2_intro/
+ ]
+ ]
+| #I #L1 #L2 #V1 #V2 #d #e2 #_ #HV21 #IHL12 #e1 #He12 elim (IHL12 … He12) -IHL12
+ #L #HL1 #HL2 elim (lift_split … HV21 d e1) -HV21 /3 width=5 by ldrop_skip, ex2_intro/
+]
+qed-.
+
+lemma ldrop_FT: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+/3 width=1 by ldrop_atom, ldrop_drop, ldrop_skip/
+qed.
+
+lemma ldrop_gen: ∀L1,L2,s,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → ⇩[s, d, e] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by ldrop_FT/
+qed-.
+
+lemma ldrop_T: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ⇩[Ⓣ, d, e] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by ldrop_FT/
+qed-.
+
+lemma l_liftable_LTC: ∀R. l_liftable R → l_liftable (LTC … R).
+#R #HR #K #T1 #T2 #H elim H -T2
+[ /3 width=10 by inj/
+| #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /4 width=12 by step/
+]
+qed-.
+
+lemma l_deliftable_sn_LTC: ∀R. l_deliftable_sn R → l_deliftable_sn (LTC … R).
+#R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #K #s #d #e #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
+| #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #s #d #e #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
+ /3 width=3 by step, ex2_intro/
+]
+qed-.
+
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #s #e #HLK2 elim (HR … HL12 … HLK2) -HR -L2
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #K2 #s #e #HLK2 elim (HR … HL2 … HLK2) -HR -L2
+ #K #HLK #HK2 elim (IHL1 … HLK) -L
+ /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma l_deliftable_sn_llstar: ∀R. l_deliftable_sn R →
+ ∀l. l_deliftable_sn (llstar … R l).
+#R #HR #l #L #U1 #U2 #H @(lstar_ind_r … l U2 H) -l -U2
+[ /2 width=3 by lstar_O, ex2_intro/
+| #l #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
+]
+qed-.
+
+(* Basic forvard lemmas *****************************************************)
+
+(* Basic_1: was: drop_S *)
+lemma ldrop_fwd_drop2: ∀L1,I2,K2,V2,s,e. ⇩[s, O, e] L1 ≡ K2. ⓑ{I2} V2 →
+ ⇩[s, O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #s #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #H
+ [ -IHL1 destruct /2 width=1 by ldrop_drop/
+ | @ldrop_drop >(plus_minus_m_m e 1) /2 width=3 by/
+ ]
+]
+qed-.
+
+lemma ldrop_fwd_length_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → |L1| ≤ d → |L2| = |L1|.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ #I #L1 #L2 #V #e #_ #_ #H elim (le_plus_xSy_O_false … H)
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma ldrop_fwd_length_le_le: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → e ≤ |L1| - d → |L2| = |L1| - e.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ /3 width=2 by le_plus_to_le_r/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 >minus_plus_plus_l
+ #Hd #He lapply (le_plus_to_le_r … Hd) -Hd
+ #Hd >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
+]
+qed-.
+
+lemma ldrop_fwd_length_le_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → |L1| - d ≤ e → |L2| = d.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e normalize
+[ /2 width=1 by le_n_O_to_eq/
+| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false … H)
+| /3 width=2 by le_plus_to_le_r/
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L1| = |L2| + e.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1 by/
+qed-.
+
+lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by plus_minus, le_n/
+qed-.
+
+lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e = |L1| - |L2|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e.
+ ⇩[Ⓕ, d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
+#L1 #I2 #K2 #V2 #d #e #H
+lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
+qed-.
+
+lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by lt_minus_to_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq1: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |L1| = |L2| → |K1| = |K2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2
+/2 width=2 by injective_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq2: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |K1| = |K2| → |L1| = |L2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2 //
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e // normalize
+[ /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #H >H -H //
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #HL12 #_ #_
+ lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
+ @(le_to_lt_to_lt … HL12) -HL12 //
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+ >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
+]
+qed-.
+
+lemma ldrop_fwd_rfw: ∀I,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#I #L #K #V #i #HLK lapply (ldrop_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact ldrop_inv_O2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → e = 0 → L1 = L2.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+[ //
+| //
+| #I #L1 #L2 #V #e #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H
+ >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e //
+]
+qed-.
+
+(* Basic_1: was: drop_gen_refl *)
+lemma ldrop_inv_O2: ∀L1,L2,s,d. ⇩[s, d, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5 by ldrop_inv_O2_aux/ qed-.
+
+lemma ldrop_inv_length_eq: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L1| = |L2| → e = 0.
+#L1 #L2 #d #e #H #HL12 lapply (ldrop_fwd_length_minus4 … H) //
+qed-.
+
+lemma ldrop_inv_refl: ∀L,d,e. ⇩[Ⓕ, d, e] L ≡ L → e = 0.
+/2 width=5 by ldrop_inv_length_eq/ qed-.
+
+fact ldrop_inv_FT_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 →
+ ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → d = 0 →
+ ⇩[Ⓕ, d, e] L1 ≡ K.ⓑ{I}V.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #_ #J #K #W #H destruct
+| #I #L #V #J #K #W #H destruct //
+| #I #L1 #L2 #V #e #_ #IHL12 #J #K #W #H1 #H2 destruct
+ /3 width=1 by ldrop_drop/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #_ #J #K #W #_ #_
+ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma ldrop_inv_FT: ∀I,L,K,V,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
+/2 width=5 by ldrop_inv_FT_aux/ qed.
+
+lemma ldrop_inv_gen: ∀I,L,K,V,s,e. ⇩[s, 0, e] L ≡ K.ⓑ{I}V → ⇩[e] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by ldrop_inv_FT/
+qed-.
+
+lemma ldrop_inv_T: ∀I,L,K,V,s,e. ⇩[Ⓣ, 0, e] L ≡ K.ⓑ{I}V → ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by ldrop_inv_FT/
+qed-.
+
+(* Basic_1: removed theorems 50:
+ drop_ctail drop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ drop_clear drop_clear_O drop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+ getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+ drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+ getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
--- /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/lift_lift.ma".
+include "basic_2/substitution/ldrop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop_mono *)
+theorem ldrop_mono: ∀L,L1,s1,d,e. ⇩[s1, d, e] L ≡ L1 →
+ ∀L2,s2. ⇩[s2, d, e] L ≡ L2 → L1 = L2.
+#L #L1 #s1 #d #e #H elim H -L -L1 -d -e
+[ #d #e #He #L2 #s2 #H elim (ldrop_inv_atom1 … H) -H //
+| #I #K #V #L2 #s2 #HL12 <(ldrop_inv_O2 … HL12) -L2 //
+| #I #L #K #V #e #_ #IHLK #L2 #s2 #H
+ lapply (ldrop_inv_drop1 … H) -H /2 width=2 by/
+| #I #L #K1 #T #V1 #d #e #_ #HVT1 #IHLK1 #X #s2 #H
+ elim (ldrop_inv_skip1 … H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
+ >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
+ >(IHLK1 … HLK2) -IHLK1 -HLK2 //
+]
+qed-.
+
+(* Basic_1: was: drop_conf_ge *)
+theorem ldrop_conf_ge: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
+ ⇩[s2, 0, e2 - e1] L1 ≡ L2.
+#L #L1 #s1 #d1 #e1 #H elim H -L -L1 -d1 -e1 //
+[ #d #e #_ #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He destruct
+ @ldrop_atom #H >He // (**) (* explicit constructor *)
+| #I #L #K #V #e #_ #IHLK #L2 #s2 #e2 #H #He2
+ lapply (ldrop_inv_drop1_lt … H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
+ <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
+| #I #L #K #V1 #V2 #d #e #_ #_ #IHLK #L2 #s2 #e2 #H #Hdee2
+ lapply (transitive_le 1 … Hdee2) // #He2
+ lapply (ldrop_inv_drop1_lt … H ?) -H // -He2 #HL2
+ lapply (transitive_le (1+e) … Hdee2) // #Hee2
+ @ldrop_drop_lt >minus_minus_comm /3 width=1 by lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem ldrop_conf_be: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. ⇩[s1, 0, d1 + e1 - e2] L2 ≡ L & ⇩[d1] L1 ≡ L.
+#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #He1 #L2 #e2 #H #Hd1 #_ elim (ldrop_inv_atom1 … H) -H #H #He2 destruct
+ >(He2 ?) in Hd1; // -He2 #Hd1 <(le_n_O_to_eq … Hd1) -d1
+ /4 width=3 by ldrop_atom, ex2_intro/
+| normalize #I #L #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_O2 … HL2) -HL2 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
+| normalize #I #L0 #K0 #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1_pair1 … H) -H * * #He2 #HL20
+ [ -IHLK0 -He21 destruct <minus_n_O /3 width=3 by ldrop_drop, ex2_intro/
+ | -HLK0 <minus_le_minus_minus_comm //
+ elim (IHLK0 … HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
+ ]
+| #I #L0 #K0 #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_drop1_lt … H ?) -H // #HL02
+ elim (IHLK0 … HL02) -L0 /3 width=3 by ldrop_drop, monotonic_pred, ex2_intro/
+]
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem ldrop_conf_le: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ⇩[s2, 0, e2] L1 ≡ L & ⇩[s1, d1 - e2, e1] L2 ≡ L.
