+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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/rt_computation/cpmuwe_cpmuwe.ma".
-include "basic_2/rt_conversion/cpce_drops.ma".
-include "basic_2/dynamic/cnv_cpmuwe.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-(* Properties with context-sensitive parallel eta-conversion for terms ******)
-
-axiom cpce_total_cnv (h) (a) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃T2. ⦃G,L⦄ ⊢ T1 ⬌η[h] T2.
-(*
-#h #a #G #L #T1 #HT1
-lapply (cnv_fwd_csx … HT1) #H
-generalize in match HT1; -HT1
-@(csx_ind_fpbg … H) -G -L -T1
-#G #L * *
-[ #s #_ #_ /2 width=2 by cpce_sort, ex_intro/
-| #i #H1i #IH #H2i
- elim (drops_ldec_dec L i) [ * #K #W #HLK | -H1i -IH #HnX ]
- [ lapply (cnv_inv_lref_pair … H2i … HLK) -H2i #H2W
- lapply (csx_inv_lref_pair_drops … HLK H1i) -H1i #H1W
- elim (cpmuwe_total_csx … H1W) -H1W #X #n #HWX
- elim (abst_dec X) [ * | -IH ]
- [ #p #V1 #U #H destruct
- lapply (cpmuwe_fwd_cpms … HWX) -HWX #HWX
- elim (IH G K V1) -IH
- [ #V2 #HV12
- elim (lifts_total V2 (𝐔❴↑i❵)) #W2 #HVW2
- /3 width=12 by cpce_eta_drops, ex_intro/
- | /3 width=6 by cnv_cpms_trans, cnv_fwd_pair_sn/
- | /4 width=6 by fqup_cpms_fwd_fpbg, fpbg_fqu_trans, fqup_lref/
- ]
- | #HnX
- @(ex_intro … (#i))
- @cpce_zero_drops #n0 #p #K0 #W0 #V0 #U0 #HLK0 #HWU0
- lapply (drops_mono … HLK0 … HLK) -i -L #H destruct
- lapply (cpmuwe_abst … HWU0) -HWU0 #HWU0
- elim (cnv_cpmuwe_mono … H2W … HWU0 … HWX) #_ #H -a -n -n0 -W
- elim (tweq_inv_abst_sn … H) -V0 -U0 #V0 #U0 #H destruct
- /2 width=4 by/
- ]
- | /5 width=3 by cpce_zero_drops, ex1_2_intro, ex_intro/
- ]
-| #l #_ #_ /2 width=2 by cpce_gref, ex_intro/
-| #p #I #V1 #T1 #_ #IH #H
- elim (cnv_inv_bind … H) -H #HV1 #HT1
- elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
- elim (IH … HT1) [| /4 width=1 by fpb_fpbg, fpb_fqu, fqu_bind_dx/ ] #T2 #HT12
- /3 width=2 by cpce_bind, ex_intro/
-| #I #V1 #T1 #_ #IH #H
- elim (cnv_fwd_flat … H) -H #HV1 #HT1
- elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
- elim (IH … HT1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_flat_dx/ ] #T2 #HT12
- /3 width=2 by cpce_flat, ex_intro/
-]
-qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/lpr_drops.ma".
-include "basic_2/rt_computation/cpms_lpr.ma".
-include "basic_2/rt_computation/fpbg_fqup.ma".
-include "basic_2/rt_conversion/cpce_drops.ma".
-include "basic_2/rt_conversion/lpce_drops.ma".
-include "basic_2/dynamic/cnv_drops.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-definition IH (h) (a): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
- ∀n,T1. ⦃G,L0⦄ ⊢ T0 ➡[n,h] T1 → ∀T2. ⦃G,L0⦄ ⊢ T0 ⬌η[h] T2 →
- ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ⬌η[h] L2 →
- ∃∃T. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T & ⦃G,L2⦄ ⊢ T2 ➡[n,h] T.
-
-(* Properties with eta-conversion for full local environments ***************)
-
-lemma pippo_aux (h) (a) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH h a G L T) →
- IH h a G0 L0 T0.
-#h #a #G0 #L0 * *
-[ #s #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
- elim (cpm_inv_sort1 … HX1) -HX1 #H #Hn destruct
- lapply (cpce_inv_sort_sn … HX2) -HX2 #H destruct
- /3 width=3 by cpce_sort, cpm_sort, ex2_intro/
-| #i #IH #Hi #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
- elim (cnv_inv_lref_drops … Hi) -Hi #I #K0 #W0 #HLK0 #HW0
- elim (lpr_drops_conf … HLK0 … HL01) [| // ] #Y1 #H1 #HLK1
- elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #HW01 #H destruct
- elim (lpce_drops_conf … HLK0 … HL02) [| // ] #Y2 #H2 #HLK2
- elim (lpce_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #HW02 #H destruct
- elim (cpm_inv_lref1_drops … HX1) -HX1 *
- [ #H1 #H2 destruct
- elim (cpce_inv_lref_sn_drops_pair … HX2 … HLK0) -HX2 *
- [ #H1 #H2 destruct -L0 -K0 -W0
- /3 width=3 by cpce_ldef_drops, ex2_intro/
- | #H1 #HW #H2 destruct -L0 -W2 -HW0 -HK02
- @(ex2_intro … (#i)) [| // ]
- @(cpce_ldec_drops … HLK1) -HLK1 #n #p #V0 #U0 #HWU0
- /4 width=10 by lpr_cpms_trans, cpms_step_sn/
- | #n #p #W01 #W02 #V0 #V01 #V02 #U0 #H1 #HWU0 #HW001 #HW012 #HV001 #HV012 #H2 destruct
- ]
- | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
- #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 -HL01 -HL02
- lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
- lapply (cpce_inv_lref_sn_drops_ldef … HX2 … HLK0) -HX2 #H destruct
- elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
- [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
- elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
- /3 width=9 by cpce_lifts_bi, cpm_delta_drops, ex2_intro/
- | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
- #m #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 #H destruct -HL01 -HL02
- lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
- elim (cpce_inv_lref_sn_drops_ldec … HX2 … HLK0) -HX2 *
- [ #_ #H destruct
- elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
- [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
- elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
- /3 width=9 by cpce_lifts_bi, cpm_ell_drops, ex2_intro/
- | lapply (drops_isuni_fwd_drop2 … HLK2) [ // ] -W2 #HLK2
- #n #p #W01 #W02 #V0 #V01 #V02 #U0 #_ #HW001 #HW012 #_ #_ #H destruct -V0 -V01 -U0
- elim (IH … HW0 … HW01 … HW001 … HK01 … HK02)
- [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
- elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
- /4 width=11 by cpce_lifts_bi, cpm_lifts_bi, cpm_ee, ex2_intro/
- ]
- ]
-| #l #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
- elim (cpm_inv_gref1 … HX1) -HX1 #H1 #H2 destruct
- lapply (cpce_inv_gref_sn … HX2) -HX2 #H destruct
- /3 width=3 by cpce_gref, cpr_refl, ex2_intro/
-
-(*
-lemma cpce_inv_eta_drops (h) (n) (G) (L) (i):
- ∀X. ⦃G,L⦄ ⊢ #i ⬌η[h] X →
- ∀K,W. ⇩*[i] L ≘ K.ⓛW →
- ∀p,V1,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
- ∀V2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 →
- ∀W2. ⇧*[↑i] V2 ≘ W2 → X = +ⓛW2.ⓐ#0.#↑i.
-
-theorem cpce_mono_cnv (h) (a) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
- ∀T1. ⦃G,L⦄ ⊢ T ⬌η[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬌η[h] T2 → T1 = T2.
-#h #a #G #L #T #HT
-*)
qed-.
(* Basic_2A1: uses: lsubsv_inv_atom2 *)
-lemma lsubv_inv_atom2 (h) (a) (G):
+lemma lsubv_inv_atom_dx (h) (a) (G):
∀L1. G ⊢ L1 ⫃![h,a] ⋆ → L1 = ⋆.
/2 width=6 by lsubv_inv_atom_dx_aux/ qed-.
--- /dev/null
+(*
+lemma cpce_inv_eta_drops (h) (n) (G) (L) (i):
+ ∀X. ⦃G,L⦄ ⊢ #i ⬌η[h] X →
+ ∀K,W. ⇩*[i] L ≘ K.ⓛW →
+ ∀p,V1,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
+ ∀V2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 →
+ ∀W2. ⇧*[↑i] V2 ≘ W2 → X = +ⓛW2.ⓐ#0.#↑i.
+
+theorem cpce_mono_cnv (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀T1. ⦃G,L⦄ ⊢ T ⬌η[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬌η[h] T2 → T1 = T2.
+#h #a #G #L #T #HT
+*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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/rt_conversion/cpce.ma".
