--- /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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+(* Note: lemma 500 *)
+lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | elim (lt_or_ge i (d + e)) #Hide
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | -Hdi
+ lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+]
+qed.
+
+lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=7/ qed.
+
+lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ 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/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/aaa_ltpss_dx.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+#L1 #T #A #HT #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
+qed.
+
+lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=5/ qed.
+
+lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/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/unfold/tpss.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Basic_1: includes: pr2_delta1 *)
+definition cpr: lenv → relation term ≝
+ λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T ▶* [0, |L|] T2.
+
+interpretation
+ "context-sensitive parallel reduction (term)"
+ 'PRed L T1 T2 = (cpr L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
+/2 width=3/ qed.
+
+lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ➡ T2.
+/3 width=5/ qed.
+
+lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
+/2 width=1/ qed.
+
+(* Note: new property *)
+(* Basic_1: was only: pr2_thin_dx *)
+lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
+ L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+lemma cpr_cast: ∀L,V,T1,T2.
+ L ⊢ T1 ➡ T2 → L ⊢ ⓝV. T1 ➡ T2.
+#L #V #T1 #T2 * /3 width=3/
+qed.
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+(* Basic_1: was only: pr2_change *)
+lemma cpr_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡ T2.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+lapply (tpss_lsubr_trans … HT2 … HL12) -HT2 -HL12 /3 width=4/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: pr2_gen_csort *)
+lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
+#T1 #T2 * #T #HT normalize #HT2
+<(tpss_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+(* Basic_1: was: pr2_gen_sort *)
+lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+#L #T2 #k * #X #H
+>(tpr_inv_atom1 … H) -H #H
+>(tpss_inv_sort1 … H) -H //
+qed-.
+
+(* Basic_1: was: pr2_gen_cast *)
+lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝV1. T1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U2 = ⓝV2. T2
+ ) ∨ L ⊢ T1 ➡ U2.
+#L #V1 #T1 #U2 * #X #H #HU2
+elim (tpr_inv_cast1 … H) -H /3 width=3/
+* #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #L #V1 #T1 #T * #X #H1 #H2 #b
+elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+qed-.
+
+lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T * #X #H1 #H2
+elim (tpr_fwd_shift1 … H1) -H1 #L0 #T0 #HL10 #H destruct
+elim (tpss_fwd_shift1 … H2) -H2 #L2 #T2 #HL02 #H destruct /2 width=4/
+qed-.
+
+(* Basic_1: removed theorems 6:
+ pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
+ pr2_gen_ctail pr2_ctail
+ Basic_1: removed local theorems 3:
+ pr2_free_free pr2_free_delta pr2_delta_delta
+*)
--- /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/unfold/tpss_lift.ma".
+include "basic_2/reducibility/tpr_lift.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
+ ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
+#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
+lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+@ex2_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
+qed.
+
+lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
+ ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
+#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a #I
+lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
+lapply (tpss_lsubr_trans … HT02 (L.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+@(ex2_intro … (ⓑ{a,I}V0.T0)) /2 width=1/ (* explicit constructors *)
+qed.
+
+lemma cpr_beta: ∀a,L,V1,V2,W,T1,T2.
+ L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
+#a #L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
+lapply (tpss_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(ex2_intro … (ⓓ{a}V.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
+qed.
+
+lemma cpr_beta_dx: ∀a,L,V1,V2,W,T1,T2.
+ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ➡ ⓓ{a}V2.T2.
+/3 width=1/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
+ ⇧[0, i + 1] T1 ≡ T2 &
+ i < |L|.
+#L #T2 #i * #X #H
+>(tpr_inv_atom1 … H) -H #H
+elim (tpss_inv_lref1 … H) -H /2 width=1/
+* /3 width=6/
+qed-.
+
+(* Basic_1: was pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1. T1 ➡ U2 →
+ (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 &
+ L. ⓓV ⊢ T1 ➡ T2 &
+ U2 = ⓓ{a}V2. T2
+ ) ∨
+ ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2 & a = true.
+#a #L #V1 #T1 #Y * #X #H1 #H2
+elim (tpr_inv_abbr1 … H1) -H1 *
+[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
+ lapply (tps_weak_full … HT0) -HT0 #HT0
+ lapply (tpss_lsubr_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02
+ lapply (tpss_weak_full … HT02) -HT02 #HT02
+ lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/
+| #T2 #HT12 #HXT2 #H destruct
+ elim (lift_total Y 0 1) #Z #HYZ
+ lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H
+ lapply (cpr_intro … HT12 … H) -T2 /3 width=3/
+]
+qed-.
+
+(* Basic_1: was: pr2_gen_abst *)
+lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡ U2 → ∀I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
+#a #L #V1 #T1 #Y * #X #H1 #H2 #I #W
+elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+lapply (tpss_lsubr_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Basic_1: was pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2
+ | ∃∃a,V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2.
+#L #V1 #U0 #Y * #X #H1 #H2
+elim (tpr_inv_appl1 … H1) -H1 *
+[ #V #U #HV1 #HU0 #H destruct
+ elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
+| #a #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=9/
+| #a #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
+ elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
+ elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=13/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
+ U2 = ⓛ{a}V2.T2.
+#a #L #V1 #T1 #U2 * #U #H #HU2 #b #I #W
+elim (tpr_fwd_abst1 … H b I W) -H #V #T #HT1 #H destruct
+elim (tpss_inv_bind1 … HU2) -HU2 #V2 #T2 #_ #HT2
+lapply (tpss_lsubr_trans … HT2 (L.ⓑ{I}W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr2_lift *)
+lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
+elim (lt_or_ge (|K|) d) #HKd
+[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
+| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
+]
+qed.
+
+(* Basic_1: was: pr2_gen_lift *)
+lemma cpr_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
+elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #T #HTU #T1
+elim (lt_or_ge (|L|) d) #HLd
+[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK /2 width=2/
+ /3 width=7 by ex2_intro, cpr_intro/
+| elim (lt_or_ge (|L|) (d + e)) #HLde
+ [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // /2 width=2/
+ /3 width=7 by ex2_intro, cpr_intro/
+ | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK //
+ /3 width=7 by ex2_intro, cpr_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_px.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+definition ltpr: relation lenv ≝ lpx tpr.
+
+interpretation
+ "context-free parallel reduction (environment)"
+ 'PRed L1 L2 = (ltpr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ltpr_refl: reflexive … ltpr.
+/2 width=1/ qed.
+
+lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2.
+/2 width=1/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: wcpr0_gen_sort *)
+lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
+/2 width=2 by lpx_inv_atom1/ qed-.
+
+(* Basic_1: was: wcpr0_gen_head *)
+lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
+ ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_inv_pair1/ qed-.
+
+lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
+/2 width=2 by lpx_inv_atom2/ qed-.
+
+lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_inv_pair2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_fwd_length/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L →
+ ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2.
+/2 width=1 by lpx_inv_append1/ qed-.
+
+lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 →
+ ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1.
+/2 width=1 by lpx_inv_append2/ qed-.
+
+(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
--- /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_lpx.ma".
+include "basic_2/substitution/fsup.ma".
+include "basic_2/reducibility/tpr_lift.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properies on local environment slicing ***********************************)
+
+(* Basic_1: was: wcpr0_drop *)
+lemma ltpr_ldrop_conf: dropable_sn ltpr.
+/3 width=3 by lpx_deliftable_dropable, tpr_inv_lift1/ qed.
+
+(* Basic_1: was: wcpr0_drop_back *)
+lemma ldrop_ltpr_trans: dedropable_sn ltpr.
+/2 width=3/ qed.
+
+lemma ltpr_ldrop_trans_O1: dropable_dx ltpr.
+/2 width=3/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsub_tpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. T2 ➡ U2 →
+ ∃∃L,U1. L1 ➡ L & T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HK2
+elim (lift_total T d e) #U #HTU
+elim (ldrop_ltpr_trans … HLK1 … HK1) -HLK1 -HK1 #L #HL1 #HLK
+lapply (tpr_lift … HT1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+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/reducibility/tpr_tpr.ma".
+include "basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Main properties **********************************************************)
+
+theorem ltpr_conf: ∀L0:lenv. ∀L1. L0 ➡ L1 → ∀L2. L0 ➡ L2 →
+ ∃∃L. L1 ➡ L & L2 ➡ L.
+#L0 #L1 #H elim H -L0 -L1 /2 width=3/
+#I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #L2 #H
+elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
+elim (IHK01 … HK02) -K0 #K #HK1 #HK2
+elim (tpr_conf … HV01 HV02) -V0 /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Basic_1: includes: csubst1_bind *)
+inductive ltpss_dx: nat → nat → relation lenv ≝
+| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆)
+| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e.
+ ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
+ ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
+ ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel unfold (local environment, dx variant)"
+ 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
+]
+qed.
+
+lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
+/2 width=4/ qed-.
+
+fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K1 #V1 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=5/ qed-.
+
+fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K2 #V2 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+qed.
+
+lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+qed.
+
+lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+(* Basic_1: was by definition: csubst1_refl *)
+lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L.
+#L elim L -L //
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
+qed.
+
+lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12
+ /3 width=5 by ltpss_dx_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
+ ]
+]
+qed.
+
+lemma ltpss_dx_weak_full: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
+qed.
+
+fact ltpss_dx_append_le_aux: ∀K1,K2,d,x. K1 ▶* [d, x] K2 → x = |K1| - d →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ▶* [d, x + e] L2 @@ K2.
+#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
+[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct //
+| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
+ lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
+ /4 width=5 by ltpss_dx_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_plus_to_le_r … H2) -H2 #Hd
+ /4 width=5 by ltpss_dx_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+]
+qed-.
+
+lemma ltpss_dx_append_le: ∀K1,K2,d. K1 ▶* [d, |K1| - d] K2 →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ▶* [d, |K1| - d + e] L2 @@ K2.
+/2 width=1 by ltpss_dx_append_le_aux/ qed.
+
+lemma ltpss_dx_append_zero: ∀K1,K2. K1 ▶* [0, |K1|] K2 →
+ ∀L1,L2,e. L1 ▶* [0, e] L2 →
+ L1 @@ K1 ▶* [0, |K1| + e] L2 @@ K2.
+/2 width=1/ qed.
+
+lemma ltpss_dx_append_ge: ∀K1,K2,d,e. K1 ▶* [d, e] K2 →
+ ∀L1,L2. L1 ▶* [d - |K1|, e] L2 → |K1| ≤ d →
+ L1 @@ K1 ▶* [d, e] L2 @@ K2.
+#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
+[ #d #e #L1 #L2 <minus_n_O //
+| #K #I #V #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
+ normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
+ lapply (le_plus_to_le_r … H) -H /3 width=1/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpss_dx_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+normalize //
+qed-.
+
+(* Basic_1: removed theorems 28:
+ csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
+ csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
+ csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
+ csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
+ csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
+ csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
+ csubst1_head csubst1_flat csubst1_gen_head
+ csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
+ fsubst0_gen_base
+*)
--- /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/fsup.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/ltpss_dx.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma ltpss_dx_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ldrop_ltpss_dx_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d1 ≤ d2 →
+ ∃∃L2. L1 ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
+ elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
+ elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
+ elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_ge … HW12 … HLK2 HWV1 … HWV2) -W1 // -Hd12
+ <le_plus_minus_comm // /4 width=5/
+]
+qed-.
+
+lemma ldrop_ltpss_dx_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 →
+ d2 ≤ d1 → d1 ≤ d2 + e2 →
+ ∃∃L2. L1 ▶* [d2, e1 + e2] L2 &
+ ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
+ elim (IHLK1 … HK12 H1 H2) -K1 -H2
+ lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
+ elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
+ [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
+ | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 // /2 width=1/
+ >plus_minus // >commutative_plus /4 width=5/
+ | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // ] /2 width=1/
+ >commutative_plus /3 width=5/
+ ]
+]
+qed-.
+
+lemma ldrop_ltpss_dx_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
+ ∃∃L2. L1 ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
+ elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
+ elim (IHLK1 … HK12 H) -K1
+ elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
+ >(ltpss_dx_inv_refl_O2 … HL12) -L1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
+ elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
+ [ -IHLK1 -Hd21 <(ltpss_dx_inv_refl_O2 … H) -X /3 width=5/
+ | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 /2 width=1/ /3 width=5/
+ | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
+ ]
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsup_tps_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶[0,|L2|] U2 →
+ ∃∃L,U1. L1 ▶*[0,|L|] L & L ⊢ T1 ▶[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+elim (le_or_ge d (|K|)) #Hd
+[ elim (ldrop_ltpss_dx_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tps_lift_be … HT1 … HL2K … HTU1 HTU … Hd) // -HT1 -HTU1 #HU1
+| elim (ldrop_ltpss_dx_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tps_lift_le … HT1 … HL2K … HTU1 HTU Hd) -HT1 -HTU1 #HU1
+]
+lapply (ltpss_dx_weak_full … HL12) -HL12 #HL12
+lapply (tps_weak_full … HU1) -HU1 #HU1
+@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
+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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_tpss.ma".
+
+(* DX PARTIAL UNFOLD ON LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_dx_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind … H) -U2 /2 width=3/
+#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2
+elim (ltpss_dx_tps_conf … HU2 … HL01) -L0 #X1 #HUX1 #HU2X1
+elim (tpss_strip_eq … HUX2 … HUX1) -U #X #HX2 #HX1
+lapply (tpss_trans_eq … HU2X1 … HX1) -X1 /3 width=3/
+qed.
+
+lemma ltpss_dx_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L0 ⊢ T ▶* [d1, e1] U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
+[ /2 width=3/
+| #U #U2 #_ #HU2 * #T #HT2 #HTU
+ elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
+ elim (ltpss_dx_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
+ lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
+ ∀L0. L0 ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
+#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
+[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
+ elim (tpss_inv_atom1 … H) -H // *
+ #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
+ lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
+ elim (ltpss_dx_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (tpss_fwd_tw … HV01) #H2
+ lapply (transitive_le (♯{K1} + ♯{V0}) … H1) -H1 /2 width=1/ -H2 #H
+ lapply (tpss_trans_eq … HV01 HV12) -V1 #HV02
+ lapply (IH … HV02 … HK01) -IH -HV02 -HK01
+ [ normalize /2 width=1/ | /2 width=6/ ]
+| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
+ [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
+ lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
+ lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
+ lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
+ | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (IH … HVW2 … HL0) -HVW2 //
+ lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ▶* [d, e] L1 →
+ L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpss_dx_conf: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ▶* [d1, e1] L.
+#L0 @(f_ind … lw … L0) -L0 #n #IH *
+[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n
+ >(ltpss_dx_inv_atom1 … H1) -L1
+ >(ltpss_dx_inv_atom1 … H2) -L2 /2 width=3/
+| #K0 #I0 #V0 #Hn #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 destruct
+ elim (eq_or_gt d1) #Hd1 [ elim (eq_or_gt e1) #He1 ] destruct
+ [ lapply (ltpss_dx_inv_refl_O2 … H1) -H1 #H1
+ | elim (ltpss_dx_inv_tpss21 … H1 … He1) -H1 #K1 #V1 #HK01 #HV01 #H1
+ | elim (ltpss_dx_inv_tpss11 … H1 … Hd1) -H1 #K1 #V1 #HK01 #HV01 #H1
+ ] destruct
+ elim (eq_or_gt d2) #Hd2 [1,3,5: elim (eq_or_gt e2) #He2 ] destruct
+ [1,3,5: lapply (ltpss_dx_inv_refl_O2 … H2) -H2 #H2
+ |2,4,6: elim (ltpss_dx_inv_tpss21 … H2 … He2) -H2 #K2 #V2 #HK02 #HV02 #H2
+ |7,8,9: elim (ltpss_dx_inv_tpss11 … H2 … Hd2) -H2 #K2 #V2 #HK02 #HV02 #H2
+ ] destruct
+ [1: -IH /2 width=3/
+ |2,3,4,7: -IH /3 width=5/
+ |5,6,8,9:
+ elim (IH … HK01 … HK02) // -K0 #K #HK1 #HK2
+ elim (ltpss_dx_tpss_conf … HV01 … HK1) -HV01 #W1 #HW1 #HVW1
+ elim (ltpss_dx_tpss_conf … HV02 … HK2) -HV02 #W2 #HW2 #HVW2
+ elim (tpss_conf_eq … HW1 … HW2) -V0 #V #HW1 #HW2
+ lapply (tpss_trans_eq … HVW1 HW1) -W1
+ lapply (tpss_trans_eq … HVW2 HW2) -W2 /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/unfold/ltpss_dx_ldrop.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial substitution on terms **********************)
+
+lemma ltpss_dx_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
+
+lemma ltpss_dx_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+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/unfold/ltpss_dx_tps.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial unfold on terms ****************************)
+
+lemma ltpss_dx_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_dx_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+(* Basic_1: was: subst1_subst1_back *)
+lemma ltpss_dx_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_dx_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_dx_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2
+ elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
+]
+qed.