+#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #He1 #L2 #s2 #e2 #H elim (ldrop_inv_atom1 … H) -H
+ #H #He2 #_ destruct /4 width=3 by ldrop_atom, ex2_intro/
+| #I #K0 #V0 #L2 #s2 #e2 #H1 #H2
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct
+ lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
+| #I #K0 #K1 #V0 #e1 #HK01 #_ #L2 #s2 #e2 #H1 #H2
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct
+ lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by ldrop_drop, ex2_intro/
+| #I #K0 #K1 #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #s2 #e2 #H #He2d1
+ elim (ldrop_inv_O1_pair1 … H) -H *
+ [ -IHK01 -He2d1 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
+ | -HK01 -HV10 #He2 #HK0L2
+ elim (IHK01 … HK0L2) -IHK01 -HK0L2 /2 width=1 by monotonic_pred/
+ >minus_le_minus_minus_comm /3 width=3 by ldrop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: with "s2", the conclusion parameter is "s1 ∨ s2" *)
+(* Basic_1: was: drop_trans_ge *)
+theorem ldrop_trans_ge: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀L2,e2. ⇩[e2] L ≡ L2 → d1 ≤ e2 → ⇩[s1, 0, e1 + e2] L1 ≡ L2.
+#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
+[ #d1 #e1 #He1 #L2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He2 destruct /4 width=1 by ldrop_atom, eq_f2/
+| /2 width=1 by ldrop_gen/
+| /3 width=1 by ldrop_drop/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 #L #e2 #H #Hde2
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
+ lapply (ldrop_inv_drop1_lt … H ?) -H // #HL2
+ @ldrop_drop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
+]
+qed.
+
+(* Basic_1: was: drop_trans_le *)
+theorem ldrop_trans_le: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → e2 ≤ d1 →
+ ∃∃L0. ⇩[s2, 0, e2] L1 ≡ L0 & ⇩[s1, d1 - e2, e1] L0 ≡ L2.
+#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
+[ #d1 #e1 #He1 #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He2 destruct /4 width=3 by ldrop_atom, ex2_intro/
+| #I #K #V #L2 #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+ #H destruct /2 width=3 by ldrop_pair, ex2_intro/
+| #I #L1 #L2 #V #e #_ #IHL12 #L #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+ #H destruct elim (IHL12 … HL2) -IHL12 -HL2 //
+ #L0 #H #HL0 lapply (ldrop_inv_O2 … H) -H #H destruct
+ /3 width=5 by ldrop_pair, ldrop_drop, ex2_intro/
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV12 #IHL12 #L #s2 #e2 #H #He2d
+ elim (ldrop_inv_O1_pair1 … H) -H *
+ [ -He2d -IHL12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
+ | -HL12 -HV12 #He2 #HL2
+ elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by ldrop_drop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma l_liftable_llstar: ∀R. l_liftable R → ∀l. l_liftable (llstar … R l).
+#R #HR #l #K #T1 #T2 #H @(lstar_ind_r … l T2 H) -l -T2
+[ #L #s #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
+ >(lift_mono … HTU2 … HTU1) -T1 -U2 -d -e //
+| #l #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /3 width=12 by lstar_dx/
+]
+qed-.
+
+(* Basic_1: was: drop_conf_lt *)
+lemma ldrop_conf_lt: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
+ ∀I,K2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ K2.ⓑ{I}V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ⇩[s2, 0, e2] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[s1, d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
+#L #L1 #s1 #d1 #e1 #H1 #I #K2 #V2 #s2 #e2 #H2 #He2d1
+elim (ldrop_conf_le … H1 … H2) -L /2 width=2 by lt_to_le/ #K #HL1K #HK2
+elim (ldrop_inv_skip1 … HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/
+#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ldrop_trans_lt: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀I,L2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ L2.ⓑ{I}V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃L0,V0. ⇩[s2, 0, e2] L1 ≡ L0.ⓑ{I}V0 &
+ ⇩[s1, d, e1] L0 ≡ L2 & ⇧[d, e1] V2 ≡ V0.
+#L1 #L #s1 #d1 #e1 #HL1 #I #L2 #V2 #s2 #e2 #HL2 #Hd21
+elim (ldrop_trans_le … HL1 … HL2) -L /2 width=1 by lt_to_le/ #L0 #HL10 #HL02
+elim (ldrop_inv_skip2 … HL02) -HL02 /2 width=1 by lt_plus_to_minus_r/ #L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma ldrop_trans_ge_comm: ∀L1,L,L2,s1,d1,e1,e2.
+ ⇩[s1, d1, e1] L1 ≡ L → ⇩[e2] L ≡ L2 → d1 ≤ e2 →
+ ⇩[s1, 0, e2 + e1] L1 ≡ L2.
+#L1 #L #L2 #s1 #d1 #e1 #e2
+>commutative_plus /2 width=5 by ldrop_trans_ge/
+qed.
+
+lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[e1] L ≡ K.ⓑ{I1}V1 →
+ ∀I2,V2,e2. ⇩[e2] L ≡ K.ⓑ{I2}V2 →
+ ∧∧ e1 = e2 & I1 = I2 & V1 = V2.
+#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2
+elim (le_or_ge e1 e2) #He
+[ lapply (ldrop_conf_ge … HLK1 … HLK2 ?)
+| lapply (ldrop_conf_ge … HLK2 … HLK1 ?)
+] -HLK1 -HLK2 // #HK
+lapply (ldrop_fwd_length_minus2 … HK) #H
+elim (discr_minus_x_xy … H) -H
+[1,3: normalize <plus_n_Sm #H destruct ]
+#H >H in HK; #HK
+lapply (ldrop_inv_O2 … HK) -HK #H destruct
+lapply (inv_eq_minus_O … H) -H /3 width=1 by le_to_le_to_eq, and3_intro/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma ldrop_fwd_be: ∀L,K,s,d,e,i. ⇩[s, d, e] L ≡ K → |K| ≤ i → i < d → |L| ≤ i.
+#L #K #s #d #e #i #HLK #HK #Hd elim (lt_or_ge i (|L|)) //
+#HL elim (ldrop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
+elim (ldrop_conf_lt … HLK … HLK0) // -HLK -HLK0 -Hd
+#K1 #V1 #HK1 #_ #_ lapply (ldrop_fwd_length_lt2 … HK1) -I -K1 -V1
+#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/grammar/leq_leq.ma".
+include "basic_2/substitution/ldrop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+definition dedropable_sn: predicate (relation lenv) ≝
+ λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
+ ∃∃L2. R L1 L2 & ⇩[s, d, e] L2 ≡ K2 & L1 ≃[d, e] L2.
+
+(* Properties on equivalence ************************************************)
+
+lemma leq_ldrop_trans_be: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
+ ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W →
+ d ≤ i → i < d + e →
+ ∃∃K1. K1 ≃[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #J #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #e #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct >ypred_succ
+ /2 width=3 by ldrop_pair, ex2_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=3 by ldrop_drop_lt, ex2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J #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=3 by ylt_O, ldrop_drop_lt, ex2_intro/
+]
+qed-.
+
+lemma leq_ldrop_conf_be: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
+ ∀I,K1,W,s,i. ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W →
+ d ≤ i → i < d + e →
+ ∃∃K2. K1 ≃[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W.
+#L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hdi #Hide
+elim (leq_ldrop_trans_be … (leq_sym … HL12) … HLK1) // -L1 -Hdi -Hide
+/3 width=3 by leq_sym, ex2_intro/
+qed-.
+
+lemma ldrop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
+ ∃∃L2. L1 ≃[0, i] L2 & ⇩[i] L2 ≡ K2.
+#K2 #i @(nat_ind_plus … i) -i
+[ /3 width=3 by leq_O2, ex2_intro/
+| #i #IHi #Y #Hi elim (ldrop_O1_lt (Ⓕ) Y 0) //
+ #I #L1 #V #H lapply (ldrop_inv_O2 … H) -H #H destruct
+ normalize in Hi; elim (IHi L1) -IHi
+ /3 width=5 by ldrop_drop, leq_pair, injective_plus_l, ex2_intro/
+]
+qed-.
+
+lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
+#R #HR #L1 #K1 #s #d #e #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
+ /3 width=4 by inj, ex3_intro/
+| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
+ /3 width=6 by leq_trans, step, ex3_intro/
+]
+qed-.
+
+(* Inversion lemmas on equivalence ******************************************)
+
+lemma ldrop_O1_inj: ∀i,L1,L2,K. ⇩[i] L1 ≡ K → ⇩[i] L2 ≡ K → L1 ≃[i, ∞] L2.
+#i @(nat_ind_plus … i) -i
+[ #L1 #L2 #K #H <(ldrop_inv_O2 … H) -K #H <(ldrop_inv_O2 … H) -L1 //
+| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
+ lapply (ldrop_fwd_length … HLK1)
+ <(ldrop_fwd_length … HLK2) [ /4 width=5 by ldrop_inv_drop1, leq_succ/ ]
+ normalize <plus_n_Sm #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/notation/relations/rdropstar_3.ma".
-include "basic_2/notation/relations/rdropstar_4.ma".
-include "basic_2/relocation/ldrop.ma".
-include "basic_2/substitution/gr2_minus.ma".
-include "basic_2/substitution/lifts.ma".
-
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
-
-inductive ldrops (s:bool): list2 nat nat → relation lenv ≝
-| ldrops_nil : ∀L. ldrops s (⟠) L L
-| ldrops_cons: ∀L1,L,L2,des,d,e.
- ldrops s des L1 L → ⇩[s, d, e] L ≡ L2 → ldrops s ({d, e} @ des) L1 L2
-.
-
-interpretation "iterated slicing (local environment) abstract"
- 'RDropStar s des T1 T2 = (ldrops s des T1 T2).
-(*
-interpretation "iterated slicing (local environment) general"
- 'RDropStar des T1 T2 = (ldrops true des T1 T2).
-*)
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ldrops_inv_nil_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → des = ⟠ → L1 = L2.