-include "basic_2/dynamic/cnv_preserve.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-theorem cnv_cpce_mono (a) (h) (G) (L) (T):
- ∀T1. ⦃G,L⦄ ⊢ T ⬌η[h] T1 → ⦃G,L⦄ ⊢ T ![a,h] →
- ∀T2. ⦃G,L⦄ ⊢ T ⬌η[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
-#h #G #L #T @(fqup_wf_ind (Ⓣ) … G L T) -G -L -T
-#G0 #L0 #T0 #IH #T1
-@(insert_eq_0 … G0) #G
-@(insert_eq_0 … L0) #L
-@(insert_eq_0 … T0) #T
-* -G -L -T
-[ #G #L1 #s #_ #_ #_ #_ #L2 #_ //
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpmuwe_cpmuwe.ma".
+include "basic_2/rt_conversion/cpce_drops.ma".
+include "basic_2/dynamic/cnv_cpmuwe.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with context-sensitive parallel eta-conversion for terms ******)
+
+axiom cpce_total_cnv (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃T2. ⦃G,L⦄ ⊢ T1 ⬌η[h] T2.
+(*
+#h #a #G #L #T1 #HT1
+lapply (cnv_fwd_csx … HT1) #H
+generalize in match HT1; -HT1
+@(csx_ind_fpbg … H) -G -L -T1
+#G #L * *
+[ #s #_ #_ /2 width=2 by cpce_sort, ex_intro/
+| #i #H1i #IH #H2i
+ elim (drops_ldec_dec L i) [ * #K #W #HLK | -H1i -IH #HnX ]
+ [ lapply (cnv_inv_lref_pair … H2i … HLK) -H2i #H2W
+ lapply (csx_inv_lref_pair_drops … HLK H1i) -H1i #H1W
+ elim (cpmuwe_total_csx … H1W) -H1W #X #n #HWX
+ elim (abst_dec X) [ * | -IH ]
+ [ #p #V1 #U #H destruct
+ lapply (cpmuwe_fwd_cpms … HWX) -HWX #HWX
+ elim (IH G K V1) -IH
+ [ #V2 #HV12
+ elim (lifts_total V2 (𝐔❴↑i❵)) #W2 #HVW2
+ /3 width=12 by cpce_eta_drops, ex_intro/
+ | /3 width=6 by cnv_cpms_trans, cnv_fwd_pair_sn/
+ | /4 width=6 by fqup_cpms_fwd_fpbg, fpbg_fqu_trans, fqup_lref/
+ ]
+ | #HnX
+ @(ex_intro … (#i))
+ @cpce_zero_drops #n0 #p #K0 #W0 #V0 #U0 #HLK0 #HWU0
+ lapply (drops_mono … HLK0 … HLK) -i -L #H destruct
+ lapply (cpmuwe_abst … HWU0) -HWU0 #HWU0
+ elim (cnv_cpmuwe_mono … H2W … HWU0 … HWX) #_ #H -a -n -n0 -W
+ elim (tweq_inv_abst_sn … H) -V0 -U0 #V0 #U0 #H destruct
+ /2 width=4 by/
+ ]
+ | /5 width=3 by cpce_zero_drops, ex1_2_intro, ex_intro/
+ ]
+| #l #_ #_ /2 width=2 by cpce_gref, ex_intro/
+| #p #I #V1 #T1 #_ #IH #H
+ elim (cnv_inv_bind … H) -H #HV1 #HT1
+ elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
+ elim (IH … HT1) [| /4 width=1 by fpb_fpbg, fpb_fqu, fqu_bind_dx/ ] #T2 #HT12
+ /3 width=2 by cpce_bind, ex_intro/
+| #I #V1 #T1 #_ #IH #H
+ elim (cnv_fwd_flat … H) -H #HV1 #HT1
+ elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
+ elim (IH … HT1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_flat_dx/ ] #T2 #HT12
+ /3 width=2 by cpce_flat, ex_intro/
+]
+qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_conversion/cpce.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+theorem cnv_cpce_mono (a) (h) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ⬌η[h] T1 → ⦃G,L⦄ ⊢ T ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌η[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
+#h #G #L #T @(fqup_wf_ind (Ⓣ) … G L T) -G -L -T
+#G0 #L0 #T0 #IH #T1
+@(insert_eq_0 … G0) #G
+@(insert_eq_0 … L0) #L
+@(insert_eq_0 … T0) #T
+* -G -L -T
+[ #G #L1 #s #_ #_ #_ #_ #L2 #_ //
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/s_computation/fqup_weight.ma".
-include "basic_2/rt_conversion/lpce.ma".
-include "basic_2/dynamic/cnv_drops.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-theorem cnv_cpce_trans_lpce (h) (G):
- ∀L1,T1,T2. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T2 → ⦃G,L1⦄ ⊢ T1 !*[h] →
- ∀L2. ⦃G,L1⦄ ⊢ ⬌η[h] L2 → ⦃G,L2⦄ ⊢ T2 !*[h].
-#h #G #L1 #T1 @(fqup_wf_ind (Ⓣ) … G L1 T1) -G -L1 -T1
-#G0 #L0 #T0 #IH #T2
-@(insert_eq_0 … G0) #G
-@(insert_eq_0 … L0) #L1
-@(insert_eq_0 … T0) #T1
-* -G -L1 -T1
-[ #G #L1 #s #_ #_ #_ #_ #L2 #_ //
-| #G #K1 #V1 #HT #HL #HG #H0 #Y2 #HY2 destruct
- elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
- elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=6 by cnv_zero, fqu_fqup, fqu_lref_O/
-| #n #G #K1 #V1 #s #_ #HT #HL #HG #H0 #Y2 #HY2 destruct
- elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
- elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=6 by cnv_zero, fqu_fqup, fqu_lref_O/
-| #n #p #G #K1 #V1 #W1 #W2 #T1 #HVT1 #HW12 #HT #HL #HG #H0 #Y2 #HY2 destruct
- elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
- elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
-| #I #G #K1 #T1 #U1 #i #H0 #HTU1 #HT #HL #HG #H0 #Y2 #HY2 destruct
- elim (cnv_inv_lref … H0) -H0 #Z1 #Y1 #Hi #H destruct
- elim (lpce_inv_bind_sn … HY2) -HY2 #Z2 #K2 #HK12 #_ #H destruct
- @(cnv_lifts … K2 … (Ⓣ) … HTU1) [| /3 width=1 by drops_refl, drops_drop/ ] -U1
- /3 width=6 by fqu_fqup/
-| #p #I #G #K1 #V1 #V2 #T1 #T2 #HV12 #HT12 #HT #HL #HG #H0 #K2 #HK12 destruct
- elim (cnv_inv_bind … H0) -H0 #HV1 #HT1
- /4 width=8 by lpce_pair, cnv_bind/
-| * #G #L1 #V1 #V2 #T1 #T2 #HV12 #HT12 #HT #HL #HG #H0 #L2 #HK12 destruct
-
\ 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/rt_conversion/cpce_drops.ma".
+include "basic_2/rt_conversion/lpce_drops.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+definition IH_cnv_cpce_cpm_conf (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n,T1. ⦃G,L0⦄ ⊢ T0 ➡[n,h] T1 → ∀T2. ⦃G,L0⦄ ⊢ T0 ⬌η[h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ⬌η[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n,h] T.
+
+definition IH_cnv_cpce_cpms_conf (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n,h] T1 → ∀T2. ⦃G,L0⦄ ⊢ T0 ⬌η[h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ⬌η[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n,h] T.
+
+(* Properties with eta-conversion for full local environments ***************)
+
+fact cnv_cpce_cpms_conf_sub (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpce_cpm_conf h a G L T) →
+ ∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpce_cpms_conf h a G L T.
+#h #a #GX #LX #TX #HX #G #L0 #T0 #HX0 #HT0 #n #X1 #HX1
+@(cpms_ind_dx … HX1) -n -X1
+[ #T2 #HT02 #L1 #HL01 #L2 #HL02
+ /2 width=8 by/
+| #n1 #n2 #T3 #T1 #HT03 #IH #HT31 #T2 #HT02 #L1 #HL01 #L2 #HL02
+ lapply (cnv_cpms_trans … HT0 … HT03) -HT0 #HT3
+ elim (IH … HT02 … L0 … HL02) -IH -HT02 [| // ] #T4 #HT34 #HT24
+ elim (HX … HT3 … HT31 … HT34 … HL01 … HL02)
+ [| /2 width=4 by fpbg_cpms_trans/ ] -GX -LX -L0 -TX -T0 -T3 #T3 #HT13 #HT43
+ /3 width=5 by cpms_trans, ex2_intro/
+]
+qed-.
+
+fact cnv_cpce_cpm_conf_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpce_cpm_conf h a G L T) →
+ IH_cnv_cpce_cpm_conf h a G0 L0 T0.