+
+lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+(* Basic_1: was: subst1_subst1 *)
+lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ T ▶* [d1, e1] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_dx_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2
+ elim (IHTU2 … HL10) -IHTU2 -HL10 /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarSn $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+inductive ltpss_sn: nat → nat → relation lenv ≝
+| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
+| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
+ ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
+ ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
+ ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel unfold (local environment, sn variant)"
+ 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
+]
+qed.
+
+lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
+/2 width=4/ qed-.
+
+fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
+ L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K1 #V1 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
+ L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
+ K1 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=5/ qed-.
+
+fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K2 #V2 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
+ K1 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+qed.
+
+lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+qed.
+
+lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
+#L elim L -L //
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
+qed.
+
+lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12
+ /3 width=5 by ltpss_sn_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
+ ]
+]
+qed.
+
+lemma ltpss_sn_weak_full: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
+qed.
+
+fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
+ ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
+#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
+[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct //
+| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
+ lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
+ /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
+ lapply (le_plus_to_le_r … H2) -H2 #Hd
+ /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
+]
+qed-.
+
+lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
+ ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
+ L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
+/2 width=1 by ltpss_sn_append_le_aux/ qed.
+
+lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
+ ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
+ L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
+#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
+[ #d #e #L1 #L2 <minus_n_O //
+| #K #I #V #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
+ lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
+| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
+ normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
+ lapply (le_plus_to_le_r … H) -H /3 width=1/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
+
+include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
+include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* alternative definition of ltpss_sn *)
+definition ltpssa: nat → nat → relation lenv ≝
+ λd,e. TC … (ltpss_dx d e).
+
+interpretation "parallel unfold (local environment, sn variant) alternative"
+ 'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
+ ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
+#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
+ ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
+#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
+/2 width=1/ qed.
+
+lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
+ ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
+ L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
+#I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
+[ /3 width=1/ | /3 width=5/ ]
+qed.
+
+lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
+ ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
+ L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
+#I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
+[ /3 width=1/ | /3 width=5/ ]
+qed.
+
+lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
+qed.
+
+lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
+ ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L #d #e #H elim H -L1 -L -d -e
+[ #d #e #X #H
+ lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
+| #L #I #V #X #H
+ lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
+| #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
+ elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
+ #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (IHL1 … HL2) -L #HL12
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
+ lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
+| #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
+ elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
+ #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (IHL1 … HL2) -L #HL12
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
+ lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
+ ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+/3 width=3/ qed.
+
+lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
+/3 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
+ ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
+#L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
+lapply (ltpss_sn_ltpssa … H) -H #HL01
+elim (ltpssa_strip … HL01 … HL02) -L0
+/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
+qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
+ L0 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
+#L #L1 #H #HL1 * #T #HT2 #HU2T
+lapply (ltpssa_ltpss_sn … H) -H #HL0
+lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
+elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
+lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
+lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
+qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
+lapply (ltpss_sn_ltpssa … H) -H #HL10
+@(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
+#L1 #L #HL1 #_ * #T #HT2 #HTU2
+elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
+lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
+lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
+ ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
+/3 width=3/ qed.
+
+(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
+theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
+ ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
+#L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
+lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
+lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
+elim (ltpssa_conf … HL01 … HL02) -L0
+/3 width=3 by ltpssa_ltpss_sn, 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/fsup.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/ltpss_sn.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma ltpss_sn_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L ⊢ ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_sn_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 ⊢ ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ltpss_sn_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L ⊢ ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ldrop_ltpss_sn_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d1 ≤ d2 →
+ ∃∃L2. L1 ⊢ ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
+ elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
+ elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
+ elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_ge … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // -Hd12
+ <le_plus_minus_comm // /4 width=5/
+]
+qed-.
+
+lemma ldrop_ltpss_sn_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 →
+ d2 ≤ d1 → d1 ≤ d2 + e2 →
+ ∃∃L2. L1 ⊢ ▶* [d2, e1 + e2] L2 &
+ ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
+ elim (IHLK1 … HK12 H1 H2) -K1 -H2
+ lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
+ elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
+ [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
+ | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // /2 width=1/
+ >plus_minus // >commutative_plus /4 width=5/
+ | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/
+ >commutative_plus /3 width=5/
+ ]
+]
+qed-.
+
+lemma ldrop_ltpss_sn_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
+ ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
+ ∃∃L2. L1 ⊢ ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
+#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
+[ #d1 #e1 #K2 #d2 #e2 #H #_
+ >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
+ elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
+| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
+ elim (IHLK1 … HK12 H) -K1
+ elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
+ >(ltpss_sn_inv_refl_O2 … HL12) -L1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
+ elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
+ [ -IHLK1 -Hd21 <(ltpss_sn_inv_refl_O2 … H) -X /3 width=5/
+ | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/
+ | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
+ elim (lift_total W2 d1 e1) #V2 #HWV2
+ lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
+ ]
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fsup_tpss_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶*[0,|L2|] U2 →
+ ∃∃L,U1. L1 ⊢ ▶*[0,|L1|] L & L ⊢ T1 ▶*[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (ltpss_sn_fwd_length … HK1) #H >H in HK1; -H #HK1
+elim (le_or_ge d (|K|)) #Hd
+[ elim (ldrop_ltpss_sn_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_be … HT1 … Hd HL2K HTU1 … HTU) // -HT1 -HTU1 #HU1
+| elim (ldrop_ltpss_sn_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_le … HT1 … Hd HL2K HTU1 … HTU) -HT1 -HTU1 #HU1
+]
+lapply (ltpss_sn_weak_full … HL12) -HL12 #HL12
+lapply (tpss_weak_full … HU1) -HU1 #HU1
+@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
+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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_sn_tpss.ma".
+
+(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_sn_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
+ ∀L0. L0 ⊢ ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
+#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
+[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
+ elim (tpss_inv_atom1 … H) -H // *
+ #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
+ lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
+ elim (ltpss_sn_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (IH … HV12 … HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12
+ lapply (tpss_trans_eq … HV01 HV12) -V1 /2 width=6/
+| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
+ [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
+ lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
+ lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
+ lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
+ | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
+ lapply (IH … HVW2 … HL0) -HVW2 //
+ lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
+]
+qed.
+
+lemma ltpss_sn_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ⊢ ▶* [d, e] L1 →
+ L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
+/3 width=3/ qed.
+
+(* Main properties **********************************************************)
+
+theorem ltpss_sn_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
+ ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
+#L1 #L #d #e #H elim H -L1 -L -d -e //
+[ #L1 #L #I #V1 #V #e #HL1 #HV1 #IHL1 #X #H
+ elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
+ lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
+| #L1 #L #I #V1 #V #d #e #HL1 #HV1 #IHL1 #X #H
+ elim (ltpss_sn_inv_tpss11 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
+ lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
+ lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
+]
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶ [d, e] T →
+ ∃∃L2,T2. L @@ L1 ⊢ ▶* [d + |L1|, e] L @@ L2 &
+ L @@ L2 ⊢ T1 ▶ [d + |L1|, e] T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #d #e #HT1
+ @ex3_2_intro [3: // |5: // |4: normalize /2 width=1/ |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T #d #e >shift_append_assoc #H
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ <append_assoc >append_length <associative_plus
+ lapply (ltpss_sn_trans_eq (L.ⓑ{I}V1@@L1) … HL12) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* 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/unfold/ltpss_sn_ldrop.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial substitution on terms **********************)
+
+lemma ltpss_sn_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
+
+lemma ltpss_sn_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶ [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+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/unfold/ltpss_sn_tps.ma".
+
+(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+(* Properties concerning partial unfold on terms ****************************)
+
+lemma ltpss_sn_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_sn_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+lemma ltpss_sn_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ U2 ▶* [d1, e1] T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_sn_ldrop_conf_le … HL01 … HLK0 ?) /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_sn_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H
+ lapply (tpss_lsubr_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2
+ elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
+]
+qed.
+
+lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
+#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
+qed.
+
+lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ T ▶* [d1, e1] U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltpss_sn_ldrop_trans_be … HL10 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
+ lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H
+ lapply (tpss_lsubr_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2
+ elim (IHTU2 … HL10) -IHTU2 -HL10 /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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_tpss.ma".
+include "basic_2/static/ssta_lift.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ssta_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
+[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H /3 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
+ lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HWV1 ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
+ ]
+| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
+| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
+]
+qed.
+
+lemma ssta_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=5/ qed.
+
+lemma ssta_ltpss_dx_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
+/2 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/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/ssta_ltpss_dx.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma ssta_ltpss_sn_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T #U1 #l #HTU1 #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(ltpssa_ind … HL12) -L2 [ /2 width=3/ ] -HTU1
+#L #L2 #HL1 #HL2 * #U #HTU #HU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+elim (ssta_ltpss_dx_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
+lapply (ltpss_dx_tpss_trans_eq … HU2 … HL2) -HU2 -HL2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL1) -HU2 -HL1 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #l #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (ssta_ltpss_sn_conf … HTU1 … HL12) -HTU1 #U #HT1U #HU1
+elim (ssta_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
+ L1 ⊢ U1 ▶* [d, e] U2.
+/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/static/ssta_ltpss_dx.ma".
+include "basic_2/unwind/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
+
+(* Properties about dx parallel unfold **************************************)
+
+lemma sstas_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
+#T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
+elim (ssta_ltpss_dx_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
+elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
+qed.
+
+lemma sstas_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=7 by step, sstas_ltpss_dx_tpss_conf/ qed. (**) (* auto fails without trace *)
+
+lemma sstas_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 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/unfold/ltpss_sn_alt.ma".
+include "basic_2/unwind/sstas_ltpss_dx.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ************************)
+
+(* Properties about sn parallel unfold *****************************************)
+
+lemma sstas_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∃∃U2. L1 ⊢ U1 ▶* [d, e] U2 & ⦃h, L2⦄ ⊢ T •*[g] U2.
+#h #g #L1 #L2 #d #e #H
+lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
+#L #L2 #HL1 #HL2 #IHL1 #T #U1 #H1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
+elim (IHL1 … H1) -IHL1 -H1 #U #HU1 #HTU
+elim (sstas_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #H2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma sstas_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (sstas_ltpss_sn_conf … HL12 … HTU1) -HTU1 #U #HU1 #HT1U
+elim (sstas_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma sstas_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+/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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( T1 ➡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRed $T1 $T2 }.
+
+include "basic_2/substitution/tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Basic_1: includes: pr0_delta1 *)
+inductive tpr: relation term ≝
+| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
+| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
+ tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| tpr_beta : ∀a,V1,V2,W,T1,T2.
+ tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
+| tpr_delta: ∀a,I,V1,V2,T1,T,T2.
+ tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
+ tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
+ tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
+ tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
+| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
+| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
+.
+
+interpretation
+ "context-free parallel reduction (term)"
+ 'PRed T1 T2 = (tpr T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
+/2 width=3/ qed.
+
+(* Basic_1: was by definition: pr0_refl *)
+lemma tpr_refl: reflexive … tpr.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
+#U1 #U2 * -U1 -U2
+[ //
+| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
+| #V #T1 #T #T2 #_ #_ #k #H destruct
+| #V #T1 #T2 #_ #k #H destruct
+]
+qed.
+
+(* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
+lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
+/2 width=3/ qed-.
+
+fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
+#U1 #U2 * -U1 -U2
+[ #J #a #I #V #T #H destruct
+| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
+| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
+| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
+| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
+| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
+| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓑ{a,I} V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_abbr *)
+lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
+ (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
+ ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
+ U2 = ⓓ{a}V2. T2
+ ) ∨
+ ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
+#a #V1 #T1 #U2 #H
+elim (tpr_inv_bind1 … H) -H * /3 width=7/
+qed-.
+
+fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2 &
+ I = Appl
+ | (U0 ➡ U2 ∧ I = Cast).
+#U1 #U2 * -U1 -U2
+[ #I #J #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
+| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
+| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
+| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
+]
+qed.
+
+lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2 &
+ I = Appl
+ | (U0 ➡ U2 ∧ I = Cast).
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_appl *)
+lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛ{a}W. T1 &
+ U2 = ⓓ{a}V2. T2
+ | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓ{a}W1. T1 &
+ U2 = ⓓ{a}W2. ⓐV. T2.
+#V1 #U0 #U2 #H
+elim (tpr_inv_flat1 … H) -H *
+/3 width=5/ /3 width=9/ /3 width=13/
+#_ #H destruct
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U = ⓐV2. T2.
+#V1 #T1 #U #H #HT1
+elim (tpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Basic_1: was: pr0_gen_cast *)
+lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
+ (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
+ ∨ T1 ➡ U2.
+#V1 #T1 #U2 #H
+elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
+[ #_ #_ #_ #_ #H destruct
+| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
+#T1 #T2 * -T1 -T2
+[ #I #i #H destruct /2 width=1/
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
+| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
+| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
+]
+qed.
+
+lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
+/2 width=3/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
+| #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #T1 #T #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #T1 #X
+ >shift_append_assoc normalize #H
+ elim (tpr_inv_bind1 … H) -H *
+ [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
+ elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
+ elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
+ >append_length >HL1 >HL2 -L1 -L
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+ | #T #_ #_ #H destruct
+ ]
+]
+qed-.
+
+(* Basic_1: removed theorems 3:
+ pr0_subst0_back pr0_subst0_fwd pr0_subst0
+*)
+(* Basic_1: removed local theorems: 1: pr0_delta_tau *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_lift.ma".
+include "basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr0_lift *)
+lemma tpr_lift: t_liftable tpr.
+#T1 #T2 #H elim H -T1 -T2
+[ * #i #U1 #d #e #HU1 #U2 #HU2
+ lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
+ [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
+ | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
+ | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
+ ]
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
+ elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
+ elim (lift_total T (d + 1) e) #U #HTU
+ @tpr_delta
+ [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+ elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+ elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
+ elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
+ elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
+]
+qed.
+
+(* Basic_1: was: pr0_gen_lift *)
+lemma tpr_inv_lift1: t_deliftable_sn tpr.
+#T1 #T2 #H elim H -T1 -T2
+[ * #i #X #d #e #HX
+ [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
+ | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
+ ]
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /3 width=5/
+| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /3 width=5/
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
+ elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
+ elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
+ elim (IHT1 … HUT1) -T1 #U #HUT #HU1
+ elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
+ @ex2_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
+ elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
+ elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
+ elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
+ @ex2_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
+ elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
+ elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
+ elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
+ elim (IHT12 … HT01) -T1 /3 width=3/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr0_gen_abst *)
+lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
+#a #V1 #T1 #U2 #H elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T #T2 #HV12 #HT1 #HT2
+ lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /2 width=5/
+| #T2 #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tpr_fwd_abst1: ∀a,V1,T1,U2. ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
+ ∃∃V2,T2. ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
+ U2 = ⓛ{a}V2.T2.