-#L1 #L2 #s #des * -L1 -L2 -des //
-#L1 #L #L2 #d #e #des #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: drop1_gen_pnil *)
-lemma ldrops_inv_nil: ∀L1,L2,s. ⇩*[s, ⟠] L1 ≡ L2 → L1 = L2.
-/2 width=4 by ldrops_inv_nil_aux/ qed-.
-
-fact ldrops_inv_cons_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 →
- ∀d,e,tl. des = {d, e} @ tl →
- ∃∃L. ⇩*[s, tl] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
-#L1 #L2 #s #des * -L1 -L2 -des
-[ #L #d #e #tl #H destruct
-| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
- /2 width=3 by ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop1_gen_pcons *)
-lemma ldrops_inv_cons: ∀L1,L2,s,d,e,des. ⇩*[s, {d, e} @ des] L1 ≡ L2 →
- ∃∃L. ⇩*[s, des] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
-/2 width=3 by ldrops_inv_cons_aux/ qed-.
-
-lemma ldrops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
- ∀L1,K2,V2. ⇩*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
- ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
- ⇩*[s, des1] K1 ≡ K2 &
- ⇧*[des1] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#I #s #des #des2 #i #H elim H -des -des2 -i
-[ #i #L1 #K2 #V2 #H
- >(ldrops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, ldrops_nil/
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (ldrops_inv_cons … H) -H #L #HL1 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
- elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, ldrops_cons/ | skip ]
- normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
- /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: drop1_skip_bind *)
-lemma ldrops_skip: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
- ∀I. ⇩*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
-#L1 #L2 #s #des #H elim H -L1 -L2 -des
-[ #L #V1 #V2 #HV12 #I
- >(lifts_inv_nil … HV12) -HV12 //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
- elim (lifts_inv_cons … H) -H /3 width=5 by ldrop_skip, ldrops_cons/
-].
-qed.
-
-(* Basic_1: removed theorems 1: drop1_getl_trans *)
+++ /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/substitution/ldrops.ma".
-
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
-
-(* Properties concerning basic local environment slicing ********************)
-
-lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[Ⓕ, des] L1 ≡ L → ∀L2,i. ⇩[i] L ≡ L2 →
- ∃∃L0,des0,i0. ⇩[i0] L1 ≡ L0 & ⇩*[Ⓕ, des0] L0 ≡ L2 &
- @⦃i, des⦄ ≡ i0 & des ▭ i ≡ des0.
-#L1 #L #des #H elim H -L1 -L -des
-[ /2 width=7 by ldrops_nil, minuss_nil, at_nil, ex4_3_intro/
-| #L1 #L0 #L #des #d #e #_ #HL0 #IHL0 #L2 #i #HL2
- elim (lt_or_ge i d) #Hid
- [ elim (ldrop_trans_le … HL0 … HL2) -L /2 width=2 by lt_to_le/
- #L #HL0 #HL2 elim (IHL0 … HL0) -L0 /3 width=7 by ldrops_cons, minuss_lt, at_lt, ex4_3_intro/
- | lapply (ldrop_trans_ge … HL0 … HL2 ?) -L // #HL02
- elim (IHL0 … HL02) -L0 /3 width=7 by minuss_ge, at_ge, ex4_3_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/substitution/ldrops_ldrop.ma".
-
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop1_trans *)
-theorem ldrops_trans: ∀L,L2,s,des2. ⇩*[s, des2] L ≡ L2 → ∀L1,des1. ⇩*[s, des1] L1 ≡ L →
- ⇩*[s, des2 @@ des1] L1 ≡ L2.
-#L #L2 #s #des2 #H elim H -L -L2 -des2 /3 width=3 by ldrops_cons/
-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/rlift_4.ma".
+include "basic_2/grammar/term_weight.ma".
+include "basic_2/grammar/term_simple.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Basic_1: includes:
+ lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
+*)
+inductive lift: relation4 nat nat term term ≝
+| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k)
+| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
+| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
+| lift_gref : ∀p,d,e. lift d e (§p) (§p)
+| lift_bind : ∀a,I,V1,V2,T1,T2,d,e.
+ lift d e V1 V2 → lift (d + 1) e T1 T2 →
+ lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| lift_flat : ∀I,V1,V2,T1,T2,d,e.
+ lift d e V1 V2 → lift d e T1 T2 →
+ lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lift_inv_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
+qed-.
+
+lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
+/2 width=4 by lift_inv_O2_aux/ qed-.
+
+fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
+/2 width=5 by lift_inv_sort1_aux/ qed-.
+
+fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #i #H destruct
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #p #d #e #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed-.
+
+lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+/2 width=3 by lift_inv_lref1_aux/ qed-.
+
+lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i.
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
+/2 width=5 by lift_inv_gref1_aux/ qed-.
+
+fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T2 = ⓑ{a,I} V2. U2.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #a #I #V1 #U1 #H destruct
+| #i #d #e #_ #a #I #V1 #U1 #H destruct
+| #i #d #e #_ #a #I #V1 #U1 #H destruct
+| #p #d #e #a #I #V1 #U1 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
+]
+qed-.
+
+lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T2 = ⓑ{a,I} V2. U2.
+/2 width=3 by lift_inv_bind1_aux/ qed-.
+
+fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #d #e #I #V1 #U1 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
+]
+qed-.
+
+lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+/2 width=3 by lift_inv_flat1_aux/ qed-.
+
+fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_sort *)
+lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k.
+/2 width=5 by lift_inv_sort2_aux/ qed-.
+
+fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #i #H destruct
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
+| #p #d #e #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lref *)
+lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+/2 width=3 by lift_inv_lref2_aux/ qed-.
+
+(* Basic_1: was: lift_gen_lref_lt *)
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_false *)
+lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+ d ≤ i → i < d + e → ⊥.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
+[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
+lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
+elim (lt_refl_false … H)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_ge *)
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
+/2 width=5 by lift_inv_gref2_aux/ qed-.
+
+fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T1 = ⓑ{a,I} V1. U1.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #a #I #V2 #U2 #H destruct
+| #i #d #e #_ #a #I #V2 #U2 #H destruct
+| #i #d #e #_ #a #I #V2 #U2 #H destruct
+| #p #d #e #a #I #V2 #U2 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_bind *)
+lemma lift_inv_bind2: ∀d,e,T1,a,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{a,I} V2. U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T1 = ⓑ{a,I} V1. U1.
+/2 width=3 by lift_inv_bind2_aux/ qed-.
+
+fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T1 = ⓕ{I} V1. U1.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_flat *)
+lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓕ{I} V2. U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T1 = ⓕ{I} V1. U1.
+/2 width=3 by lift_inv_flat2_aux/ qed-.
+
+lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → ⊥.
+#d #e #J #V elim V -V
+[ * #i #T #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+ ]
+]
+qed-.
+
+(* Basic_1: was: thead_x_lift_y_y *)
+lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → ⊥.
+#J #T elim T -T
+[ * #i #V #d #e #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #d #e #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+ ]
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lift_fwd_pair1: ∀I,T2,V1,U1,d,e. ⇧[d,e] ②{I}V1.U1 ≡ T2 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & T2 = ②{I}V2.U2.
+* [ #a ] #I #T2 #V1 #U1 #d #e #H
+[ elim (lift_inv_bind1 … H) -H /2 width=4/
+| elim (lift_inv_flat1 … H) -H /2 width=4/
+]
+qed-.
+
+lemma lift_fwd_pair2: ∀I,T1,V2,U2,d,e. ⇧[d,e] T1 ≡ ②{I}V2.U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & T1 = ②{I}V1.U1.
+* [ #a ] #I #T1 #V2 #U2 #d #e #H
+[ elim (lift_inv_bind2 … H) -H /2 width=4/
+| elim (lift_inv_flat2 … H) -H /2 width=4/
+]
+qed-.
+
+lemma lift_fwd_tw: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
+qed-.
+
+lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: lift_lref_gt *)
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ⇧[d, e] #(i - e) ≡ #i.
+#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
+qed.
+
+lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
+/2 width=1/ qed-.
+
+(* Basic_1: was: lift_r *)
+lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T.
+#T elim T -T
+[ * #i // #d elim (lt_or_ge i d) /2 width=1/
+| * /2 width=1/
+]
+qed.
+
+lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e
+ elim (IHV1 d e) -IHV1 #V2 #HV12
+ [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
+ | elim (IHT1 d e) -IHT1 /3 width=2/
+ ]
+]
+qed.
+
+(* Basic_1: was: lift_free (right to left) *)
+lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 →
+ ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
+ ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2.
+#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2
+[ /3 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+ lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
+ >(plus_minus_m_m e2 e1 ?) // /3 width=3/
+| /3 width=3/
+| #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT d2 … ? ? He12) // /3 width=5/
+]
+qed.
+
+(* Basic_1: was only: dnf_dec2 dnf_dec *)
+lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2/ ] #i #d #e
+ elim (lt_dec i d) #Hid
+ [ /4 width=2/
+ | lapply (false_lt_to_le … Hid) -Hid #Hid
+ elim (lt_dec i (d + e)) #Hide
+ [ @or_intror * #T1 #H
+ elim (lift_inv_lref2_be … H Hid Hide)
+ | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
+ ]
+ ]
+| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e
+ [ elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+ [ * #T1 #HT12 @or_introl /3 width=2/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+ [ * #T1 #HT12 /4 width=2/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ ]
+]
+qed.