+#h #a #G0 #L0 * * [|||| * ]
+[ #s #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cpm_inv_sort1 … HX1) -HX1 #H #Hn destruct
+ lapply (cpce_inv_sort_sn … HX2) -HX2 #H destruct
+ /3 width=3 by cpce_sort, ex2_intro/
+| #i #IH #H0 #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cnv_inv_lref_drops … H0) -H0 #I #K0 #W0 #HLK0 #HW0
+ elim (lpr_drops_conf … HLK0 … HL01) [| // ] #Y1 #H1 #HLK1
+ elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #HW01 #H destruct
+ elim (lpce_drops_conf … HLK0 … HL02) [| // ] #Y2 #H2 #HLK2
+ elim (lpce_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #HW02 #H destruct
+ elim (cpm_inv_lref1_drops … HX1) -HX1 *
+ [ #H1 #H2 destruct
+ elim (cpce_inv_lref_sn_drops_pair … HX2 … HLK0) -HX2 *
+ [ #H1 #H2 destruct -L0 -K0 -W0
+ /3 width=3 by cpce_ldef_drops, ex2_intro/
+ | #H1 #HW #H2 destruct -L0 -W2 -HW0 -HK02
+ @(ex2_intro … (#i)) [| // ]
+ @(cpce_ldec_drops … HLK1) -HLK1 #n #p #V0 #U0 #HWU0
+ /4 width=10 by lpr_cpms_trans, cpms_step_sn/
+ | lapply (drops_isuni_fwd_drop2 … HLK2) [ // ] -W2 #HLK2
+ #n #p #W01 #W02 #V0 #V01 #V02 #U0 #H1 #HWU0 #HW001 #HW012 #HV001 #HV012 #H2 destruct
+ lapply (cnv_cpms_trans … HW0 … HWU0) #H
+ elim (cnv_inv_bind … H) -H #HV0 #_
+ elim (cnv_cpms_conf_lpr … HW0 … HWU0 0 W1 … K0 … HK01) [| // ]
+ [| /2 width=1 by cpm_cpms/ ] <minus_O_n <minus_n_O #X #H #HWU1
+ elim (cpms_inv_abst_sn … H) -H #V1 #U1 #HV01 #HU01 #H destruct
+ elim (IH … HW0 … HW01 … HW001 … HK01 … HK02) -HW01 -HW001
+ [| /3 width=2 by fqup_fpbg, fqup_lref/ ] #W11 #HW11 #HW011
+ elim (cnv_cpce_cpms_conf_sub … IH … HV0 … HV01 … HV001 … HK01 … HK02) -HV0 -HV01 -HV001
+ [| /4 width=6 by fqup_cpms_fwd_fpbg, fpbg_fqu_trans, fqup_lref/ ] #V11 #HV11 #HV011
+ elim (cpms_lifts_sn … HW011 … HLK2 … HW012) -W01 #W12 #HW112 #HW012
+ elim (cpms_lifts_sn … HV011 … HLK2 … HV012) -V01 #V12 #HV112 #HV012
+ @(ex2_intro …(ⓝW12.+ⓛV12.ⓐ#O.#(↑i)))
+ [ /2 width=14 by cpce_eta_drops/ | /3 width=1 by cpms_cast, cpms_bind/ ]
+ ]
+ | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
+ #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 -HL01 -HL02
+ lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
+ lapply (cpce_inv_lref_sn_drops_ldef … HX2 … HLK0) -HX2 #H destruct
+ elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
+ [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
+ elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
+ /3 width=9 by cpce_lifts_bi, cpms_delta_drops, ex2_intro/
+ | lapply (drops_isuni_fwd_drop2 … HLK1) [ // ] -W1 #HLK1
+ #m #Y0 #X0 #W1 #HLY0 #HW01 #HWX1 #H destruct -HL01 -HL02
+ lapply (drops_mono … HLY0 … HLK0) -HLY0 #H destruct
+ elim (cpce_inv_lref_sn_drops_ldec … HX2 … HLK0) -HX2 *
+ [ #_ #H destruct
+ elim (IH … HW0 … HW01 … HW02 … HK01 … HK02)
+ [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
+ elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
+ /3 width=9 by cpce_lifts_bi, cpms_ell_drops, ex2_intro/
+ | lapply (drops_isuni_fwd_drop2 … HLK2) [ // ] -W2 #HLK2
+ #n #p #W01 #W02 #V0 #V01 #V02 #U0 #_ #HW001 #HW012 #_ #_ #H destruct -V0 -V01 -U0
+ elim (IH … HW0 … HW01 … HW001 … HK01 … HK02)
+ [| /3 width=2 by fqup_fpbg, fqup_lref/ ] -L0 -K0 #W #HW1 #HW2
+ elim (lifts_total W (𝐔❴↑i❵)) #V #HWV
+ /4 width=11 by cpce_lifts_bi, cpms_lifts_bi, cpms_ee, ex2_intro/
+ ]
+ ]
+| #l #_ #_ #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cpm_inv_gref1 … HX1) -HX1 #H1 #H2 destruct
+ lapply (cpce_inv_gref_sn … HX2) -HX2 #H destruct
+ /3 width=3 by cpce_gref, cpm_cpms, ex2_intro/
+| #p #I #V0 #T0 #IH #H0 #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
+ elim (cpce_inv_bind_sn … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (cpm_inv_bind1 … HX1) -HX1 *
+ [ #V1 #T1 #HV01 #HT01 #H destruct
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0
+ [| /2 width=1 by fqup_fpbg/ ] #V #HV1 #HV2
+ elim (IH … HT0 … HT01 … HT02 … (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by lpce_pair, fqup_fpbg, lpr_pair/ ] -L0 -V0 -T0 #T #HT1 #HT2
+ /3 width=5 by cpce_bind, cpms_bind, ex2_intro/
+ | #X0 #HXT0 #HX01 #H1 #H2 destruct -HV0 -HV02
+ lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HXT0) -HT0
+ [ /3 width=1 by drops_refl, drops_drop/ ] #HX0
+ elim (cpce_inv_lifts_sn … HT02 (Ⓣ) … L0 … HXT0) -HT02
+ [| /3 width=1 by drops_refl, drops_drop/ ] #X2 #HXT2 #HX02
+ elim (IH … HX0 … HX01 … HX02 … HL01 … HL02) -HX0 -HX01 -HX02
+ [| /3 width=1 by fqup_fpbg, fqup_zeta/ ] -V0 -T0 -X0 #X #HX1 #HX2
+ /3 width=5 by cpms_zeta, ex2_intro/
+ ]
+| #V0 #T0 #IH #H0 #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cnv_inv_appl … H0) -H0 #m #p #W0 #U0 #_ #HV0 #HT0 #_ #_ -m -p -W0 -U0
+ elim (cpce_inv_flat_sn … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (cpm_inv_appl1 … HX1) -HX1 *
+ [ #V1 #T1 #HV01 #HT01 #H destruct
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0 -HV01 -HV02
+ [| /2 width=1 by fqup_fpbg/ ] #V #HV1 #HV2
+ elim (IH … HT0 … HT01 … HT02 … HL01 … HL02) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by fqup_fpbg/ ] -L0 -V0 -T0 #T #HT1 #HT2
+ /3 width=5 by cpce_flat, cpms_appl, ex2_intro/
+ | #p #V1 #W0 #W1 #X0 #T1 #HV01 #HW01 #HT01 #H1 #H2 destruct
+ elim (cnv_inv_bind … HT0) -HT0 #HW0 #HT0
+ elim (cpce_inv_bind_sn … HT02) -HT02 #W2 #X2 #HW02 #HT02 #H destruct
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0 -HV01 -HV02
+ [| /2 width=1 by fqup_fpbg/ ] #V #HV1 #HV2
+ elim (IH … HW0 … HW01 … HW02 … HL01 … HL02) -HW0
+ [| /2 width=1 by fqup_fpbg/ ] #W #HW1 #HW2
+ elim (IH … HT0 … HT01 … HT02 … (L1.ⓛW1) … (L2.ⓛW2)) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by lpce_pair, fqup_fpbg, lpr_pair/ ] -L0 -V0 -X0 #T #HT1 #HT2
+ @(ex2_intro … (ⓓ{p}ⓝW.V.T))
+ [ @cpce_bind [ /2 width=1 by cpce_flat/ ]
+ | @(cpms_beta) //
+ ]
+ | #p #V1 #U1 #W0 #W1 #X0 #T1 #HV01 #HVU1 #HW01 #HT01 #H1 #H2 destruct
+ elim (cnv_inv_bind … HT0) -HT0 #HW0 #HT0
+ elim (cpce_inv_bind_sn … HT02) -HT02 #W2 #X2 #HW02 #HT02 #H destruct
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0 -HV01 -HV02
+ [| /2 width=1 by fqup_fpbg/ ] #V #HV1 #HV2
+ elim (IH … HW0 … HW01 … HW02 … HL01 … HL02) -HW0
+ [| /2 width=1 by fqup_fpbg/ ] #W #HW1 #HW2
+ elim (IH … HT0 … HT01 … HT02 … (L1.ⓓW1) … (L2.ⓓW2)) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by lpce_pair, fqup_fpbg, lpr_pair/ ] -L0 -V0 -X0 #T #HT1 #HT2
+ elim (cpce_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -V1
+ [| /3 width=1 by drops_refl, drops_drop/ ] #U #HVU #HU1
+ /4 width=9 by cpms_theta, cpce_bind, cpce_flat, ex2_intro/
+ ]
+| #V0 #T0 #IH #H0 #n #X1 #HX1 #X2 #HX2 #L1 #HL01 #L2 #HL02
+ elim (cnv_inv_cast … H0) -H0 #U0 #HV0 #HT0 #_ #_ -U0
+ elim (cpce_inv_flat_sn … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (cpm_inv_cast1 … HX1) -HX1 [ * || * ]
+ [ #V1 #T1 #HV01 #HT01 #H destruct
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0 -HV01 -HV02
+ [| /2 width=1 by fqup_fpbg/ ] #V #HV1 #HV2
+ elim (IH … HT0 … HT01 … HT02 … HL01 … HL02) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by fqup_fpbg/ ] -L0 -V0 -T0 #T #HT1 #HT2
+ /3 width=5 by cpce_flat, cpms_cast, ex2_intro/
+ | #HT01 -HV0 -HV02
+ elim (IH … HT0 … HT01 … HT02 … HL01 … HL02) -HT0 -HT01 -HT02
+ [|*: /2 width=1 by fqup_fpbg/ ] -L0 -V0 -T0 #T #HT1 #HT2
+ /3 width=3 by cpms_eps, ex2_intro/
+ | #m #HV01 #H destruct -HT0 -HT02
+ elim (IH … HV0 … HV01 … HV02 … HL01 … HL02) -HV0 -HV01 -HV02
+ [| /2 width=1 by fqup_fpbg/ ] -L0 -V0 -T0 #V #HV1 #HV2
+ /3 width=3 by cpms_ee, ex2_intro/
+ ]
+]
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 "static_2/s_computation/fqup_weight.ma".