+#a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T #T2 #HV12 #HT1 #HT2
+ lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /3 width=4/
+| #T2 #_ #_ #_ #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/reducibility/ltpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Confluence lemmas ********************************************************)
+
+fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
+/2 width=3/ qed.
+
+fact tpr_conf_flat_flat:
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃T0. ⓕ{I} V1. T1 ➡ T0 & ⓕ{I} V2. T2 ➡ T0.
+#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=5/
+qed.
+
+fact tpr_conf_flat_beta:
+ ∀a,V0,V1,T1,V2,W0,U0,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 →
+ U0 ➡ T2 → ⓛ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}V2. T2 ➡ X.
+#a #V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
+elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /3 width=5/
+qed.
+
+(* Basic-1: was:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact tpr_conf_flat_theta:
+ ∀a,V0,V1,T1,V2,V,W0,W2,U0,U2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{U0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → ⇧[O,1] V2 ≡ V →
+ W0 ➡ W2 → U0 ➡ U2 → ⓓ{a}W0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ{a}W2. ⓐV. U2 ➡ X.
+#a #V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #VV #HVV1 #HVV2
+elim (lift_total VV 0 1) #VVV #HVV
+lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
+elim (tpr_inv_abbr1 … H) -H *
+(* case 1: delta *)
+[ -HV2 -HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct
+ elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
+ elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
+ elim (tpr_tps_conf_bind … HWW2 HUUU2 … HUU2) -UU2 #UUU #HUUU2 #HUUU1
+ @ex2_intro
+ [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
+ |1:skip
+ |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
+ ] (**) (* /5 width=14/ is too slow *)
+(* case 3: zeta *)
+| -HV2 -HW02 -HVV2 #U1 #HU01 #HTU1
+ elim (IH … HU01 … HU02) -HU01 -HU02 -IH // -U0 #U #HU1 #HU2
+ elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #UU #HUU #HT1UU #H destruct
+ @(ex2_intro … (ⓐVV.UU)) /2 width=1/ /3 width=5/ (**) (* /4 width=9/ is too slow *)
+]
+qed.
+
+fact tpr_conf_flat_cast:
+ ∀X2,V0,V1,T0,T1. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → T0 ➡ T1 → T0 ➡ X2 →
+ ∃∃X. ⓝV1. T1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=3/
+qed.
+
+fact tpr_conf_beta_beta:
+ ∀a. ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃X. ⓓ{a}V1. T1 ➡X & ⓓ{a}V2. T2 ➡ X.
+#a #W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /3 width=5/
+qed.
+
+(* Basic_1: was: pr0_cong_delta pr0_delta_delta *)
+fact tpr_conf_delta_delta:
+ ∀a,I1,V0,V1,T0,T1,TT1,V2,T2,TT2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → T0 ➡ T1 → T0 ➡ T2 →
+ ⋆. ⓑ{I1} V1 ⊢ T1 ▶ [O, 1] TT1 →
+ ⋆. ⓑ{I1} V2 ⊢ T2 ▶ [O, 1] TT2 →
+ ∃∃X. ⓑ{a,I1} V1. TT1 ➡ X & ⓑ{a,I1} V2. TT2 ➡ X.
+#a #I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
+elim (IH … HV01 … HV02) -HV01 -HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // #T #HT1 #HT2
+elim (tpr_tps_conf_bind … HV1 HT1 … HTT1) -T1 #U1 #TTU1 #HTU1
+elim (tpr_tps_conf_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
+elim (tps_conf_eq … HTU1 … HTU2) -T #U #HU1 #HU2
+@ex2_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
+qed.
+
+fact tpr_conf_delta_zeta:
+ ∀X2,V0,V1,T0,T1,TT1,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → T0 ➡ T1 → ⋆. ⓓV1 ⊢ T1 ▶ [O,1] TT1 →
+ T0 ➡ T2 → ⇧[O, 1] X2 ≡ T2 →
+ ∃∃X. +ⓓV1. TT1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HT02 #HXT2
+elim (IH … HT01 … HT02) -IH -HT01 -HT02 // -V0 -T0 #T #HT1 #HT2
+elim (tpr_tps_conf_bind ? ? V1 … HT1 HTT1) -T1 // #TT #HTT1 #HTT
+elim (tpr_inv_lift1 … HT2 … HXT2) -T2 #X #HXT #HX2
+lapply (tps_inv_lift1_eq … HTT … HXT) -HTT #H destruct /3 width=3/
+qed.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact tpr_conf_theta_theta:
+ ∀a,VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + (♯{W0} + ♯{T0} + 1) + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ V0 ➡ V1 → V0 ➡ V2 → W0 ➡ W1 → W0 ➡ W2 → T0 ➡ T1 → T0 ➡ T2 →
+ ⇧[O, 1] V1 ≡ VV1 → ⇧[O, 1] V2 ≡ VV2 →
+ ∃∃X. ⓓ{a}W1. ⓐVV1. T1 ➡ X & ⓓ{a}W2. ⓐVV2. T2 ➡ X.
+#a #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
+elim (lift_total V 0 1) #VV #HVV
+lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
+lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -HVV #HVV2
+@ex2_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
+
+fact tpr_conf_zeta_zeta:
+ ∀V0:term. ∀X2,TT0,T0,T1,TT2. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{TT0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ TT0 ➡ T0 → ⇧[O, 1] T1 ≡ T0 →
+ TT0 ➡ TT2 → ⇧[O, 1] X2 ≡ TT2 →
+ ∃∃X. T1 ➡ X & X2 ➡ X.
+#V0 #X2 #TT0 #T0 #T1 #TT2 #IH #HTT0 #HT10 #HTT02 #HXTT2
+elim (IH … HTT0 … HTT02) -IH -HTT0 -HTT02 // -V0 -TT0 #T #HT0 #HTT2
+elim (tpr_inv_lift1 … HT0 … HT10) -T0 #T0 #HT0 #HT10
+elim (tpr_inv_lift1 … HTT2 … HXTT2) -TT2 #TT2 #HTT2 #HXTT2
+lapply (lift_inj … HTT2 … HT0) -HTT2 #H destruct /2 width=3/
+qed.
+
+fact tpr_conf_tau_tau:
+ ∀V0,T0:term. ∀X2,T1. (
+ ∀X0:term. ♯{X0} < ♯{V0} + ♯{T0} + 1 →
+ ∀X1,X2. X0 ➡ X1 → X0 ➡ X2 →
+ ∃∃X. X1 ➡ X & X2 ➡ X
+ ) →
+ T0 ➡ T1 → T0 ➡ X2 →
+ ∃∃X. T1 ➡ X & X2 ➡ X.
+#V0 #T0 #X2 #T1 #IH #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /2 width=3/
+qed.
+
+(* Confluence ***************************************************************)
+
+(* Basic_1: was: pr0_confluence *)
+theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ➡ T1 → T0 ➡ T2 →
+ ∃∃T. T1 ➡ T & T2 ➡ T.
+#T0 @(f_ind … tw … T0) -T0 #n #IH *
+[ #I #_ #X1 #X2 #H1 #H2 -n
+ >(tpr_inv_atom1 … H1) -X1
+ >(tpr_inv_atom1 … H2) -X2
+(* case 1: atom, atom *)
+ //
+| * [ #a ] #I #V0 #T0 #Hn #X1 #X2 #H1 #H2
+ [ elim (tpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #U1 #HV01 #HT01 #HTU1 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #U2 #HV02 #HT02 #HTU2 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+(* case 2: delta, delta *)
+ [ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 3: zeta, delta (repeated) *)
+ | @ex2_commute /3 width=10 by tpr_conf_delta_zeta/
+(* case 4: delta, zeta *)
+ | /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
+(* case 5: zeta, zeta *)
+ | /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
+ ]
+ | elim (tpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #b1 #V1 #W1 #U1 #T1 #HV01 #HUT1 #H11 #H12 #H13
+ | #b1 #V1 #Y1 #W1 #Z1 #U1 #T1 #HV01 #HWZ1 #HUT1 #HVY1 #H11 #H12 #H13
+ | #HX1 #H1
+ ]
+ elim (tpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #b2 #V2 #W2 #U2 #T2 #HV02 #HUT2 #H21 #H22 #H23
+ |3,7,11,15: #b2 #V2 #Y2 #W2 #Z2 #U2 #T2 #HV02 #HWZ2 #HUT2 #HVY2 #H21 #H22 #H23
+ |4,8,12,16: #HX2 #H2
+ ] destruct
+(* case 6: flat, flat *)
+ [ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
+(* case 7: beta, flat (repeated) *)
+ | @ex2_commute /3 width=8 by tpr_conf_flat_beta/
+(* case 8: theta, flat (repeated) *)
+ | @ex2_commute /3 width=11 by tpr_conf_flat_theta/
+(* case 9: tau, flat (repeated) *)
+ | @ex2_commute /3 width=6 by tpr_conf_flat_cast/
+(* case 10: flat, beta *)
+ | /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 11: beta, beta *)
+ | /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 12: flat, theta *)
+ | /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
+(* case 13: theta, theta *)
+ | /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+(* case 14: flat, tau *)
+ | /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
+(* case 15: tau, tau *)
+ | /3 width=5 by tpr_conf_tau_tau/
+ ]
+ ]
+]
+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 "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubst $L $T1 $d $e $T2 }.
+
+include "basic_2/substitution/ldrop_append.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+inductive tps: nat → nat → lenv → relation term ≝
+| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
+| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV → ⇧[0, i + 1] V ≡ W → tps d e L (#i) W
+| tps_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ tps d e L V1 V2 → tps (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ tps d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ tps d e L V1 V2 → tps d e L T1 T2 →
+ tps d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "parallel substritution (term)"
+ 'PSubst L T1 d e T2 = (tps d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tps_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶ [d, e] T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ //
+| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_refl: ∀T,L,d,e. L ⊢ T ▶ [d, e] T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 ▶ [d, 1] T2 & ⇧[d, 1] T ≡ T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i ? ? ?) // /3 width=4/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ ]
+]
+qed.
+
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶ [d1, e1] T2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+ L ⊢ T1 ▶ [d2, e2] T2.
+#L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
+[ //
+| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
+ lapply (transitive_le … Hd12 … Hid1) -Hd12 -Hid1 #Hid2
+ lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2 width=4/
+| /4 width=3/
+| /4 width=1/
+]
+qed.
+
+lemma tps_weak_top: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [d, |L| - d] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+ lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+ lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
+| normalize /2 width=1/
+| /2 width=1/
+]
+qed.
+
+lemma tps_weak_full: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [0, |L|] T2.
+#L #T1 #T2 #d #e #HT12
+lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tps_weak_top … HT12) //
+qed.
+
+lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶ [d, i - d] T & L ⊢ T ▶ [i, d + e - i] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
+| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j)
+ [ -Hide -Hjde
+ >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/
+ | -Hdi -Hdj #Hij
+ lapply (plus_minus_m_m … Hjde) -Hjde /3 width=8/
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T &
+ L ⊢ T ▶ [d, i - d] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
+| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+ elim (lt_or_ge i j)
+ [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=8/
+ | -Hdi -Hdj
+ >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi -Hide /3 width=5/
+]
+qed.
+
+lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 →
+ ∀L. L @@ K ⊢ T1 ▶ [d, e] T2.
+#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
+#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
+lapply (ldrop_fwd_ldrop2_length … HK0) #H
+@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
+[ #L #I #d #e #J #H destruct /2 width=1/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+]
+qed.
+
+lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+/2 width=3/ qed-.
+
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 →
+ T2 = #i ∨
+ ∃∃K,V. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2.
+#L #T2 #i #d #e #H
+elim (tps_inv_atom1 … H) -H /2 width=1/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+qed-.
+
+lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p.
+#L #T2 #p #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #a #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
+| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
+]
+qed.
+
+lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+/2 width=3/ qed-.
+
+fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+/2 width=3/ qed-.
+
+fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
+ elim (lt_refl_false … Hdd)
+| /3 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 0] T2 → T1 = T2.
+/2 width=6/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
+/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+qed-.
+
+lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #d #e #HT1
+ @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X #d #e
+ >shift_append_assoc normalize #H
+ elim (tps_inv_bind1 … H) -H
+ #V0 #T0 #_ #HT10 #H destruct
+ elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+ >append_length >HL12 -HL12
+ @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/ldrop_ldrop.ma".
+include "basic_2/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
+#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
+[ //
+| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
+ elim (lt_or_ge i (d+1)) #HiSd
+ [ -Hide1 -HV0
+ lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct
+ | -V -Hdi /2 width=4/
+ ]
+| /4 width=3/
+| /3 width=3/
+]
+qed.
+
+lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
+/2 width=3/ qed-.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #HT12 #K #V #HLK
+lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
+lapply (tps_inv_refl_O2 … HT12) -HT12 //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt + et ≤ d →
+ L ⊢ U1 ▶ [dt, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ dt ≤ d → d ≤ dt + et →
+ L ⊢ U1 ▶ [dt, et + e] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+ | -Hdti
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+ ]
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
+ ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+]
+qed.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+ ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
+ d ≤ dt →
+ L ⊢ U1 ▶ [dt + e, et] U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
+ lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
+ lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
+ lapply (transitive_le … Hdedt … Hdti) #Hdei
+ elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+ elim (le_inv_plus_l … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
+ <plus_minus // /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
+ elim (lift_inv_lref2 … H) -H * #H
+ [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
+ elim (lt_refl_false … H)
+ | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
+ elim (lt_refl_false … H)
+ ]
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >IHV12 // >IHT12 //
+]
+qed.
+(*
+ Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le (plus d h) i) ->
+ (EX u1 | (subst0 (minus i h) v u1 u2) &
+ t1 = (lift h d u1)
+ ).
+
+
+ Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le d i) -> (lt i (plus d h)) ->
+ (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
+*)
+lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
+qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
+qed.
+
+lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
+elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
+lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
+lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_lift.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
+#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (tps_inv_lref1 … H) -H
+ [ #HX destruct /3 width=6/
+ | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
+ ]
+| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
+| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
+#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+ elim (tps_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=6/
+ | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
+ [ -Hd1 -Hde2
+ lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
+ lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
+ elim (lt_refl_false … H)
+ | -Hd2 -Hde1
+ lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
+ lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
+ elim (lt_refl_false … H)
+ ]
+ ]
+| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02 ?) -T0
+ [ -H #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
+ | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
+ [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
+ ]
+| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5/
+]
+qed.
+
+(* Note: the constant 1 comes from tps_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
+ ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
+ L ⊢ T1 ▶ [d, e] T2.
+#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
+[ #L #I #d #e #T2 #H #He
+ elim (tps_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
+ ]
+| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
+ <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
+| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ lapply (IHT10 … HT02 He) -T0 #HT12
+ lapply (tps_lsubr_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
+| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
+]
+qed.
+
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
+[ /2 width=3/
+| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
+ lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=8/
+| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
+ lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02 ?) -V0 //
+ elim (IHT10 … HT02 ?) -T0 // /3 width=6/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $L $T1 $d $e $T2 }.
+
+include "basic_2/substitution/tps.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+definition tpss: nat → nat → lenv → relation term ≝
+ λd,e,L. TC … (tps d e L).
+
+interpretation "partial unfold (term)"
+ 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
+ ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
+#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
+ ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
+#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=1/ qed.
+
+lemma tpss_strap1: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+lemma tpss_strap2: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
+/3 width=3/ qed.
+
+lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
+/2 width=1/ qed.
+
+lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
+#L #V1 #V2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
+ [ /3 width=5/
+ | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+ ]
+| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
+ lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
+ lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+]
+qed.
+
+lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
+ L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
+ L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
+#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #HT12 elim HT12 -T2
+ [ /3 width=1/
+ | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+ ]
+| #V #V2 #_ #HV12 #IHV #HT12
+ lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+]
+qed.