+
+(* Basic_1: removed theorems 7:
+ lift_head lift_gen_head
+ lift_weight_map lift_weight lift_weight_add lift_weight_add_O
+ lift_tlt_dx
+*)
--- /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/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Main properies ***********************************************************)
+
+(* Basic_1: was: lift_inj *)
+theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
+#d #e #T1 #U #H elim H -d -e -T1 -U
+[ #k #d #e #X #HX
+ lapply (lift_inv_sort2 … HX) -HX //
+| #i #d #e #Hid #X #HX
+ lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_le_plus_l/
+| #p #d #e #X #HX
+ lapply (lift_inv_gref2 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lift *)
+theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T →
+ d1 ≤ d2 →
+ ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
+ [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+ | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
+ elim (le_inv_plus_l … H) -H #Hide2 #He2i
+ lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
+ >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %);
+ /4 width=3 by lift_lref_ge, ex2_intro/
+ ]
+| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
+| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+ elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+ >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+ elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
+ e ≤ e1 → e1 ≤ e + e2 →
+ ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ >(lift_inv_lref2_lt … H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+ [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
+ | >(lift_inv_lref2_ge … H ?) -H //
+ lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
+ elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
+ @ex2_intro [2: /2 width=1/ | skip ] -Hd1e12
+ @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
+ ]
+| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2) -V // >plus_plus_comm_23 in HT2; #HT2
+ elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2) -V //
+ elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2.
+#d #e #T #U1 #H elim H -d -e -T -U1
+[ #k #d #e #X #HX
+ lapply (lift_inv_sort1 … HX) -HX //
+| #i #d #e #Hid #X #HX
+ lapply (lift_inv_lref1_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #p #d #e #X #HX
+ lapply (lift_inv_gref1 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_free (left to right) *)
+theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 →
+ d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_sort1 … HT2) -HT2 //
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+ lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1 by lift_lref_lt/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2
+ [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1 by monotonic_le_plus_l/
+ | -Hd21 /2 width=1 by lift_lref_ge/
+ ]
+| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_gref1 … HT2) -HT2 //
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
+]
+qed.
+
+(* Basic_1: was: lift_d (right to left) *)
+theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 →
+ ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+ lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
+ elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
+ lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
+ >plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Basic_1: was: lift_d (left to right) *)
+theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
+ ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
+ lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
+ lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1 by le_plus_to_minus_r/ #Hid2e
+ lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+ elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
+ <plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lift_conf_O1: ∀T,T1,d1,e1. ⇧[d1, e1] T ≡ T1 → ∀T2,e2. ⇧[0, e2] T ≡ T2 →
+ ∃∃T0. ⇧[0, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T2 ≡ T0.
+#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
+elim (lift_total T1 0 e2) #T0 #HT10
+elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
+lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
+qed.
+
+lemma lift_conf_be: ∀T,T1,d,e1. ⇧[d, e1] T ≡ T1 → ∀T2,e2. ⇧[d, e2] T ≡ T2 →
+ e1 ≤ e2 → ⇧[d + e1, e2 - e1] T1 ≡ T2.
+#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
+elim (lift_split … HT2 (d+e1) e1) -HT2 // #X #H
+>(lift_mono … H … HT1) -T //
+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/lift_lift.ma".
+include "basic_2/substitution/lift_vector.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+(* Main properies ***********************************************************)
+
+theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s →
+ ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s.
+#Ts #U1s #d #e #H elim H -Ts -U1s
+[ #U2s #H >(liftv_inv_nil1 … H) -H //
+| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
+ elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
+ >(lift_mono … HTU1 … HTU2) -T /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/substitution/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Properties on negated basic relocation ***********************************)
+
+lemma nlift_lref_be_SO: ∀X,i. ⇧[i, 1] X ≡ #i → ⊥.
+/2 width=7 by lift_inv_lref2_be/ qed-.
+
+lemma nlift_bind_sn: ∀W,d,e. (∀V. ⇧[d, e] V ≡ W → ⊥) →
+ ∀a,I,U. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥).
+#W #d #e #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_bind_dx: ∀U,d,e. (∀T. ⇧[d+1, e] T ≡ U → ⊥) →
+ ∀a,I,W. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥).
+#U #d #e #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_sn: ∀W,d,e. (∀V. ⇧[d, e] V ≡ W → ⊥) →
+ ∀I,U. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥).
+#W #d #e #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_dx: ∀U,d,e. (∀T. ⇧[d, e] T ≡ U → ⊥) →
+ ∀I,W. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥).
+#U #d #e #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+(* Inversion lemmas on negated basic relocation *****************************)
+
+lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⇧[i, 1] X ≡ #j → ⊥) → j = i.
+#i #j elim (lt_or_eq_or_gt i j) // #Hij #H
+[ elim (H (#(j-1))) -H /2 width=1 by lift_lref_ge_minus/
+| elim (H (#j)) -H /2 width=1 by lift_lref_lt/
+]
+qed-.
+
+lemma nlift_inv_bind: ∀a,I,W,U,d,e. (∀X. ⇧[d, e] X ≡ ⓑ{a,I}W.U → ⊥) →
+ (∀V. ⇧[d, e] V ≡ W → ⊥) ∨ (∀T. ⇧[d+1, e] T ≡ U → ⊥).
+#a #I #W #U #d #e #H elim (is_lift_dec W d e)
+[ * /4 width=2 by lift_bind, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+qed-.
+
+lemma nlift_inv_flat: ∀I,W,U,d,e. (∀X. ⇧[d, e] X ≡ ⓕ{I}W.U → ⊥) →
+ (∀V. ⇧[d, e] V ≡ W → ⊥) ∨ (∀T. ⇧[d, e] T ≡ U → ⊥).
+#I #W #U #d #e #H elim (is_lift_dec W d e)
+[ * /4 width=2 by lift_flat, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+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/term_vector.ma".
+include "basic_2/substitution/lift.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+inductive liftv (d,e:nat) : relation (list term) ≝
+| liftv_nil : liftv d e (◊) (◊)
+| liftv_cons: ∀T1s,T2s,T1,T2.
+ ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s →
+ liftv d e (T1 @ T1s) (T2 @ T2s)
+.
+
+interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
+#T1s #T2s #d #e * -T1s -T2s //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed-.
+
+lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
+/2 width=5 by liftv_inv_nil1_aux/ qed-.
+
+fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
+ ∀U1,U1s. T1s = U1 @ U1s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 @ U2s.
+#T1s #T2s #d #e * -T1s -T2s
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 @ U1s ≡ T2s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 @ U2s.
+/2 width=3 by liftv_inv_cons1_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s.
+#d #e #T1s elim T1s -T1s
+[ /2 width=2 by liftv_nil, ex_intro/
+| #T1 #T1s * #T2s #HT12s
+ elim (lift_total T1 d e) /3 width=2 by liftv_cons, ex_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/rliftstar_3.ma".
-include "basic_2/relocation/lift.ma".
-include "basic_2/substitution/gr2_plus.ma".
-
-(* GENERIC TERM RELOCATION **************************************************)
-
-inductive lifts: list2 nat nat → relation term ≝
-| lifts_nil : ∀T. lifts (⟠) T T
-| lifts_cons: ∀T1,T,T2,des,d,e.
- ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} @ des) T1 T2
-.
-
-interpretation "generic relocation (term)"
- 'RLiftStar des T1 T2 = (lifts des T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2.
-#T1 #T2 #des * -T1 -T2 -des //
-#T1 #T #T2 #d #e #des #_ #_ #H destruct
-qed-.
-
-lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
-/2 width=3 by lifts_inv_nil_aux/ qed-.
-
-fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
- ∀d,e,tl. des = {d, e} @ tl →
- ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2.
-#T1 #T2 #des * -T1 -T2 -des
-[ #T #d #e #tl #H destruct
-| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
- /2 width=3 by ex2_intro/
-qed-.
-
-lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} @ des] T1 ≡ T2 →
- ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2.
-/2 width=3 by lifts_inv_cons_aux/ qed-.
-
-(* Basic_1: was: lift1_sort *)
-lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
-#T2 #k #des elim des -des
-[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #H
- elim (lifts_inv_cons … H) -H #X #H
- >(lift_inv_sort1 … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Basic_1: was: lift1_lref *)
-lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 →
- ∃∃i2. @⦃i1, des⦄ ≡ i2 & T2 = #i2.
-#T2 #des elim des -des
-[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
-| #d #e #des #IH #i1 #H
- elim (lifts_inv_cons … H) -H #X #H1 #H2
- elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
- elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
-]
-qed-.
-
-lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p.
-#T2 #p #des elim des -des
-[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #H
- elim (lifts_inv_cons … H) -H #X #H
- >(lift_inv_gref1 … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Basic_1: was: lift1_bind *)
-lemma lifts_inv_bind1: ∀a,I,T2,des,V1,U1. ⇧*[des] ⓑ{a,I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 &
- T2 = ⓑ{a,I} V2. U2.
-#a #I #T2 #des elim des -des
-[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
- elim (lifts_inv_cons … H) -H #X #H #HT2
- elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
- /3 width=5 by ex3_2_intro, lifts_cons/
-]
-qed-.
-
-(* Basic_1: was: lift1_flat *)
-lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 &
- T2 = ⓕ{I} V2. U2.
-#I #T2 #des elim des -des
-[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
- elim (lifts_inv_cons … H) -H #X #H #HT2
- elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
- /3 width=5 by ex3_2_intro, lifts_cons/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_dx/
-qed-.
-
-lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#T1 #T2 #des #H elim H -T1 -T2 -des /3 width=5 by lift_simple_sn/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lifts_bind: ∀a,I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
- ∀T1. ⇧*[des + 1] T1 ≡ T2 →
- ⇧*[des] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
-#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
-[ #V #T1 #H >(lifts_inv_nil … H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
- elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/
-]
-qed.