+include "basic_2/rt_conversion/lpce.ma".
+include "basic_2/dynamic/cnv_drops.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+theorem cnv_cpce_trans_lpce (h) (G):
+ ∀L1,T1,T2. ⦃G,L1⦄ ⊢ T1 ⬌η[h] T2 → ⦃G,L1⦄ ⊢ T1 !*[h] →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬌η[h] L2 → ⦃G,L2⦄ ⊢ T2 !*[h].
+#h #G #L1 #T1 @(fqup_wf_ind (Ⓣ) … G L1 T1) -G -L1 -T1
+#G0 #L0 #T0 #IH #T2
+@(insert_eq_0 … G0) #G
+@(insert_eq_0 … L0) #L1
+@(insert_eq_0 … T0) #T1
+* -G -L1 -T1
+[ #G #L1 #s #_ #_ #_ #_ #L2 #_ //
+| #G #K1 #V1 #HT #HL #HG #H0 #Y2 #HY2 destruct
+ elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
+ elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
+ /4 width=6 by cnv_zero, fqu_fqup, fqu_lref_O/
+| #n #G #K1 #V1 #s #_ #HT #HL #HG #H0 #Y2 #HY2 destruct
+ elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
+ elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
+ /4 width=6 by cnv_zero, fqu_fqup, fqu_lref_O/
+| #n #p #G #K1 #V1 #W1 #W2 #T1 #HVT1 #HW12 #HT #HL #HG #H0 #Y2 #HY2 destruct
+ elim (cnv_inv_zero … H0) -H0 #Z #Y #X #HV1 #H destruct
+ elim (lpce_inv_pair_sn … HY2) -HY2 #K2 #V2 #HK12 #HV12 #H destruct
+| #I #G #K1 #T1 #U1 #i #H0 #HTU1 #HT #HL #HG #H0 #Y2 #HY2 destruct
+ elim (cnv_inv_lref … H0) -H0 #Z1 #Y1 #Hi #H destruct
+ elim (lpce_inv_bind_sn … HY2) -HY2 #Z2 #K2 #HK12 #_ #H destruct
+ @(cnv_lifts … K2 … (Ⓣ) … HTU1) [| /3 width=1 by drops_refl, drops_drop/ ] -U1
+ /3 width=6 by fqu_fqup/
+| #p #I #G #K1 #V1 #V2 #T1 #T2 #HV12 #HT12 #HT #HL #HG #H0 #K2 #HK12 destruct
+ elim (cnv_inv_bind … H0) -H0 #HV1 #HT1
+ /4 width=8 by lpce_pair, cnv_bind/
+| * #G #L1 #V1 #V2 #T1 #T2 #HV12 #HT12 #HT #HL #HG #H0 #L2 #HK12 destruct
+
\ 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/notation/relations/pconveta_5.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
+
+(* avtivate genv *)
+inductive cpce (h): relation4 genv lenv term term ≝
+| cpce_sort: ∀G,L,s. cpce h G L (⋆s) (⋆s)
+| cpce_atom: ∀G,i. cpce h G (⋆) (#i) (#i)
+| cpce_unit: ∀I,G,K. cpce h G (K.ⓤ{I}) (#0) (#0)
+| cpce_ldef: ∀G,K,V. cpce h G (K.ⓓV) (#0) (#0)
+| cpce_ldec: ∀G,K,W. (∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ cpce h G (K.ⓛW) (#0) (#0)
+| cpce_eta : ∀n,p,G,K,W,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U →
+ cpce h G K W W1 → ⇧*[1] W1 ≘ W2 →
+ cpce h G K V V1 → ⇧*[1] V1 ≘ V2 →
+ cpce h G (K.ⓛW) (#0) (ⓝW2.+ⓛV2.ⓐ#0.#1)
+| cpce_lref: ∀I,G,K,T,U,i. cpce h G K (#i) T →
+ ⇧*[1] T ≘ U → cpce h G (K.ⓘ{I}) (#↑i) U
+| cpce_gref: ∀G,L,l. cpce h G L (§l) (§l)
+| cpce_bind: ∀p,I,G,K,V1,V2,T1,T2.
+ cpce h G K V1 V2 → cpce h G (K.ⓑ{I}V1) T1 T2 →
+ cpce h G K (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+| cpce_flat: ∀I,G,L,V1,V2,T1,T2.
+ cpce h G L V1 V2 → cpce h G L T1 T2 →
+ cpce h G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "context-sensitive parallel eta-conversion (term)"
+ 'PConvEta h G L T1 T2 = (cpce h G L T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpce_inv_sort_sn (h) (G) (L) (s):
+ ∀X2. ⦃G,L⦄ ⊢ ⋆s ⬌η[h] X2 → ⋆s = X2.
+#h #G #Y #s0 #X2
+@(insert_eq_0 … (⋆s0)) #X1 * -G -Y -X1 -X2
+[ #G #L #s #_ //
+| #G #i #_ //
+| #I #G #K #_ //
+| #G #K #V #_ //
+| #G #K #W #_ #_ //
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpce_inv_atom_sn (h) (G) (i):
+ ∀X2. ⦃G,⋆⦄ ⊢ #i ⬌η[h] X2 → #i = X2.
+#h #G #i0 #X2
+@(insert_eq_0 … LAtom) #Y
+@(insert_eq_0 … (#i0)) #X1
+* -G -Y -X1 -X2
+[ #G #L #s #_ #_ //
+| #G #i #_ #_ //
+| #I #G #K #_ #_ //
+| #G #K #V #_ #_ //
+| #G #K #W #_ #_ #_ //
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #_ #H destruct
+| #G #L #l #_ #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpce_inv_unit_sn (h) (I) (G) (K):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
+#h #I0 #G #K0 #X2
+@(insert_eq_0 … (K0.ⓤ{I0})) #Y
+@(insert_eq_0 … (#0)) #X1
+* -G -Y -X1 -X2
+[ #G #L #s #_ #_ //
+| #G #i #_ #_ //
+| #I #G #K #_ #_ //
+| #G #K #V #_ #_ //
+| #G #K #W #_ #_ #_ //
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H #_ destruct
+| #G #L #l #_ #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpce_inv_ldef_sn (h) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
+#h #G #K0 #V0 #X2
+@(insert_eq_0 … (K0.ⓓV0)) #Y
+@(insert_eq_0 … (#0)) #X1
+* -G -Y -X1 -X2
+[ #G #L #s #_ #_ //
+| #G #i #_ #_ //
+| #I #G #K #_ #_ //
+| #G #K #V #_ #_ //
+| #G #K #W #_ #_ #_ //
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H #_ destruct
+| #G #L #l #_ #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpce_inv_ldec_sn (h) (G) (K) (W):
+ ∀X2. ⦃G,K.ⓛW⦄ ⊢ #0 ⬌η[h] X2 →
+ ∨∨ ∧∧ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #0 = X2
+ | ∃∃n,p,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
+ & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[1] W1 ≘ W2
+ & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[1] V1 ≘ V2
+ & ⓝW2.+ⓛV2.ⓐ#0.#1 = X2.