+
+lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+ L ⊢ T1 ▶* [d2, e2] T2.
+#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
+]
+qed.
+
+lemma tpss_weak_top: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak_top … HT12) -HT12 /2 width=3/
+]
+qed.
+
+lemma tpss_weak_full: ∀L,T1,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
+#L #T1 #T2 #d #e #HT12
+lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tpss_weak_top … HT12) //
+qed.
+
+lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
+ ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
+#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from tpss_inv_atom1 *)
+lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
+#L #T2 #k #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_sort1 … HT2) -HT2 //
+]
+qed-.
+
+(* Note: this can be derived from tpss_inv_atom1 *)
+lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
+#L #T2 #p #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_gref1 … HT2) -HT2 //
+]
+qed-.
+
+lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
+ L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+]
+qed-.
+
+lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
+ ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
+]
+qed-.
+
+lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
+#L #T1 #T2 #d #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_fwd_tw … HT2) -HT2 #HT2
+@(transitive_le … IHT1) //
+qed-.
+
+lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
+ ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
+[ /2 width=4/
+| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
+ elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
+]
+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 "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss_lift.ma".
+
+(* PARALLEL UNFOLD ON TERMS *************************************************)
+
+(* alternative definition of tpss *)
+inductive tpssa: nat → nat → lenv → relation term ≝
+| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I})
+| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
+ ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
+| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ tpssa d e L V1 V2 → tpssa d e L T1 T2 →
+ tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "parallel unfold (term) alternative"
+ 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tpssa_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
+ ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
+#L #T1 #T #d #e #H elim H -L -T1 -T -d -e
+[ #L #I #d #e #X #H
+ elim (tps_inv_atom1 … H) -H // * /2 width=6/
+| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
+ lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
+ lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
+ elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
+| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tps_lsubr_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
+ lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
+ lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
+ lapply (tpssa_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
+| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
+]
+qed.
+
+lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
+qed-.
+
+lemma tpss_ind_alt: ∀R:nat→nat→lenv→relation term.
+ (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
+ (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
+ ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
+ ) →
+ (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
+ R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
+ R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
+#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
+// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/
+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/tps_lift.ma".
+include "basic_2/unfold/tpss.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma tpss_subst: ∀L,K,V,U1,i,d,e.
+ d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 →
+ ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2.
+#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1
+[ /3 width=4/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU … HU0) -IHU #H
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
+ lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
+]
+qed.
+
+(* Advanced inverion lemmas *************************************************)
+
+lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V1,V2,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
+ ⇧[O, i + 1] V2 ≡ T2 &
+ I = LRef i.
+#L #T2 #I #d #e #H @(tpss_ind … H) -T2
+[ /2 width=1/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
+ | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
+ lapply (ldrop_fwd_ldrop2 … HLK) #H
+ elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
+ @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
+ ]
+]
+qed-.
+
+lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶* [d, e] T2 →
+ T2 = #i ∨
+ ∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #d #e #H
+elim (tpss_inv_atom1 … H) -H /2 width=1/
+* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma tpss_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e + 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶* [d + 1, e] T2.
+#L #T1 #T2 #d #e #H #K #V #HLK @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT
+lapply (tps_inv_S2 … HT2 … HLK) -HT2 -HLK /2 width=3/
+qed-.
+
+lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 ▶* [dt, et] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_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 (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_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 (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 ▶* [dt + e, et] U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_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 (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU destruct
+<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+qed.
+
+lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /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/tps_tps.ma".
+include "basic_2/unfold/tpss_lift.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tpss_inv_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2.
+#L #T1 #T2 #d #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_trans_ge … IHT1 … HT2 ?) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2.
+#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2
+[ /2 width=3/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
+ elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ]
+ /3 width=7 by ex2_intro, step/ (**) (* just /3 width=7/ is too slow *)
+]
+qed.
+
+lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 ▶* [d2, e2] T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 ▶* [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶* [d, e] T2 →
+ L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+ ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/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/unfold/tpss.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+definition delift: nat → nat → lenv → relation term ≝
+ λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
+
+interpretation "inverse basic relocation (term)"
+ 'TSubst L T1 d e T2 = (delift d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
+ ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
+/2 width=3/ qed.
+
+lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
+/2 width=3/ qed.
+
+lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
+#L1 #T1 #T2 #d #e * /3 width=3/
+qed.
+
+lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
+/2 width=3/ qed.
+
+lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
+/3 width=3/ qed.
+
+lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
+/3 width=3/ qed.
+
+lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
+/2 width=3/ qed.
+
+lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
+qed.
+
+lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
+#L #U2 #d #e #k * #U #HU
+>(tpss_inv_sort1 … HU) -HU #HU2
+>(lift_inv_sort2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
+#L #U #d #e #p * #U #HU
+>(tpss_inv_gref1 … HU) -HU #HU2
+>(lift_inv_gref2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L ⊢ ▼*[d, e] T1 ≡ T2 &
+ U2 = ⓕ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
+qed-.
+
+lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
+#L #T1 #T2 #d * #T #HT1
+>(tpss_inv_refl_O2 … HT1) -HT1 #HT2
+>(lift_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e * #T #HT1 #HT2
+>(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
+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/unfold/delift_lift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* alternative definition of inverse basic term relocation *)
+inductive delifta: nat → nat → lenv → relation term ≝
+| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
+| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
+| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
+ ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
+| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
+| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
+| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
+ delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
+ delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ delifta d e L V1 V2 → delifta d e L T1 T2 →
+ delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "inverse basic relocation (term) alternative"
+ 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma delifta_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
+[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
+#L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
+[ * #i #Hn #T2 #d #e #H destruct
+ [ >(delift_inv_sort1 … H) -H //
+ | elim (delift_inv_lref1 … H) -H * /2 width=1/
+ #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
+ lapply (ldrop_pair2_fwd_fw … HLK) #H
+ lapply (IH … HV12) // -H /2 width=6/
+ | >(delift_inv_gref1 … H) -H //
+ ]
+| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
+ [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
+ lapply (IH … HV12) -HV12 // #HV12
+ lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
+ lapply (delifta_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
+ | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH … HV12) -HV12 //
+ lapply (IH … HT12) -IH -HT12 // /2 width=1/
+ ]
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
+qed-.
+
+lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
+ (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
+ (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
+ (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
+ ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
+ ) →
+ (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
+ (∀L,d,e,p. R d e L (§p) (§p)) →
+ (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
+ L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
+ R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
+ L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
+ R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
+// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/
+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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem delift_mono: ∀L,T,T1,T2,d,e.
+ L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2.
+#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2
+elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2
+lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct
+lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct
+lapply (lift_inj … H2TU1 … H2TU2) //
+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_lbotr.ma".
+include "basic_2/unfold/tpss_lift.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
+ ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
+#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
+elim (lift_total V 0 (i+1)) #U #HVU
+lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U
+lapply (lift_conf_be … HVU2 … HV2U ?) //
+>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
+qed.
+
+lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
+ ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 @(f2_ind … fw … L T1) -L -T1
+#n #IH #L * * /2 width=2/
+[ #i #H #d #e #Hde #HL destruct
+ elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
+ elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
+ lapply (lt_to_le_to_lt … Hide Hde) #Hi
+ elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
+ lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
+ lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
+ lapply (ldrop_fwd_O1_length … HLK0) #H
+ lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
+ [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
+ elim (IH … HKL … HK) -IH -HKL -HK
+ [2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
+ elim (lift_total V2 0 d) /3 width=7/
+| #a #I #V1 #T1 #H #d #e #Hde #HL destruct
+ elim (IH … V1 … Hde HL) // #V2 #HV12
+ elim (IH (L.ⓑ{I}V1) T1 … (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12
+ lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
+| #I #V1 #T1 #H #d #e #Hde #HL destruct
+ elim (IH … V1 … Hde HL) // #V2 #HV12
+ elim (IH … T1 … Hde HL) -IH -Hde -HL // /3 width=2/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
+#L #U2 #i #d #e * #U #HU #HU2 #Hid
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_lt … HU2) //
+| * #K #V1 #V2 #Hdi
+ lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
+
+lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
+ d ≤ i → i < d + e →
+ ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
+ ⇧[0, d] V2 ≡ U2.
+#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
+| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
+ elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
+]
+qed-.
+
+lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
+ d + e ≤ i → U2 = #(i - e).
+#L #U2 #i #d #e * #U #HU #HU2 #Hdei
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_ge … HU2) //
+| * #K #V1 #V2 #_ #Hide
+ lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
+
+lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
+ ∨∨ (i < d ∧ U2 = #i)
+ | (∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
+ ⇧[0, d] V2 ≡ U2
+ )
+ | (d + e ≤ i ∧ U2 = #(i - e)).
+#L #U2 #i #d #e #H
+elim (lt_or_ge i d) #Hdi
+[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
+| elim (lt_or_ge i (d+e)) #Hide
+ [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
+ | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
+ ]
+]
+qed-.
+
+(* Properties on basic term relocation **************************************)
+
+lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
+ L ⊢ ▼*[dt, et] U1 ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.
+
+lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ L ⊢ ▼*[dt, et + e] U1 ≡ T2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
+qed.
+
+lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ ▼*[dt + e, et] U1 ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.
+
+lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
+ ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
+#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
+lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
+lapply (lift_inj … HTU1 … HTU2) -U2 //
+qed-.
+
+lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
+ ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
+ L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
+lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
+lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
+>commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
+<minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /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/unfold/ltpss_sn_alt.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Properties on sn partial unfold on local environments ********************)
+
+lemma delift_ltpss_sn_conf_eq: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 →
+ ∀K. L ⊢ ▶* [d, e] K → K ⊢ ▼*[d, e] T1 ≡ T2.
+#L #T1 #T2 #d #e * #T #HT1 #HT2 #K #HLK
+elim (ltpss_sn_tpss_conf … HT1 … HLK) -HT1 -HLK #T0 #HT10 #HT0
+lapply (tpss_inv_lift1_eq … HT0 … HT2) -HT0 #H destruct /2 width=3/
+qed.
+
+lemma ltpss_sn_delift_trans_eq: ∀L,K,d,e. L ⊢ ▶* [d, e] K →
+ ∀T1,T2. K ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
+#L #K #d #e #HLK #T1 #T2 * #T #HT1 #HT2
+lapply (ltpss_sn_tpss_trans_eq … HT1 … HLK) -K /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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+(* Properties on partial unfold on terms ************************************)
+
+lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
+qed.
+
+lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
+qed.
+
+lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6/ qed.
+
+lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
+qed.
+
+lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
+qed.
+
+lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
+/3 width=3/ qed.
+
+lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
+qed.
+
+lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
+/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/unfold/ltpss_sn.ma".
+
+(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
+
+definition thin: nat → nat → relation lenv ≝
+ λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2.
+
+interpretation "basic thinning (local environment)"
+ 'TSubst L1 d e L2 = (thin d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e →
+ ▼*[0, e - 1] K1 ≡ L2.
+#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct
+lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /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/unfold/delift_tpss.ma".
+include "basic_2/unfold/delift_ltpss.ma".
+include "basic_2/unfold/thin.ma".
+
+(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
+
+(* Inversion lemmas on inverse basic term relocation ************************)
+
+lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
+ K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
+ L2 = K2. ⓑ{I} V2.
+#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
+elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
+qed-.
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
+ ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
+elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
+lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
+lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
+qed.
+
+lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K →
+ d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
+
+lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+ ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/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/unfold/ltpss_sn_ldrop.ma".
+include "basic_2/unfold/thin.ma".
+
+(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
+
+(* Properties on local environment slicing **********************************)
+
+lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2.
+#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/
+qed.
+
+lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
+#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1
+elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0
+elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/
+qed.
+
+lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1
+elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0
+elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/
+qed.
+
+lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+ d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2
+lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2
+lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/
+qed.
+
+lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1
+elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02
+elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /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/grammar/fsupp.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-definition fsups: bi_relation lenv term ≝ bi_star … fsup.
-
-interpretation "star-iterated structural successor (closure)"
- 'SupTermStar L1 T1 L2 T2 = (fsups L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_star_ind … IH1 IH2 ? ? H)
-qed-.
-
-lemma fsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
-@(bi_star_ind_dx … IH1 IH2 ? ? H)
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fsups_refl: bi_reflexive … fsups.
-/2 width=1/ qed.
-
-lemma fsupp_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma fsup_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=1/ qed.
-
-lemma fsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ →
- ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fsups_fsupp_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ →
- ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma fsupp_fsups_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ →
- ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fsups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 //
-/4 width=3 by fsup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *)
-qed-.
-
-(* Advanced inversion lemmas on plus-iterated supclosure ********************)
-
-lemma fsupp_inv_bind1_fsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
- ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃L2, T2⦄.
-#b #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
- elim (fsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
-]
-qed-.
-
-lemma fsupp_inv_flat1_fsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
- ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⊃* ⦃L2, T2⦄.
-#J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
- elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
-]
-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/fsups.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fsups_trans: bi_transitive … fsups.
-/2 width=4/ qed.
lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
.
-definition lpx_sn_confluent: predicate (lenv→relation term) ≝ λR.
- ∀L0,T0,T1. R L0 T0 T1 → ∀T2. R L0 T0 T2 →
- ∀L1. lpx_sn R L0 L1 → ∀L2. lpx_sn R L0 L2 →
- ∃∃T. R L1 T1 T & R L2 T2 T.
-
-definition lpx_sn_transitive: predicate (lenv→relation term) ≝ λR.
- ∀L1,T1,T. R L1 T1 T → ∀L2. lpx_sn R L1 L2 →
- ∀T2. R L2 T T2 → R L1 T1 T2.
+definition lpx_sn_confluent: relation (lenv→relation 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. R1 L1 T1 T & R2 L2 T2 T.
+definition lpx_sn_transitive: relation (lenv→relation 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.
(* Basic inversion lemmas ***************************************************)
#R #L1 #L2 #H elim H -L1 -L2 normalize //
qed-.
+(* Advanced forward lemmas **************************************************)
+
+lemma lpx_sn_fwd_append1: ∀R,L1,K1,L. lpx_sn R (K1 @@ L1) L →
+ ∃∃K2,L2. lpx_sn R K1 K2 & L = K2 @@ L2.
+#R #L1 elim L1 -L1
+[ #K1 #K2 #HK12
+ @(ex2_2_intro … K2 (⋆)) // (* explicit constructor, /2 width=4/ does not work *)
+| #L1 #I #V1 #IH #K1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #H destruct
+ @(ex2_2_intro … (L2.ⓑ{I}V2) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
+
+lemma lpx_sn_fwd_append2: ∀R,L2,K2,L. lpx_sn R L (K2 @@ L2) →
+ ∃∃K1,L1. lpx_sn R K1 K2 & L = K1 @@ L1.
+#R #L2 elim L2 -L2
+[ #K2 #K1 #HK12
+ @(ex2_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=4/ does not work *)
+| #L2 #I #V2 #IH #K2 #X #H
+ elim (lpx_sn_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #H destruct
+ @(ex2_2_intro … (L1.ⓑ{I}V1) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
+
(* Basic properties *********************************************************)
lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
#R #HR #K1 #K2 #H elim H -K1 -K2 // /3 width=1/
qed-.
-lemma lpx_sn_trans: ∀R. lpx_sn_transitive R → Transitive … (lpx_sn R).
+(* Advanced properties ******************************************************)
+
+lemma 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/
qed-.
-lemma lpx_sn_conf: ∀R. lpx_sn_confluent R → confluent … (lpx_sn R).
+lemma lpx_sn_conf: ∀R. lpx_sn_confluent R R → confluent … (lpx_sn R).
#R #HR #L0 @(f_ind … length … L0) -L0 #n #IH *
[ #_ #X1 #H1 #X2 #H2 -n
>(lpx_sn_inv_atom1 … H1) -X1
non associative with precedence 45
for @{ 'HdNormal $L $T }.