-
-lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
- ∀T1. ⇧*[des] T1 ≡ T2 →
- ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
-#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
-[ #V #T1 #H >(lifts_inv_nil … H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
- elim (lifts_inv_cons … H) -H /3 width=3 by lift_flat, lifts_cons/
-]
-qed.
-
-lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
-#des elim des -des /2 width=2 by lifts_nil, ex_intro/
-#d #e #des #IH #T1 elim (lift_total T1 d e)
-#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_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/lift_lift.ma".
-include "basic_2/substitution/gr2_minus.ma".
-include "basic_2/substitution/lifts.ma".
-
-(* GENERIC TERM RELOCATION **************************************************)
-
-(* Properties concerning basic term relocation ******************************)
-
-(* Basic_1: was: lift1_xhg (right to left) *)
-lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 →
- ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2.
-#T1 #T #des #H elim H -T1 -T -des
-[ /2 width=3/
-| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2
- elim (IHT13 … HT2) -T #T #HT3 #HT2
- elim (lift_trans_le … HT13 … HT3) -T3 // /3 width=5/
-]
-qed-.
-
-(* Basic_1: was: lift1_free (right to left) *)
-lemma lifts_lift_trans: ∀des,i,i0. @⦃i, des⦄ ≡ i0 →
- ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 →
- ∀T1,T0. ⇧*[des0] T1 ≡ T0 →
- ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 →
- ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2.
-#des elim des -des normalize
-[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2
- <(at_inv_nil … H1) -x #HT02
- lapply (minuss_inv_nil1 … H2) -H2 #H
- >(pluss_inv_nil2 … H) in HT10; -des0 #H
- >(lifts_inv_nil … H) -T1 /2 width=3/
-| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
- elim (at_inv_cons … H1) -H1 * #Hid #Hi0
- [ elim (minuss_inv_cons1_lt … H2) -H2 [2: /2 width=1/ ] #des1 #Hdes1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
- elim (pluss_inv_cons2 … H) -H #des2 #H1 #H2 destruct
- elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
- elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02
- elim (lift_trans_le … HT1 … HT0) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
- | >commutative_plus in Hi0; #Hi0
- lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hdes0
- elim (IHdes … Hi0 … Hdes0 … HT10 … HT02) -IHdes -Hi0 -Hdes0 -T0 #T0 #HT0 #HT02
- elim (lift_split … HT0 d (i+1)) -HT0 /2 width=1/ /3 width=5/
- ]
-]
-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/lift_lift_vector.ma".
-include "basic_2/substitution/lifts_lift.ma".
-include "basic_2/substitution/lifts_vector.ma".
-
-(* GENERIC RELOCATION *******************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: lifts1_xhg (right to left) *)
-lemma liftsv_liftv_trans_le: ∀T1s,Ts,des. ⇧*[des] T1s ≡ Ts →
- ∀T2s:list term. ⇧[0, 1] Ts ≡ T2s →
- ∃∃T0s. ⇧[0, 1] T1s ≡ T0s & ⇧*[des + 1] T0s ≡ T2s.
-#T1s #Ts #des #H elim H -T1s -Ts
-[ #T1s #H
- >(liftv_inv_nil1 … H) -T1s /2 width=3/
-| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H
- elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct
- elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s
- elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/
-]
-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/lifts_lift.ma".
-
-(* GENERIC RELOCATION *******************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: lift1_lift1 (left to right) *)
-theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 →
- ⇧*[des1 @@ des2] T1 ≡ T2.
-#T1 #T #des1 #H elim H -T1 -T -des1 // /3 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/relocation/lift_vector.ma".
-include "basic_2/substitution/lifts.ma".
-
-(* GENERIC TERM VECTOR RELOCATION *******************************************)
-
-inductive liftsv (des:list2 nat nat) : relation (list term) ≝
-| liftsv_nil : liftsv des (◊) (◊)
-| liftsv_cons: ∀T1s,T2s,T1,T2.
- ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s →
- liftsv des (T1 @ T1s) (T2 @ T2s)
-.
-
-interpretation "generic relocation (vector)"
- 'RLiftStar des T1s T2s = (liftsv des T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: lifts1_flat (left to right) *)
-lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 →
- ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 &
- T2 = Ⓐ V2s. U2.
-#V1s elim V1s -V1s normalize
-[ #T1 #T2 #des #HT12
- @ex3_2_intro [3,4: // |1,2: skip | // ] (**) (* explicit constructor *)
-| #V1 #V1s #IHV1s #T1 #X #des #H
- elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
- elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
- @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: lifts1_flat (right to left) *)
-lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 →
- ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
-#V1s #V2s #des #H elim H -V1s -V2s /3 width=1 by lifts_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/notation/relations/lazyeq_4.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2.
-
-definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
-
-interpretation
- "lazy equivalence (local environment)"
- 'LazyEq T d L1 L2 = (lleq d T L1 L2).
-
-definition lleq_transitive: predicate (relation3 lenv term term) ≝
- λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
- ∀L1,L2,d,k. |L1| = |L2| → R d (⋆k) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
- ) → (
- ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
- ) → (
- ∀L1,L2,d,p. |L1| = |L2| → R d (§p) L1 L2
- ) → (
- ∀a,I,L1,L2,V,T,d.
- L1 ≡[V, d]L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
- R d V L1 L2 → R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R d (ⓑ{a,I}V.T) L1 L2
- ) → (
- ∀I,L1,L2,V,T,d.
- L1 ≡[V, d]L2 → L1 ≡[T, d] L2 →
- R d V L1 L2 → R d T L1 L2 → R d (ⓕ{I}V.T) L1 L2
- ) →
- ∀d,T,L1,L2. L1 ≡[T, d] L2 → R d T L1 L2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim H -L1 -L2 -T -d /2 width=8 by/
-qed-.
-
-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] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind/ 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.
-/2 width=2 by llpx_sn_inv_flat/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ≡[T, d] L2 → |L1| = |L2|.
-/2 width=4 by llpx_sn_fwd_length/ qed-.
-
-lemma lleq_fwd_lref: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V &
- ⇩[i] L2 ≡ K2.ⓑ{I}V &
- K1 ≡[V, yinj 0] K2 & d ≤ yinj i.
-#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref … H) /2 width=1/
-* /3 width=7 by or3_intro2, ex4_4_intro/
-qed-.
-
-lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K1,i. ⇩[i] L1 ≡ K1 →
- ∃K2. ⇩[i] L2 ≡ K2.
-/2 width=7 by llpx_sn_fwd_ldrop_sn/ qed-.
-
-lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ≡[d, T] L2 → ∀K2,i. ⇩[i] L2 ≡ K2 →
- ∃K1. ⇩[i] L1 ≡ K1.
-/2 width=7 by llpx_sn_fwd_ldrop_dx/ qed-.
-
-lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
- L1 ≡[ⓑ{a,I}V.T, d] L2 → L1 ≡[V, d] L2.
-/2 width=4 by llpx_sn_fwd_bind_sn/ 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] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
-
-lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,d.
- L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[V, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
-
-lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,d.
- L1 ≡[ⓕ{I}V.T, d] L2 → L1 ≡[T, d] L2.
-/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ≡[⋆k, d] L2.
-/2 width=1 by llpx_sn_sort/ qed.
-
-lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ≡[#i, d] L2.
-/2 width=1 by llpx_sn_skip/ qed.
-
-lemma lleq_lref: ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, 0] K2 → L1 ≡[#i, d] L2.
-/2 width=9 by llpx_sn_lref/ qed.
-
-lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, d] L2.
-/2 width=1 by llpx_sn_free/ qed.
-
-lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ≡[§p, d] L2.
-/2 width=1 by llpx_sn_gref/ qed.
-
-lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
- L1 ≡[V, d] L2 → L1.ⓑ{I}V ≡[T, ⫯d] L2.ⓑ{I}V →
- L1 ≡[ⓑ{a,I}V.T, d] L2.
-/2 width=1 by llpx_sn_bind/ qed.
-
-lemma lleq_flat: ∀I,L1,L2,V,T,d.
- L1 ≡[V, d] L2 → L1 ≡[T, d] L2 → L1 ≡[ⓕ{I}V.T, d] L2.
-/2 width=1 by llpx_sn_flat/ qed.
-
-lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
-/2 width=1 by llpx_sn_refl/ qed.
-
-lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
-/2 width=1 by llpx_sn_Y/ qed.
-
-lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
-#d #T #L1 #L2 #H @(lleq_ind … H) -d -T -L1 -L2
-/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
-qed-.
-
-lemma lleq_ge_up: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
- ∀T,d,e. ⇧[d, e] T ≡ U →
- dt ≤ d + e → L1 ≡[U, d] L2.
-/2 width=6 by llpx_sn_ge_up/ qed-.
-
-lemma lleq_ge: ∀L1,L2,T,d1. L1 ≡[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ≡[T, d2] L2.
-/2 width=3 by llpx_sn_ge/ qed-.
-
-lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
- L1 ≡[ⓑ{a,I}V.T, 0] L2.
-/2 width=1 by llpx_sn_bind_O/ qed-.
-
-(* Advancded properties on lazy pointwise exyensions ************************)
-
-lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2,T,d. L1 ≡[T, d] L2 → llpx_sn R d T L1 L2.
-/2 width=3 by llpx_sn_co/ 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/llpx_sn_alt.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Alternative definition (not recursive) ***********************************)
-
-theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ V1 = V2
- ) → L1 ≡[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
-qed.
-
-theorem lleq_inv_alt: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ V1 = V2.