+#h #G #K0 #W0 #X2
+@(insert_eq_0 … (K0.ⓛW0)) #Y
+@(insert_eq_0 … (#0)) #X1
+* -G -Y -X1 -X2
+[ #G #L #s #H #_ destruct
+| #G #i #_ #H destruct
+| #I #G #K #_ #H destruct
+| #G #K #V #_ #H destruct
+| #G #K #W #HW #_ #H destruct /4 width=5 by or_introl, conj/
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #HWU #HW1 #HW12 #HV1 #HV12 #_ #H destruct
+ /3 width=14 by or_intror, ex6_8_intro/
+| #I #G #K #T #U #i #_ #_ #H #_ destruct
+| #G #L #l #H #_ destruct
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpce_inv_lref_sn (h) (I) (G) (K) (i):
+ ∀X2. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬌η[h] X2 →
+ ∃∃T2. ⦃G,K⦄ ⊢ #i ⬌η[h] T2 & ⇧*[1] T2 ≘ X2.
+#h #I0 #G #K0 #i0 #X2
+@(insert_eq_0 … (K0.ⓘ{I0})) #Y
+@(insert_eq_0 … (#↑i0)) #X1
+* -G -Y -X1 -X2
+[ #G #L #s #H #_ destruct
+| #G #i #_ #H destruct
+| #I #G #K #H #_ destruct
+| #G #K #V #H #_ destruct
+| #G #K #W #_ #H #_ destruct
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H #_ destruct
+| #I #G #K #T #U #i #Hi #HTU #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #G #L #l #H #_ destruct
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
+]
+qed-.
+
+lemma cpce_inv_gref_sn (h) (G) (L) (l):
+ ∀X2. ⦃G,L⦄ ⊢ §l ⬌η[h] X2 → §l = X2.
+#h #G #Y #l0 #X2
+@(insert_eq_0 … (§l0)) #X1 * -G -Y -X1 -X2
+[ #G #L #s #_ //
+| #G #i #_ //
+| #I #G #K #_ //
+| #G #K #V #_ //
+| #G #K #W #_ #_ //
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpce_inv_bind_sn (h) (p) (I) (G) (K) (V1) (T1):
+ ∀X2. ⦃G,K⦄ ⊢ ⓑ{p,I}V1.T1 ⬌η[h] X2 →
+ ∃∃V2,T2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,K.ⓑ{I}V1⦄ ⊢ T1 ⬌η[h] T2 & ⓑ{p,I}V2.T2 = X2.
+#h #p0 #I0 #G #Y #V0 #T0 #X2
+@(insert_eq_0 … (ⓑ{p0,I0}V0.T0)) #X1 * -G -Y -X1 -X2
+[ #G #L #s #H destruct
+| #G #i #H destruct
+| #I #G #K #H destruct
+| #G #K #V #H destruct
+| #G #K #W #_ #H destruct
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #H destruct
+| #p #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpce_inv_flat_sn (h) (I) (G) (L) (V1) (T1):
+ ∀X2. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌η[h] X2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,L⦄ ⊢ T1 ⬌η[h] T2 & ⓕ{I}V2.T2 = X2.
+#h #I0 #G #Y #V0 #T0 #X2
+@(insert_eq_0 … (ⓕ{I0}V0.T0)) #X1 * -G -Y -X1 -X2
+[ #G #L #s #H destruct
+| #G #i #H destruct
+| #I #G #K #H destruct
+| #G #K #V #H destruct
+| #G #K #W #_ #H destruct
+| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #H destruct
+| #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
+| #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/xoa/ex_7_8.ma".
+include "basic_2/rt_computation/cpms_drops.ma".
+include "basic_2/rt_conversion/cpce.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpce_ldef_drops (h) (G) (K) (V):
+ ∀i,L. ⇩*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
+#h #G #K #V #i elim i -i
+[ #L #HLK
+ lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
+ /2 width=1 by cpce_ldef/
+| #i #IH #L #HLK
+ elim (drops_inv_succ … HLK) -HLK #Z #Y #HYK #H destruct
+ /3 width=3 by cpce_lref/
+]
+qed.
+
+lemma cpce_ldec_drops (h) (G) (K) (W):
+ (∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ ∀i,L. ⇩*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
+#h #G #K #W #HW #i elim i -i
+[ #L #HLK
+ lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
+ /3 width=5 by cpce_ldec/
+| #i #IH #L #HLK
+ elim (drops_inv_succ … HLK) -HLK #Z #Y #HYK #H destruct
+ /3 width=3 by cpce_lref/
+]
+qed.
+
+lemma cpce_eta_drops (h) (G) (K) (W):
+ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U →
+ ∀W1. ⦃G,K⦄ ⊢ W ⬌η[h] W1 → ∀V1. ⦃G,K⦄ ⊢ V ⬌η[h] V1 →
+ ∀i,L. ⇩*[i] L ≘ K.ⓛW → ∀W2. ⇧*[↑i] W1 ≘ W2 →
+ ∀V2. ⇧*[↑i] V1 ≘ V2 → ⦃G,L⦄ ⊢ #i ⬌η[h] ⓝW2.+ⓛV2.ⓐ#0.#↑i.
+#h #G #K #W #n #p #V #U #HWU #W1 #HW1 #V1 #HV1 #i elim i -i
+[ #L #HLK #W2 #HW12 #V2 #HV12
+ lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
+ /2 width=8 by cpce_eta/
+| #i #IH #L #HLK #W2 #HW12 #V2 #HV12
+ elim (drops_inv_succ … HLK) -HLK #I #Y #HYK #H destruct
+ elim (lifts_split_trans … HW12 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #XW #HXW1 #HXW2
+ elim (lifts_split_trans … HV12 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #XV #HXV1 #HXV2
+ /6 width=9 by cpce_lref, lifts_push_lref, lifts_bind, lifts_flat/
+]
+qed.
+
+lemma cpce_lref_drops (h) (G) (K) (i):
+ ∀T. ⦃G,K⦄ ⊢ #i ⬌η[h] T → ∀j,L. ⇩*[j] L ≘ K →
+ ∀U. ⇧*[j] T ≘ U → ⦃G,L⦄ ⊢ #(j+i) ⬌η[h] U.
+#h #G #K #i #T #Hi #j elim j -j
+[ #L #HLK #U #HTU
+ lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
+ lapply (lifts_fwd_isid … HTU ?) -HTU [ // ] #H destruct //
+| #j #IH #Y #HYK #X #HTX -Hi
+ elim (drops_inv_succ … HYK) -HYK #I #L #HLK #H destruct
+ elim (lifts_split_trans … HTX (𝐔❴j❵) (𝐔❴1❵)) [| // ] #U #HTU #HUX
+ /3 width=3 by cpce_lref/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+axiom cpce_inv_lref_sn_drops_pair (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀I,K,W. ⇩*[i] L ≘ K.ⓑ{I}W →
+ ∨∨ ∧∧ Abbr = I & #i = X2
+ | ∧∧ Abst = I & ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
+ | ∃∃n,p,W1,W2,V,V1,V2,U. Abst = I & ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
+ & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
+ & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
+ & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
+
+axiom cpce_inv_lref_sn_drops_ldef (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀K,V. ⇩*[i] L ≘ K.ⓓV → #i = X2.
+
+axiom cpce_inv_lref_sn_drops_ldec (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀K,W. ⇩*[i] L ≘ K.ⓛW →
+ ∨∨ ∧∧ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
+ | ∃∃n,p,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
+ & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
+ & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
+ & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
+(*
+#h #G #i elim i -i
+[ #L #X2 #HX2 #I #K #HLK
+ lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
+ /2 width=1 by cpce_inv_zero_sn/
+| #i #IH #L0 #X0 #HX0 #J #K #H0
+ elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
+ elim (cpce_inv_lref_sn … HX0) -HX0 #X2 #HX2 #HX20
+ elim (IH … HX2 … HLK) -IH -I -L *
+ [ #HJ #H destruct
+ lapply (lifts_inv_lref1_uni … HX20) -HX20 #H destruct
+ /4 width=7 by or_introl, conj/
+ | #n #p #W #V1 #V2 #W2 #U #HWU #HV12 #HVW2 #H1 #H2 destruct
+ elim (lifts_inv_bind1 … HX20) -HX20 #X2 #X #HWX2 #HX #H destruct
+ elim (lifts_inv_flat1 … HX) -HX #X0 #X1 #H0 #H1 #H destruct
+ lapply (lifts_inv_push_zero_sn … H0) -H0 #H destruct
+ elim (lifts_inv_push_succ_sn … H1) -H1 #j #Hj #H destruct
+ lapply (lifts_inv_lref1_uni … Hj) -Hj #H destruct
+ /4 width=12 by lifts_trans_uni, ex5_7_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma cpce_inv_zero_sn_drops (h) (G) (i) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
+ ∀I,K. ⇩*[i] L ≘ K.ⓘ{I} →
+ (∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ #i = X2.