-notation "hvbox( T1 ➡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRed $T1 $T2 }.
-
notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )"
non associative with precedence 45
for @{ 'PRed $L $T1 $T2 }.
-notation "hvbox( ⦃ term 46 L1 ⦄ ➡ break ⦃ term 46 L2 ⦄ )"
+notation "hvbox( L1 ⊢ ➡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSn $L1 $L2 }.
+
+notation "hvbox( L1 ⊢ ➡ ➡ break term 46 L2 )"
non associative with precedence 45
- for @{ 'FocalizedPRed $L1 $L2 }.
+ for @{ 'PRedSnAlt $L1 $L2 }.
notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
non associative with precedence 45
for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
-notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )"
- non associative with precedence 45
- for @{ 'FocalizedPRedAlt $L1 $L2 }.
-
(* Computation **************************************************************)
notation "hvbox( T1 ➡ * break term 46 T2 )"
elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // -Hi
<H2i -H2i <minus_plus_m_m /3 width=6/
| /4 width=1/
-| /3 width=1/
+|4,6: /3 width=1/
| /4 width=3/
-| /3 width=1/
]
qed-.
(* Basic properties *********************************************************)
-(* Basic_1: was by definition: csubst1_refl *)
lemma lpqs_refl: ∀L. L ⊢ ➤* L.
/2 width=1 by lpx_sn_refl/ qed.
lemma lpqs_fwd_length: ∀L1,L2. L1 ⊢ ➤* L2 → |L1| = |L2|.
/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpqs_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➤* L →
+ ∃∃K2,L2. K1 ⊢ ➤* K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpqs_fwd_append2: ∀L,K2,L2. L ⊢ ➤* K2 @@ L2 →
+ ∃∃K1,L1. K1 ⊢ ➤* K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
qed-.
-theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs.
+theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
elim (cpqs_inv_atom1 … H1) -H1
theorem cpqs_conf: ∀L. confluent … (cpqs L).
/2 width=6 by cpqs_conf_lpqs/ qed-.
-theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs.
+theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
elim (cpqs_inv_atom1 … H1) -H1
elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
lapply (lpqs_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
@(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
+ | #T #_ #_ #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/unfold/lpss_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+(* Note: lemma 500 *)
+lemma aaa_cpss_lpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ▶* T2 →
+ ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #X #H
+ >(cpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
+ | * #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
+]
+qed-.
+
+lemma aaa_cpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ▶* T2 → L ⊢ T2 ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ qed-.
+
+lemma aaa_lpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ 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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_ldrop.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-(* Note: lemma 500 *)
-lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #H elim H -L1 -T1 -A
-[ #L1 #k #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H //
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lt_or_ge i d) #Hdi
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | elim (lt_or_ge i (d + e)) #Hide
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | -Hdi
- lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
- ]
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-]
-qed.
-
-lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=7/ qed.
-
-lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ 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/unfold/ltpss_sn_alt.ma".
-include "basic_2/static/aaa_ltpss_dx.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
-#L1 #T #A #HT #L2 #d #e #HL12
-lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
-@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
-qed.
-
-lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=5/ qed.
-
-lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/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/unfold/lpss_ldrop.ma".
+include "basic_2/static/ssta_lift.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ssta_tpss_lpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
+[ #L1 #k1 #l1 #Hkl1 #X #H
+ >(cpss_inv_sort1 … H) -H /3 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHVW1 … HV12 … HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
+ | * #Y #Z #V2 #H #HV12 #HV2
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #Z #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHVW1 … HV12 … HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (ssta_lift … HVW2 … HLK2 … HV2 … HWU2) -HVW2 -HLK2 -HV2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 -HWU2 /3 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #X #H #L2 #HL12
+ elim (cpss_inv_lref1 … H) -H [ | -IHWV1 -HWU1 -HL12 ]
+ [ #H destruct
+ elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
+ elim (IHWV1 … HW12 … HK12) -IHWV1 -HK12 #V2 #HWV2 #_
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
+ | * #K2 #V2 #W2 #HLK2 #_ #_
+ lapply (ldrop_mono … HLK2 … HLK1) -HLK1 -HLK2 #H destruct
+ ]
+| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12 (L2.ⓑ{I}V2)) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=5/
+| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
+ elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+ elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=3/
+]
+qed-.
+
+lemma ssta_cpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+ ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* U2.
+/2 width=3 by ssta_tpss_lpss_conf/ qed-.
+
+lemma ssta_lpss_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
+ ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
+/2 width=3 by ssta_tpss_lpss_conf/ 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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_tpss.ma".
-include "basic_2/static/ssta_lift.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about dx parallel unfold **************************************)
-
-(* Note: apparently this was missing in basic_1 *)
-lemma ssta_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
-[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H /3 width=3/
-| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
- elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
- lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HWV1 ]
- [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
- [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
- elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
- ]
-| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
-| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
-| #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
-]
-qed.
-
-lemma ssta_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=5/ qed.
-
-lemma ssta_ltpss_dx_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* [d, e] U2.
-/2 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/unfold/ltpss_sn_alt.ma".
-include "basic_2/static/ssta_ltpss_dx.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma ssta_ltpss_sn_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T #U1 #l #HTU1 #L2 #d #e #HL12
-lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
-@(ltpssa_ind … HL12) -L2 [ /2 width=3/ ] -HTU1
-#L #L2 #HL1 #HL2 * #U #HTU #HU1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-elim (ssta_ltpss_dx_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
-lapply (ltpss_dx_tpss_trans_eq … HU2 … HL2) -HU2 -HL2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL1) -HU2 -HL1 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma ssta_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #l #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (ssta_ltpss_sn_conf … HTU1 … HL12) -HTU1 #U #HT1U #HU1
-elim (ssta_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma ssta_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ &
- L1 ⊢ U1 ▶* [d, e] U2.
-/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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 break ▶ [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubst $L $T1 $d $e $T2 }.
-
-include "basic_2/substitution/ldrop_append.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-inductive tps: nat → nat → lenv → relation term ≝
-| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
-| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV → ⇧[0, i + 1] V ≡ W → tps d e L (#i) W
-| tps_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- tps d e L V1 V2 → tps (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- tps d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
- tps d e L V1 V2 → tps d e L T1 T2 →
- tps d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "parallel substritution (term)"
- 'PSubst L T1 d e T2 = (tps d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tps_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶ [d, e] T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ //
-| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_refl: ∀T,L,d,e. L ⊢ T ▶ [d, e] T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 ▶ [d, 1] T2 & ⇧[d, 1] T ≡ T2.
-#K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
- elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i ? ? ?) // /3 width=4/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
- ]
-]
-qed.
-
-lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶ [d1, e1] T2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L ⊢ T1 ▶ [d2, e2] T2.
-#L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
-[ //
-| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
- lapply (transitive_le … Hd12 … Hid1) -Hd12 -Hid1 #Hid2
- lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2 width=4/
-| /4 width=3/
-| /4 width=1/
-]
-qed.
-
-lemma tps_weak_top: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [d, |L| - d] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
- lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
- lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
-| normalize /2 width=1/
-| /2 width=1/
-]
-qed.
-
-lemma tps_weak_full: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶ [0, |L|] T2.
-#L #T1 #T2 #d #e #HT12
-lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
-lapply (tps_weak_top … HT12) //
-qed.
-
-lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶ [d, i - d] T & L ⊢ T ▶ [i, d + e - i] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide -Hjde
- >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/
- | -Hdi -Hdj #Hij
- lapply (plus_minus_m_m … Hjde) -Hjde /3 width=8/
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
-]
-qed.
-
-lemma tps_split_down: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶ [i, d + e - i] T &
- L ⊢ T ▶ [d, i - d] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=8/
- | -Hdi -Hdj
- >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi -Hide /3 width=5/
-]
-qed.
-
-lemma tps_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶ [d, e] T2 →
- ∀L. L @@ K ⊢ T1 ▶ [d, e] T2.
-#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
-#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_ldrop2_length … HK0) #H
-@(tps_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
-[ #L #I #d #e #J #H destruct /2 width=1/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-]
-qed.
-
-lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶ [d, e] T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-/2 width=3/ qed-.
-
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶ [d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H
-elim (tps_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: subst1_gen_lref *)
-lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶ [d, e] T2 →
- T2 = #i ∨
- ∃∃K,V. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2.
-#L #T2 #i #d #e #H
-elim (tps_inv_atom1 … H) -H /2 width=1/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
-qed-.
-
-lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶ [d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H
-elim (tps_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #a #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
-| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
-]
-qed.
-
-lemma tps_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶ [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶ [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-/2 width=3/ qed-.
-
-fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 ▶ [d, e] U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-]
-qed.
-
-lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶ [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶ [d, e] V2 & L ⊢ T1 ▶ [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-/2 width=3/ qed-.
-
-fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
- elim (lt_refl_false … Hdd)
-| /3 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 0] T2 → T1 = T2.
-/2 width=6/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
-qed-.
-
-lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #d #e #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X #d #e
- >shift_append_assoc normalize #H
- elim (tps_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_ldrop.ma".
-include "basic_2/substitution/tps.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
-#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
-[ //
-| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
- elim (lt_or_ge i (d+1)) #HiSd
- [ -Hide1 -HV0
- lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
- lapply (ldrop_mono … HLK0 … HLK) #H destruct
- | -V -Hdi /2 width=4/
- ]
-| /4 width=3/
-| /3 width=3/
-]
-qed.
-
-lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
-/2 width=3/ qed-.
-
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #HT12 #K #V #HLK
-lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
-lapply (tps_inv_refl_O2 … HT12) -HT12 //
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt + et ≤ d →
- L ⊢ U1 ▶ [dt, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
- lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-]
-qed.
-
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- dt ≤ d → d ≤ dt + et →
- L ⊢ U1 ▶ [dt, et + e] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
- elim (lift_inv_lref1 … H) -H * #Hid #H destruct
- [ -Hdtd
- lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
- | -Hdti
- lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
- ]
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
- ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-]
-qed.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
- ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- d ≤ dt →
- L ⊢ U1 ▶ [dt + e, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
- lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
- lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
- lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
- /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ -Hdtd -Hidet
- lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
- | -Hdti -Hdedet
- lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
- elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
- ]
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
- /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 //
- elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d + e ≤ dt →
- ∃∃T2. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
- lapply (transitive_le … Hdedt … Hdti) #Hdei
- elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
- elim (le_inv_plus_l … Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (le_inv_plus_l … Hdetd) #_ #Hedt
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
- <plus_minus // /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
- L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
- elim (lift_inv_lref2 … H) -H * #H
- [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
- elim (lt_refl_false … H)
- ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
-]
-qed.
-(*
- Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le (plus d h) i) ->
- (EX u1 | (subst0 (minus i h) v u1 u2) &
- t1 = (lift h d u1)
- ).
-
-
- Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le d i) -> (lt i (plus d h)) ->
- (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
-*)
-lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
-qed.
-
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
-elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
-qed.
-
-lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
-elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
-lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
-lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/tps_lift.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶ [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶ [d1, e1] T.
-#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (tps_inv_lref1 … H) -H
- [ #HX destruct /3 width=6/
- | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
- ]
-| #L #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
- elim (IHV01 … HV02) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V) ?) -HT2 /3 width=5/
-| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶ [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶ [d1, e1] T.
-#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
- elim (tps_inv_lref1 … H1) -H1
- [ #H destruct /3 width=6/
- | -HLK1 -HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
- [ -Hd1 -Hde2
- lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
- lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
- elim (lt_refl_false … H)
- | -Hd2 -Hde1
- lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
- lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
- elim (lt_refl_false … H)
- ]
- ]
-| #L1 #a #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02 ?) -T0
- [ -H #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /3 width=5/
- | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H #H
- [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3 / is too slow *)
- ]
-| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -H /3 width=5/
-]
-qed.
-
-(* Note: the constant 1 comes from tps_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 ▶ [d, e] T0 →
- ∀T2. L ⊢ T0 ▶ [d, 1] T2 → 1 ≤ e →
- L ⊢ T1 ▶ [d, e] T2.
-#L #T1 #T0 #d #e #H elim H -L -T1 -T0 -d -e
-[ #L #I #d #e #T2 #H #He
- elim (tps_inv_atom1 … H) -H
- [ #H destruct //
- | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
- ]
-| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
- <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
-| #L #a #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- lapply (IHT10 … HT02 He) -T0 #HT12
- lapply (tps_lsubr_trans … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
-| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
-]
-qed.
-
-theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
-#L #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
-[ /2 width=3/
-| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
- lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
- <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=8/
-| #L #a #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (tps_lsubr_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
- elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
- lapply (tps_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubr_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /3 width=6/
-| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 … HV02 ?) -V0 //
- elim (IHT10 … HT02 ?) -T0 // /3 width=6/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-definition delift: nat → nat → lenv → relation term ≝
- λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
-
-interpretation "inverse basic relocation (term)"
- 'TSubst L T1 d e T2 = (delift d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
- ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
-/2 width=3/ qed.
-
-lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
-/2 width=3/ qed.
-
-lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
-#L1 #T1 #T2 #d #e * /3 width=3/
-qed.
-
-lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
-/2 width=3/ qed.
-
-lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
-/3 width=3/ qed.
-
-lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
-/3 width=3/ qed.
-
-lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
-/2 width=3/ qed.
-
-lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
- L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
- L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
-#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
-qed.
-
-lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
- L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
- L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
-#L #U2 #d #e #k * #U #HU
->(tpss_inv_sort1 … HU) -HU #HU2
->(lift_inv_sort2 … HU2) -HU2 //
-qed-.
-
-lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
-#L #U #d #e #p * #U #HU
->(tpss_inv_gref1 … HU) -HU #HU2
->(lift_inv_gref2 … HU2) -HU2 //
-qed-.
-
-lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
- ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
- L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
- U2 = ⓑ{a,I} V2. T2.
-#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
-lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-qed-.
-
-lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
- ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
- L ⊢ ▼*[d, e] T1 ≡ T2 &
- U2 = ⓕ{I} V2. T2.
-#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
-qed-.
-
-lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
-#L #T1 #T2 #d * #T #HT1
->(tpss_inv_refl_O2 … HT1) -HT1 #HT2
->(lift_inv_refl_O2 … HT2) -HT2 //
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e * #T #HT1 #HT2
->(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
-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/unfold/delift_lift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* alternative definition of inverse basic term relocation *)
-inductive delifta: nat → nat → lenv → relation term ≝
-| delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
-| delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
-| delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
- ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
-| delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
-| delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
-| delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
- delifta d e L V1 V2 → delifta d e L T1 T2 →
- delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "inverse basic relocation (term) alternative"
- 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma delifta_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
-[ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
-#L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
-[ * #i #Hn #T2 #d #e #H destruct
- [ >(delift_inv_sort1 … H) -H //
- | elim (delift_inv_lref1 … H) -H * /2 width=1/
- #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
- lapply (ldrop_pair2_fwd_fw … HLK) #H
- lapply (IH … HV12) // -H /2 width=6/
- | >(delift_inv_gref1 … H) -H //
- ]
-| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
- [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
- lapply (IH … HV12) -HV12 // #HV12
- lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
- lapply (delifta_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
- | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV12) -HV12 //
- lapply (IH … HT12) -IH -HT12 // /2 width=1/
- ]
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
-qed-.
-
-lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
- (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
- (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
- (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
- ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
- ) →
- (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
- (∀L,d,e,p. R d e L (§p) (§p)) →
- (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
- L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
- R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
- L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
- R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
-// /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/
-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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Main properties **********************************************************)
-
-theorem delift_mono: ∀L,T,T1,T2,d,e.
- L ⊢ ▼*[d, e] T ≡ T1 → L ⊢ ▼*[d, e] T ≡ T2 → T1 = T2.