-#L1 #L2 #T #d #H elim (llpx_sn_llpx_sn_alt … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
-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/llpx_sn_alt_rec.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Alternative definition (recursive) ***************************************)
-
-theorem lleq_intro_alt_r: ∀L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2
- ) → L1 ≡[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt_r // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
-qed.
-
-theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv.
- (∀L1,L2,T,d. |L1| = |L2| → (
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2
- ) → S d T L1 L2) →
- ∀L1,L2,T,d. L1 ≡[T, d] L2 → S d T L1 L2.
-#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
-qed-.
-
-theorem lleq_inv_alt_r: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
-#L1 #L2 #T #d #H elim (llpx_sn_inv_alt_r … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_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/substitution/fqus_alt.ma".
-include "basic_2/substitution/lleq_ldrop.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on supclosure *************************************************)
-
-lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ →
- ∀L1. L1 ≡[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
-[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
- #K1 #H1 #H2 lapply (ldrop_inv_O2 … H1) -H1
- #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
-| * [ #a ] #I #G #L2 #V #T #L1 #H
- [ elim (lleq_inv_bind … H)
- | elim (lleq_inv_flat … H)
- ] -H
- /2 width=3 by fqu_pair_sn, ex2_intro/
-| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
- #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
-| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
- /2 width=3 by fqu_flat_dx, ex2_intro/
-| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
- elim (ldrop_O1_le (Ⓕ) (e+1) L1)
- [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
- | lapply (ldrop_fwd_length_le2 … HLK2) -K2
- lapply (lleq_fwd_length … HL12) -T -U //
- ]
-]
-qed-.
-
-lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ →
- ∀L1. L1 ≡[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
-[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
-| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ →
- ∀L1. L1 ≡[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
-[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
- /3 width=3 by fqu_fqup, ex2_intro/
-| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
- #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
- /3 width=5 by fqup_strap1, ex2_intro/
-]
-qed-.
-
-lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ →
- ∀L1. L1 ≡[T, 0] L2 →
- ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
-[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
-| * #HG #HL #HT 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/substitution/llpx_sn_ldrop.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
- ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W.
-/2 width=7 by llpx_sn_bind_repl_O/ qed-.
-
-lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ≡[T, d] L2).
-/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
-
-lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
- ∀L1,L2,T,d. L1 ≡[T, d] L2 →
- ∀L. llpx_sn R d T L2 L → llpx_sn R d T L1 L.
-#R #HR #L1 #L2 #T #d #H @(lleq_ind … H) -L1 -L2 -T -d
-[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
-|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
-| #I #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
- /3 width=11 by llpx_sn_lref/
-| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
- /3 width=1 by llpx_sn_bind/
-| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-]
-qed-.
-
-lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
- ∀L1,L2,T,d. L1 ≡[T, d] L2 →
- ∀L. llpx_sn R d T L1 L → llpx_sn R d T L2 L.
-/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
- ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
- ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2.
-#L1 #L2 #d #i #H #Hdi #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
- ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
- ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2.
-#L1 #L2 #d #i #H #Hdi #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lleq_inv_lref_ge_bi: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2.
-/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
-
-lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ≡[#i, d] L2 → d ≤ i →
- ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
-elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
-qed-.
-
-lemma lleq_inv_S: ∀L1,L2,T,d. L1 ≡[T, d+1] L2 →
- ∀I,K1,K2,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 ≡[V, 0] K2 → L1 ≡[T, d] L2.
-/2 width=9 by llpx_sn_inv_S/ qed-.
-
-lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
- L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind_O/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
- ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
- i < d ∨
- ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
-[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
-qed-.
-
-lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ≡[#i, d] L2 →
- ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
- i < d ∨
- ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & d ≤ i.
-#L1 #L2 #d #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
-[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
-qed-.
-
-lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
- L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ≡[T, dt] K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ≡[U, dt] L2.
-/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
-
-lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ≡[T, dt] K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ≡[U, dt+e] L2.
-/2 width=9 by llpx_sn_lift_ge/ qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ≡[T, dt] K2.
-/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
-
-lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ≡[T, d] K2.
-/2 width=11 by llpx_sn_inv_lift_be/ qed-.
-
-lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ≡[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ≡[T, dt-e] K2.
-/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_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] L2.ⓑ{I}V → ⊥).
-/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
-
-lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ≡[ⓕ{I}V.T, d] L2 → ⊥) →
- (L1 ≡[V, d] L2 → ⊥) ∨ (L1 ≡[T, d] L2 → ⊥).
-/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
-
-lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
- (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥).
-/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ 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/llpx_sn_leq.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma leq_lleq_trans: ∀L2,L,T,d. L2 ≡[T, d] L →
- ∀L1. L1 ≃[d, ∞] L2 → L1 ≡[T, d] L.
-/2 width=3 by leq_llpx_sn_trans/ qed-.
-
-lemma lleq_leq_trans: ∀L,L1,T,d. L ≡[T, d] L1 →
- ∀L2. L1 ≃[d, ∞] L2 → L ≡[T, d] L2.
-/2 width=3 by llpx_sn_leq_trans/ qed-.
-
-lemma lleq_leq_repl: ∀L1,L2,T,d. L1 ≡[T, d] L2 → ∀K1. K1 ≃[d, ∞] L1 →
- ∀K2. L2 ≃[d, ∞] K2 → K1 ≡[T, d] K2.
-/2 width=5 by llpx_sn_leq_repl/ qed-.
-
-lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 →
- ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2.
-/2 width=5 by llpx_sn_bind_repl_SO/ 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/lleq_ldrop.ma".
-
-(* Main properties **********************************************************)
-
-theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
-/2 width=3 by lleq_llpx_sn_trans/ qed-.
-
-theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ≡[d, T] L1→ L ≡[d, T] L2 → L1 ≡[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ≡[d, T] L → L2 ≡[d, T] L → L1 ≡[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-(* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1≡[T, d] L →
- ∀L2. (L ≡[T, d] L2 → ⊥) → (L1 ≡[T, d] L2 → ⊥).
-/3 width=3 by lleq_canc_sn/ qed-.
-works with /4 width=8/ so lleq_canc_sn is more convenient
-*)
+++ /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/llor.ma".
-include "basic_2/substitution/lleq_alt.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on poinwise union for local environments **********************)
-
-lemma llpx_sn_llor_dx: ∀R,L1,L2.
- (∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄) →
- ∀T. llpx_sn R 0 T L1 L2 → ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
-#R #L1 #L2 #HR #T #H1 #L #H2
-elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
-elim H2 -H2 #_ #HL1 #IH2
-@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
-lapply (ldrop_fwd_length_lt2 … HLK) #HiL
-elim (ldrop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
-elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct
-elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
-elim H -H /2 width=1 by/
-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/lazyor_4.ma".
-include "basic_2/substitution/frees.ma".
-
-(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-
-definition llor: relation4 term lenv lenv lenv ≝ λT,L2,L1,L.
- ∧∧ |L1| ≤ |L2| & |L1| = |L|
- & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → ⇩[i] L ≡ K.ⓑ{I}V →
- (∧∧ (L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄ → ⊥) & I1 = I & V1 = V) ∨
- (∧∧ L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄ & I1 = I & V2 = V)
- ).
-
-interpretation
- "lazy union (local environment)"
- 'LazyOr L1 T L2 L = (llor T L2 L1 L).
-
-(* Basic properties *********************************************************)
-
-lemma llor_atom: ∀T,L2. ⋆ ⩖[T] L2 ≡ ⋆.
-#T #L2 @and3_intro //
-#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
-elim (ldrop_inv_atom1 … HLK1) -HLK1 #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/frees_lift.ma".
-include "basic_2/substitution/llor.ma".
-
-(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Advanced properties ******************************************************)
-
-axiom llor_total: ∀L1,L2,T. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
+++ /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/relocation/ldrop.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_sort: ∀L1,L2,d,k. |L1| = |L2| → llpx_sn R d (⋆k) L1 L2
-| llpx_sn_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn R d (#i) L1 L2
-| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R d (#i) L1 L2
-| llpx_sn_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R d (#i) L1 L2
-| llpx_sn_gref: ∀L1,L2,d,p. |L1| = |L2| → llpx_sn R d (§p) L1 L2
-| llpx_sn_bind: ∀a,I,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 → llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R d (ⓑ{a,I}V.T) L1 L2
-| llpx_sn_flat: ∀I,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 → llpx_sn R d T L1 L2 → llpx_sn R d (ⓕ{I}V.T) L1 L2
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
- ∀a,I,V,T. X = ⓑ{a,I}V.T →
- llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #b #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #b #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
-| #I #L1 #L2 #V #T #d #_ #_ #b #J #W #U #H destruct
-]
-qed-.
-
-lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R d V L1 L2 ∧ llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
-
-fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 →
- ∀I,V,T. X = ⓕ{I}V.T →
- llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #V #T #d #HV #HT #J #W #U #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2 ∧ llpx_sn R d T L1 L2.
-/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpx_sn_fwd_length: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d //
-#I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #HLK1 #HLK2 #_ #_ #HK12
-lapply (ldrop_fwd_length … HLK1) -HLK1
-lapply (ldrop_fwd_length … HLK2) -HLK2
-normalize //
-qed-.
-
-lemma llpx_sn_fwd_ldrop_sn: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- ∀K1,i. ⇩[i] L1 ≡ K1 → ∃K2. ⇩[i] L2 ≡ K2.
-#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
-qed-.
-
-lemma llpx_sn_fwd_ldrop_dx: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- ∀K2,i. ⇩[i] L2 ≡ K2 → ∃K1. ⇩[i] L1 ≡ K1.
-#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by ldrop_O1_le/
-qed-.