+#h #G #i #L #X2 #HX2 #I #K #HLK #HI
+elim (cpce_inv_lref_sn_drops_bind … HX2 … HLK) -L *
+[ #_ #H //
+| #n #p #W #V1 #V2 #W2 #U #HWU #_ #_ #H destruct
+ elim (HI … HWU) -n -p -K -X2 -V1 -V2 -W2 -U -i //
+]
+qed-.
+*)
+(* Properties with uniform slicing for local environments *******************)
+
+axiom cpce_lifts_sn (h) (G):
+ d_liftable2_sn … lifts (cpce h G).
+(*
+#h #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #G #K #s #b #f #L #HLK #X #HX
+ lapply (lifts_inv_sort1 … HX) -HX #H destruct
+ /2 width=3 by cpce_sort, lifts_sort, ex2_intro/
+| #G #i #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
+ @(ex2_intro … (#j))
+ [ /2 width=1 by lifts_lref/
+ | @cpce_zero_drops #n #p #Y #W #V #U #HY #_
+ elim (drops_inv_atom2 … HLK) -HLK #j1 #g #HLK #Hg
+ elim (after_at_fwd … Hf … Hg) -f #j2 #_ #Hj -g -i
+ lapply (at_inv_uni … Hj) -Hj #H destruct
+ lapply (drops_conf … HLK … HY ??) -L [3:|*: // ] #H
+ elim (drops_inv_atom1 … H) -H #H #_ destruct
+ ]
+| #G #K #I #HI #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
+ @(ex2_intro … (#j))
+ [ /2 width=1 by lifts_lref/
+ | elim (drops_split_trans_bind2 … HLK … Hf) -HLK -Hf #J #Y1 #HY1 #HK #HIJ
+ @cpce_zero_drops #n #p #Y2 #W #V #U #HY2 #HWU
+ lapply (drops_mono … HY2 … HY1) -L #H destruct
+ elim (liftsb_inv_pair_dx … HIJ) -HIJ #X #HXW #H destruct
+ elim (cpms_inv_lifts_sn … HWU … HK … HXW) -b -Y1 -W #X0 #H #HXU
+ elim (lifts_inv_bind2 … H) -H #V0 #U0 #_ #_ #H destruct -f -j -V -U
+ /2 width=7 by/
+ ]
+| #n #p #G #K #W #V1 #V2 #W2 #U #HWU #_ #HVW2 #IH #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … HLK … Hf) -HLK #J #Y #HY #HK #HIJ
+ elim (liftsb_inv_pair_sn … HIJ) -HIJ #W0 #HW0 #H destruct
+ elim (cpms_lifts_sn … HWU … HK … HW0) -HWU -HW0 #X #H #HWU0
+ elim (lifts_inv_bind1 … H) -H #V0 #U0 #HV10 #HU0 #H destruct
+ elim (IH … HK … HV10) -IH -HK -HV10 #VX #HV2X #HV0X
+ elim (lifts_total W2 f) #WX2 #HWX2
+ lapply (lifts_trans … HVW2 … HWX2 ??) [3:|*: // ] -HVW2 #HVX2
+ @(ex2_intro … (+ⓛWX2.ⓐ#O.#(↑j)))
+ [ /5 width=1 by lifts_lref, lifts_bind, lifts_flat, at_S1/
+ | /4 width=18 by cpce_eta_drops, lifts_conf, after_uni_succ_dx/
+ ]
+| #I #G #K #T #U #i #_ #HTU #IH #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref1 … HX) -HX #x #Hf #H destruct
+ elim (at_inv_nxx … Hf) -Hf [|*: // ] #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … HLK) -HLK [|*: // ] #Z #Y #HLY #HYK #_ -I
+ lapply (drops_isuni_fwd_drop2 … HLY) -HLY [ // ] #HLY
+ elim (IH … HYK) -IH -HYK [|*: /2 width=2 by lifts_lref/ ] -i #T0 #HT0 #Hj
+ elim (lifts_total U f) #U0 #HU0
+ lapply (lifts_trans … HTU … HU0 ??) [3:|*: // ] -HTU #HTU0
+ lapply (lifts_conf … HT0 … HTU0 ??) -T
+ [3:|*: /2 width=3 by after_uni_succ_dx/ ] #HTU0 >plus_S1
+ /3 width=7 by cpce_lref_drops, ex2_intro/
+| #G #K #l #b #f #L #HLK #X #HX
+ lapply (lifts_inv_gref1 … HX) -HX #H destruct
+ /2 width=3 by cpce_gref, lifts_gref, ex2_intro/
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
+ elim (lifts_inv_bind1 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HLK … HVW1) -IHV #W2 #HVW2 #HW12
+ elim (IHT … HTU1) -IHT -HTU1 [|*: /3 width=3 by drops_skip, ext2_pair/ ] -HVW1 #U2 #HTU2 #HU12
+ /3 width=5 by cpce_bind, lifts_bind, ex2_intro/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
+ elim (lifts_inv_flat1 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HLK … HVW1) -IHV -HVW1 #W2 #HVW2 #HW12
+ elim (IHT … HLK … HTU1) -IHT -HTU1 -HLK #U2 #HTU2 #HU12
+ /3 width=5 by cpce_flat, lifts_flat, ex2_intro/
+]
+qed-.
+*)
+lemma cpce_lifts_bi (h) (G):
+ d_liftable2_bi … lifts (cpce h G).
+/3 width=12 by cpce_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
+
+(* Inversion lemmas with uniform slicing for local environments *************)
+
+axiom cpce_inv_lifts_sn (h) (G):
+ d_deliftable2_sn … lifts (cpce h G).
+(*
+#h #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #G #K #s #b #f #L #HLK #X #HX
+ lapply (lifts_inv_sort2 … HX) -HX #H destruct
+ /2 width=3 by cpce_sort, lifts_sort, ex2_intro/
+| #G #i #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
+ @(ex2_intro … (#j))
+ [ /2 width=1 by lifts_lref/
+ | @cpce_zero_drops #n #p #Y #W #V #U #HY #_ -n -p -G -V -U -i
+ elim (drops_inv_atom1 … HLK) -HLK #H #_ destruct -b -f
+ elim (drops_inv_atom1 … HY) -HY #H #_ destruct
+ ]
+| #G #K #I #HI #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
+ @(ex2_intro … (#j))
+ [ /2 width=1 by lifts_lref/
+ | elim (at_inv_xxp … Hf) -Hf [| // ] #g #H1 #H2 destruct
+ elim (drops_inv_skip1 … HLK) -HLK #J #Y #HKY #HIJ #H destruct
+ @cpce_zero #n #p #W #V #U #H #HWU destruct
+ elim (liftsb_inv_pair_sn … HIJ) -HIJ #X #HXW #H destruct
+ elim (cpms_lifts_sn … HWU … HKY … HXW) -b -Y -W #X0 #H #HXU
+ elim (lifts_inv_bind1 … H) -H #V0 #U0 #_ #_ #H destruct -V -U
+ /2 width=7 by/
+ ]
+| #n #p #G #K #W #V1 #V2 #W2 #U #HWU #_ #HVW2 #IH #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
+ elim (at_inv_xxp … Hf) -Hf [| // ] #g #H1 #H2 destruct
+ elim (drops_inv_skip1 … HLK) -HLK #J #Y #HKY #HIJ #H destruct
+ elim (liftsb_inv_pair_dx … HIJ) -HIJ #W0 #HW0 #H destruct
+ elim (cpms_inv_lifts_sn … HWU … HKY … HW0) -HWU -HW0 #X #H #HWU0
+ elim (lifts_inv_bind2 … H) -H #V0 #U0 #HV10 #HU0 #H destruct
+ elim (IH … HKY … HV10) -IH -HKY -HV10 #VX #HV2X #HV0X
+ lapply (lifts_trans … HV2X … HVW2 (↑g) ?)
+ [ /3 width=5 by after_isid_sn, after_next/ ] -V2 #H
+ elim (lifts_split_trans … H 𝐔❴1❵ (⫯g) ?)