-#L #T #T1 #T2 #d #e * #U1 #H1TU1 #H2TU1 * #U2 #H1TU2 #H2TU2
-elim (tpss_conf_eq … H1TU1 … H1TU2) -T #U #HU1 #HU2
-lapply (tpss_inv_lift1_eq … HU1 … H2TU1) -HU1 #H destruct
-lapply (tpss_inv_lift1_eq … HU2 … H2TU2) -HU2 #H destruct
-lapply (lift_inj … H2TU1 … H2TU2) //
-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_lbotr.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
- ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
-#L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
-elim (lift_total V 0 (i+1)) #U #HVU
-lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U
-lapply (lift_conf_be … HVU2 … HV2U ?) //
->commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
-qed.
-
-lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
- ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 @(f2_ind … fw … L T1) -L -T1
-#n #IH #L * * /2 width=2/
-[ #i #H #d #e #Hde #HL destruct
- elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
- elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
- lapply (lt_to_le_to_lt … Hide Hde) #Hi
- elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
- lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
- lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
- lapply (ldrop_fwd_O1_length … HLK0) #H
- lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
- [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
- elim (IH … HKL … HK) -IH -HKL -HK
- [2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
- elim (lift_total V2 0 d) /3 width=7/
-| #a #I #V1 #T1 #H #d #e #Hde #HL destruct
- elim (IH … V1 … Hde HL) // #V2 #HV12
- elim (IH (L.ⓑ{I}V1) T1 … (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12
- lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
-| #I #V1 #T1 #H #d #e #Hde #HL destruct
- elim (IH … V1 … Hde HL) // #V2 #HV12
- elim (IH … T1 … Hde HL) -IH -Hde -HL // /3 width=2/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
-#L #U2 #i #d #e * #U #HU #HU2 #Hid
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_lt … HU2) //
-| * #K #V1 #V2 #Hdi
- lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
-
-lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
- d ≤ i → i < d + e →
- ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
- ⇧[0, d] V2 ≡ U2.
-#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
-| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
- elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
-]
-qed-.
-
-lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
- d + e ≤ i → U2 = #(i - e).
-#L #U2 #i #d #e * #U #HU #HU2 #Hdei
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_ge … HU2) //
-| * #K #V1 #V2 #_ #Hide
- lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
-
-lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
- ∨∨ (i < d ∧ U2 = #i)
- | (∃∃K,V1,V2. d ≤ i & i < d + e &
- ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
- ⇧[0, d] V2 ≡ U2
- )
- | (d + e ≤ i ∧ U2 = #(i - e)).
-#L #U2 #i #d #e #H
-elim (lt_or_ge i d) #Hdi
-[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
-| elim (lt_or_ge i (d+e)) #Hide
- [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
- | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
- ]
-]
-qed-.
-
-(* Properties on basic term relocation **************************************)
-
-lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
- L ⊢ ▼*[dt, et] U1 ≡ U2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
->(lift_mono … HTU2 … HT2) -T2 /2 width=3/
-qed.
-
-lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
- ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- L ⊢ ▼*[dt, et + e] U1 ≡ T2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
-qed.
-
-lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
- ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ ▼*[dt + e, et] U1 ≡ U2.
-#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
-elim (lift_total T d e) #U #HTU
-lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
-elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
->(lift_mono … HTU2 … HT2) -T2 /2 width=3/
-qed.
-
-lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
- ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
-#L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
-lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
-lapply (lift_inj … HTU1 … HTU2) -U2 //
-qed-.
-
-lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
- ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
- L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
-lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
-lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
->commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
-<minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /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/unfold/ltpss_sn_alt.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Properties on sn partial unfold on local environments ********************)
-
-lemma delift_ltpss_sn_conf_eq: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 →
- ∀K. L ⊢ ▶* [d, e] K → K ⊢ ▼*[d, e] T1 ≡ T2.
-#L #T1 #T2 #d #e * #T #HT1 #HT2 #K #HLK
-elim (ltpss_sn_tpss_conf … HT1 … HLK) -HT1 -HLK #T0 #HT10 #HT0
-lapply (tpss_inv_lift1_eq … HT0 … HT2) -HT0 #H destruct /2 width=3/
-qed.
-
-lemma ltpss_sn_delift_trans_eq: ∀L,K,d,e. L ⊢ ▶* [d, e] K →
- ∀T1,T2. K ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
-#L #K #d #e #HLK #T1 #T2 * #T #HT1 #HT2
-lapply (ltpss_sn_tpss_trans_eq … HT1 … HLK) -K /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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/delift.ma".
-
-(* INVERSE BASIC TERM RELOCATION *******************************************)
-
-(* Properties on partial unfold on terms ************************************)
-
-lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
-qed.
-
-lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
-qed.
-
-lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=6/ qed.
-
-lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
-qed.
-
-lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
-#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
-elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
-lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
-qed.
-
-lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
-/3 width=3/ qed.
-
-lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
-#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
-lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
-qed.
-
-lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
-/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/unfold/fsupp.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+definition fsups: bi_relation lenv term ≝ bi_star … fsup.
+
+interpretation "star-iterated structural successor (closure)"
+ 'SupTermStar L1 T1 L2 T2 = (fsups L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L2 T2.
+#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
+@(bi_star_ind … IH1 IH2 ? ? H)
+qed-.
+
+lemma fsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L1 T1.
+#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
+@(bi_star_ind_dx … IH1 IH2 ? ? H)
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fsups_refl: bi_reflexive … fsups.
+/2 width=1/ qed.
+
+lemma fsupp_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma fsup_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma fsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ →
+ ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fsups_fsupp_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ →
+ ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fsupp_fsups_fsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ →
+ ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fsups_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
+#L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 //
+/4 width=3 by fsup_fwd_cw, lt_to_le_to_lt, lt_to_le/ (**) (* slow even with trace *)
+qed-.
+(*
+(* Advanced inversion lemmas on plus-iterated supclosure ********************)
+
+lemma fsupp_inv_bind1_fsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
+ ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃L2, T2⦄.
+#b #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
+[ #L2 #T2 #H
+ elim (fsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
+| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+]
+qed-.
+
+lemma fsupp_inv_flat1_fsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
+ ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⊃* ⦃L2, T2⦄.
+#J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
+[ #L2 #T2 #H
+ elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
+| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+]
+qed-.
+*)
\ No newline at end of file
--- /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/unfold/fsups.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fsups_trans: bi_transitive … fsups.
+/2 width=4/ qed.
/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
lemma lpss_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* L2 →
- ∃∃K2,V2. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L2 = K2. ⓑ{I} V2.
+ ∃∃K2,V2. K1 ⊢ ▶* K2 & K1 ⊢ V1 ▶* V2 & L2 = K2. ⓑ{I} V2.
/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
lemma lpss_inv_atom2: ∀L1. L1 ⊢ ▶* ⋆ → L1 = ⋆.
lemma lpss_fwd_length: ∀L1,L2. L1 ⊢ ▶* L2 → |L1| = |L2|.
/2 width=2 by lpx_sn_fwd_length/ qed-.
+(* Advanced forward lemmas **************************************************)
+
+lemma lpss_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ▶* L →
+ ∃∃K2,L2. K1 ⊢ ▶* K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpss_fwd_append2: ∀L,K2,L2. L ⊢ ▶* K2 @@ L2 →
+ ∃∃K1,L1. K1 ⊢ ▶* K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
+
(* Basic_1: removed theorems 28:
csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
qed-.
-theorem cpss_conf_lpss: lpx_sn_confluent cpss.
+theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss.
#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
elim (cpss_inv_atom1 … H1) -H1
theorem cpss_conf: ∀L. confluent … (cpss L).
/2 width=6 by cpss_conf_lpss/ qed-.
-theorem cpss_trans_lpss: lpx_sn_transitive cpss.
+theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss.
#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
elim (cpss_inv_atom1 … H1) -H1
+++ /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 "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Basic_1: includes: csubst1_bind *)
-inductive ltpss_dx: nat → nat → relation lenv ≝
-| ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆)
-| ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e.
- ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
- ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-| ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e.
- ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
- ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "parallel unfold (local environment, dx variant)"
- 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
- >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
-]
-qed.
-
-lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
-/2 width=4/ qed-.
-
-fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2.
- L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
- ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K1 #V1 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
- ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2.
- L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
- K2 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=5/ qed-.
-
-fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K2 #V2 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
- K2 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
- L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
- L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2 width=1/
-qed.
-
-lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2 width=1/
-qed.
-
-lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-(* Basic_1: was by definition: csubst1_refl *)
-lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L.
-#L elim L -L //
-#L #I #V #IHL * /2 width=1/ * /2 width=1/
-qed.
-
-lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
-#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
-[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
- lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
- >plus_plus_comm_23 in Hde12; #Hde12
- elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
- [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
- lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
- | -Hd21 normalize in Hde12;
- lapply (lt_to_le_to_lt 0 … Hde12) // #He2
- lapply (le_plus_to_minus_r … Hde12) -Hde12
- /3 width=5 by ltpss_dx_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
- ]
-]
-qed.
-
-lemma ltpss_dx_weak_full: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-// /3 width=2/ /3 width=3/
-qed.
-
-fact ltpss_dx_append_le_aux: ∀K1,K2,d,x. K1 ▶* [d, x] K2 → x = |K1| - d →
- ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ▶* [d, x + e] L2 @@ K2.
-#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
-[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_n_O_to_eq … H2) -H2 #H destruct //
-| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
- lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
- /4 width=5 by ltpss_dx_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_plus_to_le_r … H2) -H2 #Hd
- /4 width=5 by ltpss_dx_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-]
-qed-.
-
-lemma ltpss_dx_append_le: ∀K1,K2,d. K1 ▶* [d, |K1| - d] K2 →
- ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ▶* [d, |K1| - d + e] L2 @@ K2.
-/2 width=1 by ltpss_dx_append_le_aux/ qed.
-
-lemma ltpss_dx_append_zero: ∀K1,K2. K1 ▶* [0, |K1|] K2 →
- ∀L1,L2,e. L1 ▶* [0, e] L2 →
- L1 @@ K1 ▶* [0, |K1| + e] L2 @@ K2.
-/2 width=1/ qed.
-
-lemma ltpss_dx_append_ge: ∀K1,K2,d,e. K1 ▶* [d, e] K2 →
- ∀L1,L2. L1 ▶* [d - |K1|, e] L2 → |K1| ≤ d →
- L1 @@ K1 ▶* [d, e] L2 @@ K2.
-#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
-[ #d #e #L1 #L2 <minus_n_O //
-| #K #I #V #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
- normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
- lapply (le_plus_to_le_r … H) -H /3 width=1/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpss_dx_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-normalize //
-qed-.
-
-(* Basic_1: removed theorems 28:
- csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
- csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
- csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
- csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
- csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
- csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
- csubst1_head csubst1_flat csubst1_gen_head
- csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
- fsubst0_gen_base
-*)
+++ /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/fsup.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/ltpss_dx.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma ltpss_dx_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ltpss_dx_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ldrop_ltpss_dx_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d1 ≤ d2 →
- ∃∃L2. L1 ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
- elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
- elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
- elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_ge … HW12 … HLK2 HWV1 … HWV2) -W1 // -Hd12
- <le_plus_minus_comm // /4 width=5/
-]
-qed-.
-
-lemma ldrop_ltpss_dx_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 →
- d2 ≤ d1 → d1 ≤ d2 + e2 →
- ∃∃L2. L1 ▶* [d2, e1 + e2] L2 &
- ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
- elim (IHLK1 … HK12 H1 H2) -K1 -H2
- lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #_ #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
- elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
- [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
- | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 // /2 width=1/
- >plus_minus // >commutative_plus /4 width=5/
- | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // ] /2 width=1/
- >commutative_plus /3 width=5/
- ]
-]
-qed-.
-
-lemma ldrop_ltpss_dx_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
- ∃∃L2. L1 ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_dx_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
- elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
- elim (IHLK1 … HK12 H) -K1
- elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
- >(ltpss_dx_inv_refl_O2 … HL12) -L1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
- elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
- [ -IHLK1 -Hd21 <(ltpss_dx_inv_refl_O2 … H) -X /3 width=5/
- | elim (ltpss_dx_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 /2 width=1/ /3 width=5/
- | elim (ltpss_dx_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK2 HWV1 … HWV2) -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
- ]
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsup_tps_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶[0,|L2|] U2 →
- ∃∃L,U1. L1 ▶*[0,|L|] L & L ⊢ T1 ▶[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-elim (le_or_ge d (|K|)) #Hd
-[ elim (ldrop_ltpss_dx_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tps_lift_be … HT1 … HL2K … HTU1 HTU … Hd) // -HT1 -HTU1 #HU1
-| elim (ldrop_ltpss_dx_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tps_lift_le … HT1 … HL2K … HTU1 HTU Hd) -HT1 -HTU1 #HU1
-]
-lapply (ltpss_dx_weak_full … HL12) -HL12 #HL12
-lapply (tps_weak_full … HU1) -HU1 #HU1
-@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
-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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_dx_tpss.ma".
-
-(* DX PARTIAL UNFOLD ON LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_dx_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
- L1 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind … H) -U2 /2 width=3/
-#U #U2 #_ #HU2 * #X2 #HTX2 #HUX2
-elim (ltpss_dx_tps_conf … HU2 … HL01) -L0 #X1 #HUX1 #HU2X1
-elim (tpss_strip_eq … HUX2 … HUX1) -U #X #HX2 #HX1
-lapply (tpss_trans_eq … HU2X1 … HX1) -X1 /3 width=3/
-qed.
-
-lemma ltpss_dx_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
- L1 ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L0 ⊢ T ▶* [d1, e1] U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
-[ /2 width=3/
-| #U #U2 #_ #HU2 * #T #HT2 #HTU
- elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
- elim (ltpss_dx_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
- lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/
-]
-qed.
-
-lemma ltpss_dx_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
- ∀L0. L0 ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
-#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
-[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
- elim (tpss_inv_atom1 … H) -H // *
- #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
- lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
- elim (ltpss_dx_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
- elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
- lapply (tpss_fwd_tw … HV01) #H2
- lapply (transitive_le (♯{K1} + ♯{V0}) … H1) -H1 /2 width=1/ -H2 #H
- lapply (tpss_trans_eq … HV01 HV12) -V1 #HV02
- lapply (IH … HV02 … HK01) -IH -HV02 -HK01
- [ normalize /2 width=1/ | /2 width=6/ ]
-| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
- [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
- lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
- lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
- lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
- | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (IH … HVW2 … HL0) -HVW2 //
- lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ▶* [d, e] L1 →
- L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpss_dx_conf: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ▶* [d1, e1] L.
-#L0 @(f_ind … lw … L0) -L0 #n #IH *
-[ #_ #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 -n
- >(ltpss_dx_inv_atom1 … H1) -L1
- >(ltpss_dx_inv_atom1 … H2) -L2 /2 width=3/
-| #K0 #I0 #V0 #Hn #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2 destruct
- elim (eq_or_gt d1) #Hd1 [ elim (eq_or_gt e1) #He1 ] destruct
- [ lapply (ltpss_dx_inv_refl_O2 … H1) -H1 #H1
- | elim (ltpss_dx_inv_tpss21 … H1 … He1) -H1 #K1 #V1 #HK01 #HV01 #H1
- | elim (ltpss_dx_inv_tpss11 … H1 … Hd1) -H1 #K1 #V1 #HK01 #HV01 #H1
- ] destruct
- elim (eq_or_gt d2) #Hd2 [1,3,5: elim (eq_or_gt e2) #He2 ] destruct
- [1,3,5: lapply (ltpss_dx_inv_refl_O2 … H2) -H2 #H2
- |2,4,6: elim (ltpss_dx_inv_tpss21 … H2 … He2) -H2 #K2 #V2 #HK02 #HV02 #H2
- |7,8,9: elim (ltpss_dx_inv_tpss11 … H2 … Hd2) -H2 #K2 #V2 #HK02 #HV02 #H2
- ] destruct
- [1: -IH /2 width=3/
- |2,3,4,7: -IH /3 width=5/
- |5,6,8,9:
- elim (IH … HK01 … HK02) // -K0 #K #HK1 #HK2
- elim (ltpss_dx_tpss_conf … HV01 … HK1) -HV01 #W1 #HW1 #HVW1
- elim (ltpss_dx_tpss_conf … HV02 … HK2) -HV02 #W2 #HW2 #HVW2
- elim (tpss_conf_eq … HW1 … HW2) -V0 #V #HW1 #HW2
- lapply (tpss_trans_eq … HVW1 HW1) -W1
- lapply (tpss_trans_eq … HVW2 HW2) -W2 /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/unfold/ltpss_dx_ldrop.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial substitution on terms **********************)
-
-lemma ltpss_dx_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
-
-lemma ltpss_dx_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_dx_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-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/unfold/ltpss_dx_tps.ma".