-
-fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn R d X L1 L2 → ∀i. X = #i →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #j #H destruct
-| #L1 #L2 #d #i #_ #Hid #j #H destruct /2 width=1 by or3_intro1/
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
- /3 width=9 by or3_intro2, ex5_5_intro/
-| #L1 #L2 #d #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
-| #L1 #L2 #d #p #_ #j #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
-| #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
-]
-qed-.
-
-lemma llpx_sn_fwd_lref: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ≤ yinj i.
-/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
-
-lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,d. llpx_sn R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,d. llpx_sn R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,d. llpx_sn R d (②{I}V.T) L1 L2 →
- llpx_sn R d V L1 L2.
-#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
-qed-.
-
-(* Basic_properties *********************************************************)
-
-lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,d. llpx_sn R d T L L.
-#R #HR #T #L @(f2_ind … rfw … L T) -L -T
-#n #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
-#i #Hn elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
-#HiL #d elim (ylt_split i d) /2 width=1 by llpx_sn_skip/
-elim (ldrop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
-qed-.
-
-lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
-#a #I #V1 #T1 #Hn #L2 #HL12
-@llpx_sn_bind /2 width=1/ (**) (* explicit constructor *)
-@IH -IH // normalize /2 width=1 by eq_f2/
-qed-.
-
-lemma llpx_sn_ge_up: ∀R,L1,L2,U,dt. llpx_sn R dt U L1 L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
- dt ≤ d + e → llpx_sn R d U L1 L2.
-#R #L1 #L2 #U #dt #H elim H -L1 -L2 -U -dt
-[ #L1 #L2 #dt #k #HL12 #X #d #e #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
- elim (ylt_yle_false … Hidt) -Hidt
- @(yle_trans … Hdtde) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
-| #I #L1 #L2 #K1 #K2 #W1 #W2 #dt #i #Hdti #HLK1 #HLK2 #HW1 #HW12 #_ #X #d #e #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
- lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I -W1 -W2 -dt /3 width=1 by llpx_sn_skip, ylt_inj/
- | /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
- ]
-| /2 width=1 by llpx_sn_free/
-| #L1 #L2 #dt #p #HL12 #X #d #e #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
- elim (lift_inv_bind2 … H) -H #V #T #HVW >commutative_plus #HTU #H destruct
- @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
- @(IHT … HTU) /2 width=1 by yle_succ/
-| #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
- elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
- /3 width=4 by llpx_sn_flat/
-]
-qed-.
-
-(**) (* the minor premise comes first *)
-lemma llpx_sn_ge: ∀R,L1,L2,T,d1,d2. d1 ≤ d2 →
- llpx_sn R d1 T L1 L2 → llpx_sn R d2 T L1 L2.
-#R #L1 #L2 #T #d1 #d2 * -d1 -d2 (**) (* destructed yle *)
-/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
-qed-.
-
-lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
-/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
-
-lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T,d. llpx_sn R1 d T L1 L2 → llpx_sn R2 d T L1 L2.
-#R1 #R2 #HR12 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_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/frees.ma".
-include "basic_2/substitution/llpx_sn_alt_rec.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* alternative definition of llpx_sn (not recursive) *)
-definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝
- λR,d,T,L1,L2. |L1| = |L2| ∧
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2
- ).
-
-(* Main properties **********************************************************)
-
-theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
-#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
-#HL12 #IHU @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv … H) -H
-[ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
-| * #J1 #K10 #W10 #j #Hdj #Hji #HnU #HLK10 #HnW10 destruct
- lapply (ldrop_fwd_drop2 … HLK10) #H
- lapply (ldrop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
- elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
- lapply (ldrop_fwd_drop2 … HLK20) #H
- lapply (ldrop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
- elim (IHn K10 W10 … K20 0) -IHn -HL12 /3 width=6 by ldrop_fwd_rfw/
- elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
-]
-qed.
-
-theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
-#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
-elim (IHU … HLK1 HLK2) /3 width=2 by frees_eq/
-#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
-lapply (ldrop_fwd_drop2 … HLK1) #H1
-lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
-@conj [ @(ldrop_fwd_length_eq1 … H1 H2) // ] -HL12
-#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
->(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
-#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
-lapply (ldrop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
-lapply (ldrop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
-/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
-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/lift_neg.ma".
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* alternative definition of llpx_sn (recursive) *)
-inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_alt_r_intro: ∀L1,L2,T,d.
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
- ) →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
- ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
-.
-
-(* Compact definition of llpx_sn_alt_r **************************************)
-
-lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
- ) → llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
-lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
- (∀L1,L2,T,d. |L1| = |L2| → (
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) → S d T L1 L2) →
- ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
-#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
-qed-.
-
-lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
-/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
-qed-.
-
-lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
-[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
- /3 width=9 by nlift_bind_sn, and3_intro/
-| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
- lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
- lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
- elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
- @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
-qed-.
-
-lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
-#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
-#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
-/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
-qed.
-
-lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
-qed.
-
-lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
-/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
-qed.
-
-lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
-lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
-/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
-qed.
-
-lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn_alt_r R 0 V1 K1 K2 → R K1 V1 V2 →
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
-[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
- @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
-| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
- elim (lt_or_eq_or_gt i j) #Hij destruct
- [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
- | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
- | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
- ]
-]
-qed.
-
-lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
- llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
-]
-qed.
-
-lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 →
- llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #T #d #HV #HT
-elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
-elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT #H
-[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
-| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
-[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
-| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
- #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
- [ * /2 width=1 by llpx_sn_free/
- | /2 width=1 by llpx_sn_skip/
- | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
- ]
-| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
- /3 width=1 by llpx_sn_bind/
-| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
- /3 width=1 by llpx_sn_flat/
-]
-qed-.
-
-(* Alternative definition of llpx_sn (recursive) ****************************)
-
-lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
- ) → llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
-@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
-qed.
-
-lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
- (∀L1,L2,T,d. |L1| = |L2| → (
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) → S d T L1 L2) →
- ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
-#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
-#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
-qed-.
-
-lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
-#H elim (llpx_sn_alt_r_inv_alt … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_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/substitution/llpx_sn_leq.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- i < d ∨
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
- R K1 V1 V2 & d ≤ i.
-#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
-[ #_ #H elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
- lapply (ldrop_mono … HLK22 … HLK2) -L2 #H destruct
- /3 width=5 by ex4_2_intro, or_intror/
-]
-qed-.
-
-lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- i < d ∨
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
- R K1 V1 V2 & d ≤ i.
-#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
-[ #H #_ elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
- lapply (ldrop_mono … HLK11 … HLK1) -L1 #H destruct
- /3 width=5 by ex4_2_intro, or_intror/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
-[ #H elim (ylt_yle_false … H Hdi)
-| * /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
-elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
-#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
-qed-.
-
-fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
- ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
-/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
- elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=9 by llpx_sn_lref/
-| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
- /3 width=9 by llpx_sn_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
- /4 width=9 by llpx_sn_bind, ldrop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
- /3 width=9 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
- ∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2.
-/2 width=9 by llpx_sn_inv_S_aux/ qed-.
-
-lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
-/3 width=9 by ldrop_pair, conj, llpx_sn_inv_S/
-qed-.
-
-(* More advanced forward lemmas *********************************************)
-
-lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
- ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
-/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
-
-lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
-#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
-| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
- [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/
- #Hdi elim (lt_or_ge i (|L1|)) #HiL1
- elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
- elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
- elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
- elim (eq_bind2_dec I2 I1)
- [ #H2 elim (HR K1 V1 V2) -HR
- [ #H3 elim (IH K1 V1 … K2 0) destruct
- /3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
- ]
- ]
- -IH #H3 @or_intror
- #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
- [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
- |7,8,9: /2 width=4 by ylt_yle_false/
- ]
- #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
- lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2
- #H #H0 destruct /2 width=1 by/
- ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
-| #a #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/
- #H1 #H2 @or_intror
- #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V … L2 d) /2 width=1 by/
- elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/
- #H1 #H2 @or_intror
- #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
-]
--n /4 width=4 by llpx_sn_fwd_length, or_intror/
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma llpx_sn_lift_le: ∀R. l_liftable R →
- ∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
-#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
-[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hdi #H destruct
- [ lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_skip/
- | elim (ylt_yle_false … Hid0) -L1 -L2 -K1 -K2 -e -Hid0
- /3 width=3 by yle_trans, yle_inj/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hdi #H destruct [ -HK12 | -IHK12 ]
- [ elim (ldrop_trans_lt … HLK1 … HK11) // -K1
- elim (ldrop_trans_lt … HLK2 … HK22) // -Hdi -K2
- /3 width=18 by llpx_sn_lref/
- | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hdi -Hd0 -K2
- /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
- ]
-| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_lift_ge: ∀R,K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
- ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀U. ⇧[d, e] T ≡ U → d ≤ d0 → llpx_sn R (d0+e) U L1 L2.
-#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
-[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
- [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
- | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (ldrop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (ldrop_trans_ge_comm … HLK2 … HK22 ?) // -Hid -Hd0 -K2
- /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
- ]
-| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H
- * #Hid #H destruct
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (ldrop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma llpx_sn_inv_lift_le: ∀R. l_deliftable_sn R →
- ∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
-#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /2 width=1 by llpx_sn_skip/
- | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct [ -HK12 | -IHK12 ]
- [ elim (ldrop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
- elim (ldrop_conf_lt … HLK2 … HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
- elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
- lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
- /3 width=10 by llpx_sn_lref/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0
- elim (le_inv_plus_l … Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d ≤ d0 → d0 ≤ yinj d + e → llpx_sn R d T K1 K2.