+ [| /3 width=7 by after_isid_dx, after_push/ ] #VX2 #HVX2 #HVW2
+ /5 width=10 by cpce_eta, lifts_flat, lifts_bind, lifts_lref, ex2_intro/
+| #I #G #K #T #U #i #_ #HTU #IH #b #f #L #HLK #X #HX
+ elim (lifts_inv_lref2 … HX) -HX #x #Hf #H destruct
+(**) (* this part should be a lemma *)
+ elim (at_inv_xxn … Hf) -Hf [2,4: // ] *
+ [ #g #j #Hij #H1 #H2 destruct
+ elim (drops_inv_skip1 … HLK) -HLK #J #Y #HLK #_ #H destruct -I
+ | #g #Hij #H destruct
+ lapply (drops_inv_drop1 … HLK) -HLK #HLK
+ ]
+(**)
+ elim (IH … HLK) -IH -HLK [1,4:|*: /2 width=2 by lifts_lref/ ] -i #T0 #HT0 #Hj
+ lapply (lifts_trans … HT0 … HTU (↑g) ?)
+ [1,3: /3 width=5 by after_isid_sn, after_next/ ] -T #H
+ elim (lifts_split_trans … H 𝐔❴1❵ (⫯g) ?)
+ [2,4: /3 width=7 by after_isid_dx, after_push/ ] #U0 #HTU0 #HU0
+ /3 width=5 by cpce_lref, ex2_intro/
+| #G #K #l #b #f #L #HLK #X #HX
+ lapply (lifts_inv_gref2 … HX) -HX #H destruct
+ /2 width=3 by cpce_gref, lifts_gref, ex2_intro/
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
+ elim (lifts_inv_bind2 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HLK … HVW1) -IHV #W2 #HVW2 #HW12
+ elim (IHT … HTU1) -IHT -HTU1 [|*: /3 width=3 by drops_skip, ext2_pair/ ] -HVW1 #U2 #HTU2 #HU12
+ /3 width=5 by cpce_bind, lifts_bind, ex2_intro/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
+ elim (lifts_inv_flat2 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HLK … HVW1) -IHV -HVW1 #W2 #HVW2 #HW12
+ elim (IHT … HLK … HTU1) -IHT -HTU1 -HLK #U2 #HTU2 #HU12
+ /3 width=5 by cpce_flat, lifts_flat, ex2_intro/
+]
+qed-.
+*)
+lemma cpce_inv_lifts_bi (h) (G):
+ d_deliftable2_bi … lifts (cpce h G).
+/3 width=12 by cpce_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/cext2.ma".
+include "basic_2/rt_conversion/cpce.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR BINDERS ********************)
+
+definition cpce_ext (h) (G): relation3 lenv bind bind ≝ cext2 (cpce h G).
+
+interpretation
+ "context-sensitive parallel eta-conversion (binder)"
+ 'PConvEta h G L I1 I2 = (cpce_ext h G L I1 I2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 "static_2/relocation/lex.ma".
+include "basic_2/notation/relations/pconveta_4.ma".
+include "basic_2/rt_conversion/cpce_ext.ma".
+
+(* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
+
+definition lpce (h) (G): relation lenv ≝ lex (cpce h G).
+
+interpretation
+ "parallel eta-conversion on all entries (local environment)"
+ 'PConvEta h G L1 L2 = (lpce h G L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lpce_bind (h) (G):
+ ∀K1,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 →
+ ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] K2.ⓘ{I2}.
+/2 width=1 by lex_bind/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lpce_pair (h) (G):
+ ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 →
+ ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2.
+/2 width=1 by lex_pair/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpce_inv_atom_sn (h) (G):
+ ∀L2. ⦃G,⋆⦄ ⊢ ⬌η[h] L2 → L2 = ⋆.
+/2 width=2 by lex_inv_atom_sn/ qed-.
+
+lemma lpce_inv_bind_sn (h) (G):
+ ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L2 = K2.ⓘ{I2}.
+/2 width=1 by lex_inv_bind_sn/ qed-.
+
+lemma lpce_inv_atom_dx (h) (G):
+ ∀L1. ⦃G,L1⦄ ⊢ ⬌η[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
+
+lemma lpce_inv_bind_dx (h) (G):
+ ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L1 = K1.ⓘ{I1}.
+/2 width=1 by lex_inv_bind_dx/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lpce_inv_unit_sn (h) (G):
+ ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
+/2 width=1 by lex_inv_unit_sn/ qed-.
+
+lemma lpce_inv_pair_sn (h) (G):
+ ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L2 = K2.ⓑ{I}V2.
+/2 width=1 by lex_inv_pair_sn/ qed-.
+
+lemma lpce_inv_unit_dx (h) (G):
+ ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L1 = K1.ⓤ{I}.
+/2 width=1 by lex_inv_unit_dx/ qed-.
+
+lemma lpce_inv_pair_dx (h) (G):
+ ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L1 = K1.ⓑ{I}V1.
+/2 width=1 by lex_inv_pair_dx/ qed-.
+
+lemma lpce_inv_pair (h) (G):
+ ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬌η[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬌η[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬌η[h] V2 & I1 = I2.
+/2 width=1 by lex_inv_pair/ 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 "static_2/relocation/drops_lex.ma".
+include "basic_2/rt_conversion/lpce.ma".
+
+(* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma lpce_drops_conf (h) (G): dropable_sn (cpce h G).
+/2 width=3 by lex_dropable_sn/ qed-.
+
+lemma lpce_drops_trans (h) (G): dropable_dx (cpce h G).
+/2 width=3 by lex_dropable_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_conversion/cpce.ma".
+include "basic_2/rt_equivalence/cpcs.ma".
+include "basic_2/dynamic/lsubv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE VALIDITY *************************)
+
+lemma lsubv_inv_unit_dx (h) (a) (G):
+ ∀I,L1,K2. G ⊢ L1 ⫃![h,a] K2.ⓤ{I} →
+ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓤ{I}.
+#h #a #G #I #L1 #K2 #H
+elim (lsubv_inv_bind_dx … H) -H // *
+#K1 #XW #XV #_ #_ #H1 #H2 destruct
+qed-.
+
+lemma lsubv_inv_abbr_dx (h) (a) (G):
+ ∀L1,K2,V. G ⊢ L1 ⫃![h,a] K2.ⓓV →
+ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓓV.
+#h #a #G #L1 #K2 #V #H
+elim (lsubv_inv_bind_dx … H) -H // *
+#K1 #XW #XV #_ #_ #H1 #H2 destruct
+qed-.
+
+lemma lsubv_cpce_trans_cpcs (h) (a) (G) (T0):
+ ∀L2,T2. ⦃G,L2⦄ ⊢ T0 ⬌η[h] T2 → ∀L1. G ⊢ L1 ⫃![h,a] L2 →
+ ∃∃T1. ⦃G,L1⦄ ⊢ T0 ⬌η[h] T1 & ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
+#h #a #G #T0 #L2 #T2 #H elim H -G -L2 -T0 -T2
+[ #G #L2 #s #L1 #HL12
+ /2 width=3 by cpce_sort, ex2_intro/
+| #G #i #Y1 #HY1
+ lapply (lsubv_inv_atom2 … HY1) -HY1 #H destruct
+ /2 width=3 by cpce_atom, ex2_intro/
+| #I #G #K2 #Y1 #HY1
+ elim (lsubv_inv_unit_dx … HY1) -HY1 #K2 #_ #H destruct
+ /2 width=3 by cpce_unit, ex2_intro/
+| #G #K2 #V2 #Y1 #HY1
+ elim (lsubv_inv_abbr_dx … HY1) -HY1 #K2 #_ #H destruct
+ /2 width=3 by cpce_ldef, ex2_intro/
+| #G #K2 #W2 #HW2 #Y1 #HY1
+ elim (lsubv_inv_bind_dx … HY1) -HY1 *
+ [ #K1 #HK12 #H destruct
+ @(ex2_intro … (#0)) [| // ]
+ @cpce_ldec #n #p #V2 #U2 #HWU2
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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/rt_equivalence/cpes.ma". *)
+include "basic_2/dynamic/cnv_cpmuwe.ma".
+(* include "basic_2/dynamic/lsubv.ma". *)
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Properties with restricted refinement for local environments *************)
+
+lemma lsubr_cnuw_trans (h) (G):
+ ∀L2,T. ⦃G,L2⦄ ⊢ ➡𝐍𝐖*[h] T → ∀L1. L1 ⫃ L2 → ⦃G,L1⦄ ⊢ ➡𝐍𝐖*[h] T.
+#h #G #L2 #T1 #HT1 #L1 #HL12 #n #T2 #HT12
+
+lemma lsubv_cpmuwe_trans (h) (a) (n) (G):
+ lsub_trans … (cpmuwe h n G) (lsubv h a G).
+#h #a #n #G #L2 #T1 #T2 * #HT12 #HT2 #L1 #HL12
+lapply (lsubv_cpms_trans … HT12 … HL12) -HT12 #HT12
+@(cpmuwe_intro … HT12) -HT12
+
+lemma cnv_cpmuwe_cpms_conf (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n1,T1. ⦃G,L⦄ ⊢ T ➡*[n1,h] T1 →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n2] T2 →
+ ∃∃T0. ⦃G,L⦄ ⊢ T1 ➡*[n2-n1,h] T0 & T0 ≅ T2 & ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T2.