-
-(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial unfold on terms ****************************)
-
-lemma ltpss_dx_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_dx_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-(* Basic_1: was: subst1_subst1_back *)
-lemma ltpss_dx_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L1 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_dx_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_dx_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW0 … HVW1) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_dx_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2
- elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
-]
-qed.
-
-lemma ltpss_dx_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_dx_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-(* Basic_1: was: subst1_subst1 *)
-lemma ltpss_dx_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L0 ⊢ T ▶* [d1, e1] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_dx_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_dx_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
- elim (ltpss_dx_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_dx_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2
- elim (IHTU2 … HL10) -IHTU2 -HL10 /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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 break ⊢ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSn $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-inductive ltpss_sn: nat → nat → relation lenv ≝
-| ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
-| ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
- ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
- ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-| ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
- ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
- ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "parallel unfold (local environment, sn variant)"
- 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
- >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
-]
-qed.
-
-lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
-/2 width=4/ qed-.
-
-fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
- L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
- ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K1 #V1 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
- L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #_ #K1 #I #V1 #H destruct
-| #L1 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
- K1 ⊢ V1 ▶* [0, e - 1] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=5/ qed-.
-
-fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K2 #V2 #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
- K1 ⊢ V1 ▶* [d - 1, e] V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
- L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
- L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2 width=1/
-qed.
-
-lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2 width=1/
-qed.
-
-lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
- 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
- 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
-/3 width=1/ qed.
-
-lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
-#L elim L -L //
-#L #I #V #IHL * /2 width=1/ * /2 width=1/
-qed.
-
-lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
-#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
-[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
- lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
- >plus_plus_comm_23 in Hde12; #Hde12
- elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
- [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
- lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
- | -Hd21 normalize in Hde12;
- lapply (lt_to_le_to_lt 0 … Hde12) // #He2
- lapply (le_plus_to_minus_r … Hde12) -Hde12
- /3 width=5 by ltpss_sn_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
- ]
-]
-qed.
-
-lemma ltpss_sn_weak_full: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-// /3 width=2/ /3 width=3/
-qed.
-
-fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
- ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
-#K1 #K2 #d #x #H elim H -K1 -K2 -d -x
-[ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_n_O_to_eq … H2) -H2 #H destruct //
-| #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
- lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
- /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-| #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
- lapply (le_plus_to_le_r … H2) -H2 #Hd
- /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
-]
-qed-.
-
-lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
- ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
- L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
-/2 width=1 by ltpss_sn_append_le_aux/ qed.
-
-lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
- ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
- L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
-#K1 #K2 #d #e #H elim H -K1 -K2 -d -e
-[ #d #e #L1 #L2 <minus_n_O //
-| #K #I #V #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
- lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
-| #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
- normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
- lapply (le_plus_to_le_r … H) -H /3 width=1/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
-
-include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
-include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* alternative definition of ltpss_sn *)
-definition ltpssa: nat → nat → relation lenv ≝
- λd,e. TC … (ltpss_dx d e).
-
-interpretation "parallel unfold (local environment, sn variant) alternative"
- 'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
-#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
- ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
-#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
-/2 width=1/ qed.
-
-lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
-qed.
-
-lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
- ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L #d #e #H elim H -L1 -L -d -e
-[ #d #e #X #H
- lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
-| #L #I #V #X #H
- lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
-| #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-| #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
- ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-/3 width=3/ qed.
-
-lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
-lapply (ltpss_sn_ltpssa … H) -H #HL01
-elim (ltpssa_strip … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
- L0 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
-#L #L1 #H #HL1 * #T #HT2 #HU2T
-lapply (ltpssa_ltpss_sn … H) -H #HL0
-lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
-elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
-lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
-lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
- L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
-lapply (ltpss_sn_ltpssa … H) -H #HL10
-@(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
-#L1 #L #HL1 #_ * #T #HT2 #HTU2
-elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
-lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
-lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
-lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
-lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
-elim (ltpssa_conf … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, 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/fsup.ma".
-include "basic_2/unfold/tpss_lift.ma".
-include "basic_2/unfold/ltpss_sn.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma ltpss_sn_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 ⊢ ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L ⊢ ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltpss_sn_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 ⊢ ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ltpss_sn_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L ⊢ ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ldrop_ltpss_sn_trans_le: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d1 ≤ d2 →
- ∃∃L2. L1 ⊢ ▶* [d2 + e1, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #Hd
- elim (IHLK1 … HK12 Hd) -K1 -Hd /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd12
- elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hd2
- elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 … Hd12) -IHLK1 -HK12 <le_plus_minus_comm // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_ge … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // -Hd12
- <le_plus_minus_comm // /4 width=5/
-]
-qed-.
-
-lemma ldrop_ltpss_sn_trans_be: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 →
- d2 ≤ d1 → d1 ≤ d2 + e2 →
- ∃∃L2. L1 ⊢ ▶* [d2, e1 + e2] L2 &
- ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_ #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H #_
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H1 #H2
- elim (IHLK1 … HK12 H1 H2) -K1 -H2
- lapply (le_n_O_to_eq … H1) -H1 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21 #Hd12
- elim (eq_or_gt d2) #Hd2 [ -Hd21 elim (eq_or_gt e2) #He2 ] destruct
- [ lapply (le_n_O_to_eq … Hd12) -Hd12 <plus_n_Sm #H destruct
- | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 // /2 width=1/ >plus_minus_commutative // #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 // /2 width=1/
- >plus_minus // >commutative_plus /4 width=5/
- | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // ] /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_be … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // ] /2 width=1/
- >commutative_plus /3 width=5/
- ]
-]
-qed-.
-
-lemma ldrop_ltpss_sn_trans_ge: ∀L1,K1,d1,e1. ⇩[d1, e1] L1 ≡ K1 →
- ∀K2,d2,e2. K1 ⊢ ▶* [d2, e2] K2 → d2 + e2 ≤ d1 →
- ∃∃L2. L1 ⊢ ▶* [d2, e2] L2 & ⇩[d1, e1] L2 ≡ K2.
-#L1 #K1 #d1 #e1 #H elim H -L1 -K1 -d1 -e1
-[ #d1 #e1 #K2 #d2 #e2 #H #_
- >(ltpss_sn_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #K2 #d2 #e2 #HK12 #H
- elim (plus_le_0 … H) -H #H1 #H2 destruct /2 width=3/
-| #L1 #K1 #I #V #e1 #_ #IHLK1 #K2 #d2 #e2 #HK12 #H
- elim (IHLK1 … HK12 H) -K1
- elim (plus_le_0 … H) -H #H1 #H2 destruct #L2 #HL12
- >(ltpss_sn_inv_refl_O2 … HL12) -L1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d1 #e1 #HLK1 #HWV1 #IHLK1 #X #d2 #e2 #H #Hd21
- elim (eq_or_gt d2) #Hd2 [ elim (eq_or_gt e2) #He2 ] destruct
- [ -IHLK1 -Hd21 <(ltpss_sn_inv_refl_O2 … H) -X /3 width=5/
- | elim (ltpss_sn_inv_tpss21 … H He2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 /2 width=1/ #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 /2 width=1/ /3 width=5/
- | elim (ltpss_sn_inv_tpss11 … H Hd2) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHLK1 … HK12 …) -IHLK1 [2: >plus_minus // /2 width=1/ ] #L2 #HL12 #HLK2
- elim (lift_total W2 d1 e1) #V2 #HWV2
- lapply (tpss_lift_le … HW12 … HLK1 HWV1 … HWV2) -HLK1 -W1 [ >plus_minus // /2 width=1/ ] /3 width=5/
- ]
-]
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fsup_tpss_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶*[0,|L2|] U2 →
- ∃∃L,U1. L1 ⊢ ▶*[0,|L1|] L & L ⊢ T1 ▶*[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (ltpss_sn_fwd_length … HK1) #H >H in HK1; -H #HK1
-elim (le_or_ge d (|K|)) #Hd
-[ elim (ldrop_ltpss_sn_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tpss_lift_be … HT1 … Hd HL2K HTU1 … HTU) // -HT1 -HTU1 #HU1
-| elim (ldrop_ltpss_sn_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
- lapply (tpss_lift_le … HT1 … Hd HL2K HTU1 … HTU) -HT1 -HTU1 #HU1
-]
-lapply (ltpss_sn_weak_full … HL12) -HL12 #HL12
-lapply (tpss_weak_full … HU1) -HU1 #HU1
-@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
-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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_sn_tpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_sn_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
- ∀L0. L0 ⊢ ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
-#L1 #T2 @(f2_ind … fw … L1 T2) -L1 -T2 #n #IH #L1 *
-[ #I #Hn #W2 #d #e #H #L0 #HL01 destruct
- elim (tpss_inv_atom1 … H) -H // *
- #K1 #V1 #V2 #i #Hdi #Hide #HLK1 #HV12 #HVW2 #H destruct
- lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
- elim (ltpss_sn_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
- elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
- lapply (IH … HV12 … HK01) -IH -HV12 -HK01 [ normalize /2 width=1/ ] #HV12
- lapply (tpss_trans_eq … HV01 HV12) -V1 /2 width=6/
-| * [ #a ] #I #V2 #T2 #Hn #X #d #e #H #L0 #HL0 destruct
- [ elim (tpss_inv_bind1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (tpss_lsubr_trans … HTU2 (L1. ⓑ{I} V2) ?) -HTU2 /2 width=1/ #HTU2
- lapply (IH … HVW2 … HL0) -HVW2 [ /2 width=2/ ] #HVW2
- lapply (IH … HTU2 (L0. ⓑ{I} V2) ?) -IH -HTU2 // /2 width=2/ -HL0 #HTU2
- lapply (tpss_lsubr_trans … HTU2 (L0. ⓑ{I} W2) ?) -HTU2 /2 width=1/
- | elim (tpss_inv_flat1 … H) -H #W2 #U2 #HVW2 #HTU2 #H destruct
- lapply (IH … HVW2 … HL0) -HVW2 //
- lapply (IH … HTU2 … HL0) -IH -HTU2 // -HL0 /2 width=1/
-]
-qed.
-
-lemma ltpss_sn_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ⊢ ▶* [d, e] L1 →
- L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
-/3 width=3/ qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpss_sn_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
- ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L #d #e #H elim H -L1 -L -d -e //
-[ #L1 #L #I #V1 #V #e #HL1 #HV1 #IHL1 #X #H
- elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
- lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
-| #L1 #L #I #V1 #V #d #e #HL1 #HV1 #IHL1 #X #H
- elim (ltpss_sn_inv_tpss11 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
- lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
-]
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶ [d, e] T →
- ∃∃L2,T2. L @@ L1 ⊢ ▶* [d + |L1|, e] L @@ L2 &
- L @@ L2 ⊢ T1 ▶ [d + |L1|, e] T2 &
- T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #d #e #HT1
- @ex3_2_intro [3: // |5: // |4: normalize /2 width=1/ |1,2: skip ] (**) (* /2 width=4/ does not work *)
-| #I #L1 #V1 #IH #L #T1 #T #d #e >shift_append_assoc #H
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- <append_assoc >append_length <associative_plus
- lapply (ltpss_sn_trans_eq (L.ⓑ{I}V1@@L1) … HL12) -HL12 /3 width=1/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* 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/unfold/ltpss_sn_ldrop.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial substitution on terms **********************)
-
-lemma ltpss_sn_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
-
-lemma ltpss_sn_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶ [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltpss_sn_tpss1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-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/unfold/ltpss_sn_tps.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* Properties concerning partial unfold on terms ****************************)
-
-lemma ltpss_sn_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_sn_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-lemma ltpss_sn_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L0 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_sn_ldrop_conf_le … HL01 … HLK0 ?) /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // >minus_plus <plus_minus_m_m // /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_sn_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 // /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H
- lapply (tpss_lsubr_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2
- elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
-]
-qed.
-
-lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 ▶* [d2, e2] U2.
-#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU
-lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
-qed.
-
-lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
- ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
- ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
- L1 ⊢ T ▶* [d1, e1] U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltpss_sn_ldrop_trans_be … HL10 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
- elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
- lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H
- lapply (tpss_lsubr_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2
- elim (IHTU2 … HL10) -IHTU2 -HL10 /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/unfold/ltpss_sn.ma".
-
-(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
-
-definition thin: nat → nat → relation lenv ≝
- λd,e,L1,L2. ∃∃L. L1 ⊢ ▶* [d, e] L & ⇩[d, e] L ≡ L2.
-
-interpretation "basic thinning (local environment)"
- 'TSubst L1 d e L2 = (thin d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ▼*[d, e] L1 ≡ L2.
-/2 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma thin_inv_thin1: ∀I,K1,V1,L2,e. ▼*[0, e] K1.ⓑ{I} V1 ≡ L2 → 0 < e →
- ▼*[0, e - 1] K1 ≡ L2.
-#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e
-elim (ltpss_sn_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct
-lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /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/unfold/delift_tpss.ma".
-include "basic_2/unfold/delift_ltpss.ma".
-include "basic_2/unfold/thin.ma".
-
-(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
-
-(* Inversion lemmas on inverse basic term relocation ************************)
-
-lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
- K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
- L2 = K2. ⓑ{I} V2.
-#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
-elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
-elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
-qed-.
-
-(* Properties on inverse basic term relocation ******************************)
-
-lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
- ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
-#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
-elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
-lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
-lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
-qed.
-
-lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=3/ qed.
-
-lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K →
- d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
-
-lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
-lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
-elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
-elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
-lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
-lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
-qed.
-
-lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
- ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
- ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
- L ⊢ ▼*[dd, ee] U2 ≡ T2.
-/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/unfold/ltpss_sn_ldrop.ma".
-include "basic_2/unfold/thin.ma".
-
-(* BASIC LOCAL ENVIRONMENT THINNING *****************************************)
-
-(* Properties on local environment slicing **********************************)
-
-lemma thin_ldrop_conf_ge: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ⇩[0, e2 - e1] L1 ≡ L2.
-#L0 #L1 #d1 #e1 * /3 width=8 by ltpss_sn_ldrop_conf_ge, ldrop_conf_ge/
-qed.
-
-lemma thin_ldrop_conf_be: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. ▼*[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
-#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #Hd1e2 #He2de1
-elim (ltpss_sn_ldrop_conf_be … HL0 … HL02 ? ?) -L0 // #L0 #HL20 #HL0
-elim (ldrop_conf_be … HL1 … HL0 ? ?) -L // -Hd1e2 -He2de1 /3 width=3/
-qed.
-
-lemma thin_ldrop_conf_le: ∀L0,L1,d1,e1. ▼*[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ▼*[d1 - e2, e1] L2 ≡ L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 * #L #HL0 #HL1 #L2 #e2 #HL02 #He2d1
-elim (ltpss_sn_ldrop_conf_le … HL0 … HL02 ?) -L0 // #L0 #HL20 #HL0
-elim (ldrop_conf_le … HL1 … HL0 ?) -L // -He2d1 /3 width=3/
-qed.