-#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/
- | elim (ylt_yle_false … Hid0) -L1 -L2 -Hd0 -Hid0
- /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
- elim (le_inv_plus_l … Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 … H) -H
- >commutative_plus #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ d0 → llpx_sn R (d0-e) T K1 K2.
-#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
-[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct [ -Hid0 | -Hded0 ]
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /4 width=3 by llpx_sn_skip, yle_plus_to_minus_inj2, ylt_yle_trans, ylt_inj/
- | elim (le_inv_plus_l … Hid) -Hid #_
- /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #Hid #H destruct
- [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0
- /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
- | lapply (ldrop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hded0 -Hid
- /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
- ]
-| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (ldrop_fwd_length_le4 … HLK1) -HLK1
- lapply (ldrop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
- @(IHU … HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
-| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,K2,e. ⇩[e] L1 ≡ K1 → ⇩[e] L2 ≡ K2 →
- ∀T. ⇧[0, e] T ≡ U → llpx_sn R 0 T K1 K2.
-/2 width=11 by llpx_sn_inv_lift_be/ qed-.
-
-lemma llpx_sn_ldrop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,e. ⇩[e] L1 ≡ K1 → ∀T. ⇧[0, e] T ≡ U →
- ∃∃K2. ⇩[e] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K1 #e #HLK1 #T #HTU elim (llpx_sn_fwd_ldrop_sn … HU … HLK1)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
-lemma llpx_sn_ldrop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K2,e. ⇩[e] L2 ≡ K2 → ∀T. ⇧[0, e] T ≡ U →
- ∃∃K1. ⇩[e] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K2 #e #HLK2 #T #HTU elim (llpx_sn_fwd_ldrop_dx … HU … HLK2)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
-(* Inversion lemmas on negated lazy pointwise extension *********************)
-
-lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
-/4 width=1 by llpx_sn_bind, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
- (llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
-#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
-/4 width=1 by llpx_sn_flat, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
-/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
-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_leq.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma leq_llpx_sn_trans: ∀R,L2,L,T,d. llpx_sn R d T L2 L →
- ∀L1. L1 ≃[d, ∞] L2 → llpx_sn R d T L1 L.
-#R #L2 #L #T #d #H elim H -L2 -L -T -d
-/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
-[ #I #L2 #L #K2 #K #V2 #V #d #i #Hdi #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
- elim (leq_ldrop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
- lapply (leq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
-| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_trans_aux, trans_eq/
-| /4 width=1 by llpx_sn_bind, leq_succ/
-]
-qed-.
-
-lemma llpx_sn_leq_trans: ∀R,L,L1,T,d. llpx_sn R d T L L1 →
- ∀L2. L1 ≃[d, ∞] L2 → llpx_sn R d T L L2.
-#R #L #L1 #T #d #H elim H -L -L1 -T -d
-/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
-[ #I #L #L1 #K #K1 #V #V1 #d #i #Hdi #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
- elim (leq_ldrop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
- lapply (leq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
-| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_conf_aux, trans_eq/
-| /4 width=1 by llpx_sn_bind, leq_succ/
-]
-qed-.
-
-lemma llpx_sn_leq_repl: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → ∀K1. K1 ≃[d, ∞] L1 →
- ∀K2. L2 ≃[d, ∞] K2 → llpx_sn R d T K1 K2.
-/3 width=4 by llpx_sn_leq_trans, leq_llpx_sn_trans/ qed-.
-
-lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
- ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2).
-#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT
-/3 width=7 by llpx_sn_leq_repl, leq_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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lpx_sn_alt.ma".
-include "basic_2/substitution/llor.ma".
-include "basic_2/substitution/lleq_alt.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Inversion lemmas on poinwise union for local environments ****************)
-
-lemma llpx_sn_llor_fwd_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2,T. llpx_sn R 0 T L1 L2 →
- ∀L. L1 ⩖[T] L2 ≡ L → lpx_sn R L1 L.
-#R #HR #L1 #L2 #T #H1 #L #H2
-elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
-elim H2 -H2 #_ #HL1 #IH2
-@lpx_sn_intro_alt // #I1 #I #K1 #K #V1 #V #i #HLK1 #HLK
-lapply (ldrop_fwd_length_lt2 … HLK) #HiL
-elim (ldrop_O1_lt (Ⓕ) L2 i) // -HiL -HL1 -HL12 #I2 #K2 #V2 #HLK2
-elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK * [ /2 width=1 by conj/ ]
-#HnT #H1 #H2 elim (IH1 … HnT … HLK1 HLK2) -IH1 -HnT -HLK1 -HLK2 /2 width=1 by 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/relocation/lpx_sn_ldrop.ma".
-include "basic_2/substitution/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties on pointwise extensions ***************************************)
-
-lemma lpx_sn_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀T,L1,L2,d. lpx_sn R L1 L2 → llpx_sn R d T L1 L2.
-#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
-| -HR #i elim (lt_or_ge i (|L1|))
- [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
- #Hi #Hn #L2 #d elim (ylt_split i d)
- [ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
- #Hdi #HL12 elim (ldrop_O1_lt (Ⓕ) L1 i) //
- #I #K1 #V1 #HLK1 elim (lpx_sn_ldrop_conf … HL12 … HLK1) -HL12
- /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
-| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
-| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
-| -HR /3 width=1 by llpx_sn_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/llpx_sn_ldrop.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties about transitive closure **************************************)
-
-lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L,V1,V2,T. LTC … R L V1 V2 →
- LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
-#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
-/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
-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".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+inductive lpx_sn (R:relation3 lenv term term): relation lenv ≝
+| lpx_sn_atom: lpx_sn R (⋆) (⋆)
+| lpx_sn_pair: ∀I,K1,K2,V1,V2.
+ lpx_sn R K1 K2 → R K1 V1 V2 →
+ lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
+#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
+ lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+ ∧∧ lpx_sn R L1 L2 & R L1 V1 V2 & I1 = I2.
+#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
+#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+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.ma".
+include "basic_2/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* alternative definition of lpx_sn *)
+definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
+ λR,L1,L2. |L1| = |L2| ∧
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2
+ ).
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
+#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_sn/
+qed-.
+
+lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I1 #L2 #K1 #V1 #H elim H -H
+#H #IH elim (length_inv_pos_sn … H) -H
+#I2 #K2 #V2 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by ldrop_drop, conj/
+qed-.
+
+lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
+#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_dx/
+qed-.
+
+lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I2 #L1 #K2 #V2 #H elim H -H
+#H #IH elim (length_inv_pos_dx … H) -H
+#I1 #K1 #V1 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by ldrop_drop, conj/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
+#R @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
+#H destruct
+qed.
+
+lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
+ lpx_sn_alt R L1 L2 → R L1 V1 V2 →
+ lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
+#HL12 #IH @conj normalize //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
+[ #HLK1 #HLK2
+ lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
+ lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
+ /2 width=1 by conj/
+| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
+#R #L1 elim L1 -L1
+[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
+| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
+]
+qed-.
+
+(* alternative definition of lpx_sn *****************************************)
+
+lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2
+ ) → lpx_sn R L1 L2.
+/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
+
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2.
+#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H elim H -H /3 width=4 by 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/substitution/ldrop_leq.ma".
+include "basic_2/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properies on dropping ****************************************************)
+
+lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_ldrop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K2,V2,i. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HLK1 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
+]
+qed-.
+
+lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
+ l_liftable R → dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (lpx_sn_fwd_length … HK12)
+ #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
+ /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
+ @leq_O2 normalize //
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
+ /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1
+ /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
+]
+qed-.
+
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+ #K1 #V1 #HK12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+ #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
+/2 width=5 by lpx_sn_dropable_aux/ 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/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
+#R #HR #L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
+qed-.
+
+theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+ confluent2 … (lpx_sn R1) (lpx_sn R2).
+#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
+[ #_ #X1 #H1 #X2 #H2 -n
+ >(lpx_sn_inv_atom1 … H1) -X1
+ >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
+| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
+ elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+ elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+ elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, 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/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀V1,V2. LTC … R L1 V1 V2 →
+ TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+ TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+ elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+ lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀S:relation lenv.
+ S (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+ S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+ elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+ #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ lpx_sn (LTC … R) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+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/substitution/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 elim (le_plus_xSy_O_false … H)
+| #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/substitution/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/unfold_4.ma".
include "basic_2/grammar/lenv_append.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/substitution/ldrop.ma".
(* CONTEXT-SENSITIVE UNFOLD FOR TERMS ***************************************)
}
]
class "yellow"
- [ { "substitution" * } {
+ [ { "multiple substitution" * } {
[ { "lazy equivalence" * } {
[ "fleq ( ⦃?,?,?⦄ ⋕[?] ⦃?,?,?⦄ )" "fleq_fleq" * ]
[ "lleq ( ? ⋕[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ]
}
]
class "orange"
- [ { "relocation" * } {
+ [ { "substitution" * } {
[ { "structural successor for closures" * } {
[ "fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )" "fquq_alt ( ⦃?,?,?⦄ ⊐⊐⸮ ⦃?,?,?⦄ )" * ]
[ "fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )" * ]
(* Forward lemmas on minus **************************************************)
-lemma yle_plus_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
/2 width=1 by monotonic_yle_minus_dx/ qed-.
-lemma yle_plus_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
-/2 width=1 by yle_plus_to_minus_inj2/ qed-.
+lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
+
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
#x *
(* Inversion lemmas on minus ************************************************)
lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
-/3 width=3 by yle_plus_to_minus_inj2, yle_trans, conj/ qed-.
+/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
/2 width=1 by yle_inv_plus_inj2/ qed-.
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