-
-lemma thin_ldrop_trans_ge: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
- d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #Hd1e2
-lapply (ldrop_trans_ge … HL0 … HL02 ?) -L0 // #HL2
-lapply (ltpss_sn_ldrop_trans_ge … HL1 … HL2 ?) -L // /2 width=1/
-qed.
-
-lemma thin_ldrop_trans_le: ∀L1,L0,d1,e1. ▼*[d1, e1] L1 ≡ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ▼*[d1 - e2, e1] L ≡ L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 * #L #HL1 #HL0 #L2 #e2 #HL02 #He2d1
-elim (ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL0 #HL02
-elim (ltpss_sn_ldrop_trans_le … HL1 … HL0 He2d1) -L -He2d1 /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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $L $T1 $d $e $T2 }.
-
-include "basic_2/substitution/tps.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-definition tpss: nat → nat → lenv → relation term ≝
- λd,e,L. TC … (tps d e L).
-
-interpretation "partial unfold (term)"
- 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
- ∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
-#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
- ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
-#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=1/ qed.
-
-lemma tpss_strap1: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-lemma tpss_strap2: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-lemma tpss_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
-/3 width=3/ qed.
-
-lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
-/2 width=1/ qed.
-
-lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
- L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
-#L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
- [ /3 width=5/
- | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
- ]
-| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
- lapply (tpss_lsubr_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
- lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
-]
-qed.
-
-lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
- L ⊢ V1 ▶* [d, e] V2 → L ⊢ T1 ▶* [d, e] T2 →
- L ⊢ ⓕ{I} V1. T1 ▶* [d, e] ⓕ{I} V2. T2.
-#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #HT12 elim HT12 -T2
- [ /3 width=1/
- | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
- ]
-| #V #V2 #_ #HV12 #IHV #HT12
- lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
-]
-qed.
-
-lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 ▶* [d1, e1] T2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L ⊢ T1 ▶* [d2, e2] T2.
-#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
-]
-qed.
-
-lemma tpss_weak_top: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [d, |L| - d] T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak_top … HT12) -HT12 /2 width=3/
-]
-qed.
-
-lemma tpss_weak_full: ∀L,T1,T2,d,e.
- L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶* [0, |L|] T2.
-#L #T1 #T2 #d #e #HT12
-lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
-lapply (tpss_weak_top … HT12) //
-qed.
-
-lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
- ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
-#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k ▶* [d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_sort1 … HT2) -HT2 //
-]
-qed-.
-
-(* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p ▶* [d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_gref1 … HT2) -HT2 //
-]
-qed-.
-
-lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
- L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
- U2 = ⓑ{a,I} V2. T2.
-#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-]
-qed-.
-
-lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* [d, e] U2 →
- ∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 & L ⊢ T1 ▶* [d, e] T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
-]
-qed-.
-
-lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 0] T2 → T1 = T2.
-#L #T1 #T2 #d #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-lapply (tps_fwd_tw … HT2) -HT2 #HT2
-@(transitive_le … IHT1) //
-qed-.
-
-lemma tpss_fwd_shift1: ∀L,L1,T1,T,d,e. L ⊢ L1 @@ T1 ▶*[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L #L1 #T1 #T #d #e #H @(tpss_ind … H) -T
-[ /2 width=4/
-| #T #X #_ #H0 * #L0 #T0 #HL10 #H destruct
- elim (tps_fwd_shift1 … H0) -H0 #L2 #T2 #HL02 #H destruct /2 width=4/
-]
-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 "hvbox( L ⊢ break term 46 T1 break ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarAlt $L $T1 $d $e $T2 }.
-
-include "basic_2/unfold/tpss_lift.ma".
-
-(* PARALLEL UNFOLD ON TERMS *************************************************)
-
-(* alternative definition of tpss *)
-inductive tpssa: nat → nat → lenv → relation term ≝
-| tpssa_atom : ∀L,I,d,e. tpssa d e L (⓪{I}) (⓪{I})
-| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
- ⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
-| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
- tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
- tpssa d e L V1 V2 → tpssa d e L T1 T2 →
- tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "parallel unfold (term) alternative"
- 'PSubstStarAlt L T1 d e T2 = (tpssa d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpssa_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
- ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
- ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
-#L #T1 #T #d #e #H elim H -L -T1 -T -d -e
-[ #L #I #d #e #X #H
- elim (tps_inv_atom1 … H) -H // * /2 width=6/
-| #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
- lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
- lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
- elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
-| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tps_lsubr_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
- lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
- lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
- lapply (tpssa_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
-| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
-#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
-qed-.
-
-lemma tpss_ind_alt: ∀R:nat→nat→lenv→relation term.
- (∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
- (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
- ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
- ) →
- (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
- R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
- ) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
- R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) →
- ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
-#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
-// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/
-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/tps_lift.ma".
-include "basic_2/unfold/tpss.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma tpss_subst: ∀L,K,V,U1,i,d,e.
- d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 →
- ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2.
-#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1
-[ /3 width=4/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU … HU0) -IHU #H
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
- lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
-]
-qed.
-
-(* Advanced inverion lemmas *************************************************)
-
-lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V1,V2,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-#L #T2 #I #d #e #H @(tpss_ind … H) -T2
-[ /2 width=1/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
- | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (ldrop_fwd_ldrop2 … HLK) #H
- elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
- @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
- ]
-]
-qed-.
-
-lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶* [d, e] T2 →
- T2 = #i ∨
- ∃∃K,V1,V2. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV1 &
- K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #d #e #H
-elim (tpss_inv_atom1 … H) -H /2 width=1/
-* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma tpss_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e + 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶* [d + 1, e] T2.
-#L #T1 #T2 #d #e #H #K #V #HLK @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT
-lapply (tps_inv_S2 … HT2 … HLK) -HT2 -HLK /2 width=3/
-qed-.
-
-lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 ▶* [dt, et] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_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 (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
- ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_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 (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
- ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 ▶* [dt + e, et] U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_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 (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d + e ≤ dt →
- ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
- L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU destruct
-<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
-qed.
-
-lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /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/tps_tps.ma".
-include "basic_2/unfold/tpss_lift.ma".
-
-(* PARTIAL UNFOLD ON TERMS **************************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tpss_inv_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2.
-#L #T1 #T2 #d #H @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1
-lapply (tps_trans_ge … IHT1 … HT2 ?) //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
-/3 width=3/ qed.
-
-lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
-/3 width=3/ qed.
-
-lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2.
-#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2
-[ /2 width=3/
-| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
- elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ]
- /3 width=7 by ex2_intro, step/ (**) (* just /3 width=7/ is too slow *)
-]
-qed.
-
-lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
- ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 ▶* [d2, e2] T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 ▶* [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
-/3 width=3/ qed.
-
-theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶* [d, e] T2 →
- L ⊢ T1 ▶* [d, e] T2.
-/2 width=3/ qed.
-
-theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
- ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
-/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/static/ssta_lpss.ma".
+include "basic_2/unwind/sstas.ma".
+
+(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma sstas_tpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ▶* U2.
+#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
+#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12
+elim (ssta_tpss_lpss_conf … HTU0 … HT0 … HL12) -HTU0 -HT0 #U #HTU #HU0
+elim (IHU01 … HU0 … HL12) -IHU01 -U0 -HL12 /3 width=4/
+qed-.
+
+lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
+ ∀T2. L ⊢ T1 ▶* T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* U2.
+/2 width=3 by sstas_tpss_lpss_conf/ qed-.
+
+lemma sstas_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
+ ∀L2. L1 ⊢ ▶* L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ▶* U2.
+/2 width=3 by sstas_tpss_lpss_conf/ 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/static/ssta_ltpss_dx.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Properties about dx parallel unfold **************************************)
-
-lemma sstas_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
-#T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
-elim (ssta_ltpss_dx_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
-elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
-qed.
-
-lemma sstas_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=7 by step, sstas_ltpss_dx_tpss_conf/ qed. (**) (* auto fails without trace *)
-
-lemma sstas_ltpss_dx_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 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/unfold/ltpss_sn_alt.ma".
-include "basic_2/unwind/sstas_ltpss_dx.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ************************)
-
-(* Properties about sn parallel unfold *****************************************)
-
-lemma sstas_ltpss_sn_conf: ∀h,g,L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∃∃U2. L1 ⊢ U1 ▶* [d, e] U2 & ⦃h, L2⦄ ⊢ T •*[g] U2.
-#h #g #L1 #L2 #d #e #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L2 /2 width=3/
-#L #L2 #HL1 #HL2 #IHL1 #T #U1 #H1
-lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
-lapply (ltpss_sn_dx_trans_eq … HL1 … HL2) -HL1 #HL12
-elim (IHL1 … H1) -IHL1 -H1 #U #HU1 #HTU
-elim (sstas_ltpss_dx_conf … HTU … HL2) -HTU -HL2 #U2 #H2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma sstas_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L1 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U1 #HTU1 #L2 #d #e #HL12 #T2 #HT12
-elim (sstas_ltpss_sn_conf … HL12 … HTU1) -HTU1 #U #HU1 #HT1U
-elim (sstas_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
-lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
-lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
-qed.
-
-lemma sstas_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
- L1 ⊢ U1 ▶* [d, e] U2.
-/3 width=3/ qed.
Extended context-sensitive strong normalization
for simply typed terms.
</news>
+ <news date="2013 April 16.">
+ Reaxiomatized substitution and reduction
+ commute with respect to subclosure
+ (anniversary milestone).
+ </news>
<news date="2013 March 16.">
Mutual recursive preservation of stratified native validity
for hyper computation on closures.
[ "cnf ( ? ⊢ 𝐍⦃?⦄ )" "cnf_lift" + "cnf_cif" * ]
}
]
+ [ { "revised context-sensitive reduction" * } {
+ [ "lpr ( ? ⊢ ➡ ? )" "lpr_ldrop" + "lpr_cpr" + "lpr_lpr" * ]
+ [ "cpr ( ? ⊢ ? ➡ ? )" "cpr_lift" * ]
+ }
+ ]
[ { "context-sensitive reduction" * } {
[ "cpr ( ? ⊢ ? ➡ ? )" "cpr_lift" + "cpr_tpss" + "cpr_ltpss_dx" + "cpr_ltpss_sn" + "cpr_delift" + "cpr_aaa" + "cpr_ltpr" + "cpr_cpr" * ]
}
[ "lpqs ( ? ⊢ ➤* ? )" "lpqs_ldrop" + "lpqs_cpqs" + "lpqs_lpqs" * ]
[ "cpqs ( ? ⊢ ? ➤* ? )" "cpqs_lift" * ]
}
- ]
+ ]
}
]
class "green"
[ { "unwind" * } {
[ { "iterated stratified static type assignment" * } {
- [ "sstas ( ⦃?,?⦄ ⊢ ? •*[?] ? )" "sstas_lift" + "sstas_ltpss_dx" + "sstas_ltpss_sn" + "sstas_aaa" + "sstas_sstas" * ]
+ [ "sstas ( ⦃?,?⦄ ⊢ ? •*[?] ? )" "sstas_lift" + "sstas_lpss" + "sstas_aaa" + "sstas_sstas" * ]
}
]
}
class "grass"
[ { "static typing" * } {
[ { "stratified static type assignment" * } {
- [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ? )" "ssta_lift" + "ssta_ltpss_dx" + "ssta_ltpss_sn" + "ssta_aaa" + "ssta_ssta" * ]
+ [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ? )" "ssta_lift" + "ssta_lpss" + "ssta_aaa" + "ssta_ssta" * ]
}
]
[ { "local env. ref. for atomic arity assignment" * } {
}
]
[ { "atomic arity assignment" * } {
- [ "aaa ( ? ⊢ ? ⁝ ? )" "aaa_lift" + "aaa_lifts" + "aaa_ltpss_dx" + "aaa_ltpss_sn" + "aaa_aaa" * ]
+ [ "aaa ( ? ⊢ ? ⁝ ? )" "aaa_lift" + "aaa_lifts" + "aaa_lpss" + "aaa_aaa" * ]
}
]
[ { "parameters" * } {
]
class "yellow"
[ { "unfold" * } {
- [ { "basic local env. thinning" * } {
+(*
+ [ { "basic local env. thinning" * } {
[ "thin ( ? ▼*[?,?] ≡ ? )" "thin_ldrop" + "thin_delift" * ]
}
]
[ "delift ( ? ⊢ ? ▼*[?,?] ≡ ? )" "delift_alt ( ? ⊢ ? ▼▼*[?,?] ≡ ? )" "delift_lift" + "delift_tpss" + "delift_ltpss" + "delift_delift" * ]
}
]
- [ { "revised parallel substitution" * } {
+*)
+ [ { "parallel substitution" * } {
[ "lpss ( ? ⊢ ▶* ? )" "lpss_ldrop" + "lpss_cpss" + "lpss_lpss" * ]
[ "cpss ( ? ⊢ ? ▶* ? )" "cpss_lift" * ]
}
]
- [ { "partial unfold" * } {
- [ "ltpss_sn ( ? ⊢ ▶*[?,?] ? )" "ltpss_sn_alt ( ? ⊢ ▶▶*[?,?] ? )" "ltpss_sn_ldrop" + "ltpss_sn_tps" + "ltpss_sn_tpss" + "ltpss_sn_ltpss_sn" * ]
- [ "ltpss_dx ( ? ▶*[?,?] ? )" "ltpss_dx_ldrop" + "ltpss_dx_tps" + "ltpss_dx_tpss" + "ltpss_dx_ltpss_dx" * ]
- [ "tpss ( ? ⊢ ? ▶*[?,?] ? )" "tpss_alt ( ? ⊢ ? ▶▶*[?,?] ? )" "tpss_lift" "tpss_tpss" * ]
- }
- ]
[ { "iterated structural successor for closures" * } {
+ [ "fsups ( ⦃?,?⦄ ⊃* ⦃?,?⦄ )" "fsups_fsups" * ]
[ "fsupp ( ⦃?,?⦄ ⊃+ ⦃?,?⦄ )" "fsupp_fsupp" * ]
}
]
]
class "orange"
[ { "substitution" * } {
- [ { "parallel substitution" * } {
- [ "tps ( ? ⊢ ? ▶[?,?] ? )" "tps_lift" + "tps_tps" * ]
- }
- ]
[ { "structural successor for closures" * } {
[ "fsup ( ⦃?,?⦄ ⊃ ⦃?,?⦄ )" * ]
}
["#"; "♯"; "⌘"; ];
["-"; "÷"; "⊢"; "⊩"; ];
["="; "≃"; "≈"; "≝"; "≡"; "≅"; "≐"; "≑"; ];
- ["→"; "↦"; "⇝"; "⇾"; "⤍"; "⤏"; "⤳"; ] ;
+ ["â\86\92"; "â\86¦"; "â\87\9d"; "â¤\9e"; "â\87¾"; "â¤\8d"; "â¤\8f"; "⤳"; ] ;
["⇒"; "⤇"; "➾"; "⇨"; "➡"; "➤"; "➸"; "⇉"; "⥰"; ] ;
["^"; "↑"; ] ;
["⇑"; "⇧"; "⬆"; ] ;
["□"; "◽"; "▪"; "◾"; ];
["◊"; "♢"; "⧫"; "♦"; "⟐"; "⟠"; ] ;
[">"; "⭃"; "⧁"; "〉"; "»"; "❭"; "❯"; "❱"; "▸"; "►"; "▶"; ] ;
- ["≥"; "≽"; "⥸"; ];
+ ["â\89¥"; "âª\80"; "â\89½"; "⥸"; ];
["a"; "α"; "𝕒"; "𝐚"; "𝛂"; "ⓐ"; ] ;
["A"; "ℵ"; "𝔸"; "𝐀"; "Ⓐ"; ] ;
["b"; "β"; "ß"; "𝕓"; "𝐛"; "𝛃"; "ⓑ"; ] ;
projects / helm.git / commitdiff