+#h #a #G #L #T0 #HT0 #n1 #T1 #HT01 #n2 #T2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (HT2 … HT20) -HT20 #HT20
+/3 width=3 by tweq_sym, ex3_intro/
+qed-.
+
+lemma lsubv_cpms_abst_conf_cnv (h) (a) (G) (L1) (T0):
+ ∀n1,p1,W1,T1. ⦃G,L1⦄ ⊢ T0 ➡*[n1,h] ⓛ{p1}W1.T1 →
+ ∀L2. ⦃G,L2⦄ ⊢ T0 ![h,a] → G ⊢ L1 ⫃![h,a] L2 →
+ ∃∃n2,p2,W2,T2. ⦃G,L2⦄ ⊢ T0 ➡*[n2,h] ⓛ{p2}W2.T2.
+#h #a #G #L1 #T0 #n1 #p1 #W1 #T1 #HT01 #L2 #HT0 #HL12
+elim (cnv_R_cpmuwe_total … HT0) #n2 * #X2 #HT02
+elim (abst_dec X2) [ * | #HnX2 ]
+[ #p2 #W2 #T2 #H destruct
+ /3 width=5 by cpmuwe_fwd_cpms, ex1_4_intro/
+| lapply (lsubv_cnv_trans … HT0 … HL12) -HT0 #HT0
+ lapply (lsubv_cpmuwe_trans … HT02 … HL12) -HT02 -HL12 #HT02
+ elim (cnv_cpmuwe_cpms_conf … HT0 … HT01 … HT02) -HT0 -HT01 -HT02 #U2 #H1 #H2 #_
+ elim (cpms_inv_abst_sn … H1) -H1 #W2 #T2 #_ #_ #H destruct
+ elim (tweq_inv_abst_sn … H2) -W2 -T2 #W2 #T2 #H destruct
+ elim (HnX2 p1 W2 T2) -HnX2 //
+]
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/pconveta_5.ma".
-include "basic_2/rt_computation/cpms.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
-
-(* avtivate genv *)
-inductive cpce (h): relation4 genv lenv term term ≝
-| cpce_sort: ∀G,L,s. cpce h G L (⋆s) (⋆s)
-| cpce_atom: ∀G,i. cpce h G (⋆) (#i) (#i)
-| cpce_unit: ∀I,G,K. cpce h G (K.ⓤ{I}) (#0) (#0)
-| cpce_ldef: ∀G,K,V. cpce h G (K.ⓓV) (#0) (#0)
-| cpce_ldec: ∀G,K,W. (∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
- cpce h G (K.ⓛW) (#0) (#0)
-| cpce_eta : ∀n,p,G,K,W,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U →
- cpce h G K W W1 → ⇧*[1] W1 ≘ W2 →
- cpce h G K V V1 → ⇧*[1] V1 ≘ V2 →
- cpce h G (K.ⓛW) (#0) (ⓝW2.+ⓛV2.ⓐ#0.#1)
-| cpce_lref: ∀I,G,K,T,U,i. cpce h G K (#i) T →
- ⇧*[1] T ≘ U → cpce h G (K.ⓘ{I}) (#↑i) U
-| cpce_gref: ∀G,L,l. cpce h G L (§l) (§l)
-| cpce_bind: ∀p,I,G,K,V1,V2,T1,T2.
- cpce h G K V1 V2 → cpce h G (K.ⓑ{I}V1) T1 T2 →
- cpce h G K (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
-| cpce_flat: ∀I,G,L,V1,V2,T1,T2.
- cpce h G L V1 V2 → cpce h G L T1 T2 →
- cpce h G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation
- "context-sensitive parallel eta-conversion (term)"
- 'PConvEta h G L T1 T2 = (cpce h G L T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpce_inv_sort_sn (h) (G) (L) (s):
- ∀X2. ⦃G,L⦄ ⊢ ⋆s ⬌η[h] X2 → ⋆s = X2.
-#h #G #Y #s0 #X2
-@(insert_eq_0 … (⋆s0)) #X1 * -G -Y -X1 -X2
-[ #G #L #s #_ //
-| #G #i #_ //
-| #I #G #K #_ //
-| #G #K #V #_ //
-| #G #K #W #_ #_ //
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H destruct
-| #G #L #l #_ //
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma cpce_inv_atom_sn (h) (G) (i):
- ∀X2. ⦃G,⋆⦄ ⊢ #i ⬌η[h] X2 → #i = X2.
-#h #G #i0 #X2
-@(insert_eq_0 … LAtom) #Y
-@(insert_eq_0 … (#i0)) #X1
-* -G -Y -X1 -X2
-[ #G #L #s #_ #_ //
-| #G #i #_ #_ //
-| #I #G #K #_ #_ //
-| #G #K #V #_ #_ //
-| #G #K #W #_ #_ #_ //
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #_ #H destruct
-| #G #L #l #_ #_ //
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-]
-qed-.
-
-lemma cpce_inv_unit_sn (h) (I) (G) (K):
- ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
-#h #I0 #G #K0 #X2
-@(insert_eq_0 … (K0.ⓤ{I0})) #Y
-@(insert_eq_0 … (#0)) #X1
-* -G -Y -X1 -X2
-[ #G #L #s #_ #_ //
-| #G #i #_ #_ //
-| #I #G #K #_ #_ //
-| #G #K #V #_ #_ //
-| #G #K #W #_ #_ #_ //
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H #_ destruct
-| #G #L #l #_ #_ //
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-]
-qed-.
-
-lemma cpce_inv_ldef_sn (h) (G) (K) (V):
- ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
-#h #G #K0 #V0 #X2
-@(insert_eq_0 … (K0.ⓓV0)) #Y
-@(insert_eq_0 … (#0)) #X1
-* -G -Y -X1 -X2
-[ #G #L #s #_ #_ //
-| #G #i #_ #_ //
-| #I #G #K #_ #_ //
-| #G #K #V #_ #_ //
-| #G #K #W #_ #_ #_ //
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H #_ destruct
-| #G #L #l #_ #_ //
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-]
-qed-.
-
-lemma cpce_inv_ldec_sn (h) (G) (K) (W):
- ∀X2. ⦃G,K.ⓛW⦄ ⊢ #0 ⬌η[h] X2 →
- ∨∨ ∧∧ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #0 = X2
- | ∃∃n,p,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
- & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[1] W1 ≘ W2
- & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[1] V1 ≘ V2
- & ⓝW2.+ⓛV2.ⓐ#0.#1 = X2.
-#h #G #K0 #W0 #X2
-@(insert_eq_0 … (K0.ⓛW0)) #Y
-@(insert_eq_0 … (#0)) #X1
-* -G -Y -X1 -X2
-[ #G #L #s #H #_ destruct
-| #G #i #_ #H destruct
-| #I #G #K #_ #H destruct
-| #G #K #V #_ #H destruct
-| #G #K #W #HW #_ #H destruct /4 width=5 by or_introl, conj/
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #HWU #HW1 #HW12 #HV1 #HV12 #_ #H destruct
- /3 width=14 by or_intror, ex6_8_intro/
-| #I #G #K #T #U #i #_ #_ #H #_ destruct
-| #G #L #l #H #_ destruct
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-]
-qed-.
-
-lemma cpce_inv_lref_sn (h) (I) (G) (K) (i):
- ∀X2. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬌η[h] X2 →
- ∃∃T2. ⦃G,K⦄ ⊢ #i ⬌η[h] T2 & ⇧*[1] T2 ≘ X2.
-#h #I0 #G #K0 #i0 #X2
-@(insert_eq_0 … (K0.ⓘ{I0})) #Y
-@(insert_eq_0 … (#↑i0)) #X1
-* -G -Y -X1 -X2
-[ #G #L #s #H #_ destruct
-| #G #i #_ #H destruct
-| #I #G #K #H #_ destruct
-| #G #K #V #H #_ destruct
-| #G #K #W #_ #H #_ destruct
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H #_ destruct
-| #I #G #K #T #U #i #Hi #HTU #H1 #H2 destruct /2 width=3 by ex2_intro/
-| #G #L #l #H #_ destruct
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
-]
-qed-.
-
-lemma cpce_inv_gref_sn (h) (G) (L) (l):
- ∀X2. ⦃G,L⦄ ⊢ §l ⬌η[h] X2 → §l = X2.
-#h #G #Y #l0 #X2
-@(insert_eq_0 … (§l0)) #X1 * -G -Y -X1 -X2
-[ #G #L #s #_ //
-| #G #i #_ //
-| #I #G #K #_ //
-| #G #K #V #_ //
-| #G #K #W #_ #_ //
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H destruct
-| #G #L #l #_ //
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma cpce_inv_bind_sn (h) (p) (I) (G) (K) (V1) (T1):
- ∀X2. ⦃G,K⦄ ⊢ ⓑ{p,I}V1.T1 ⬌η[h] X2 →
- ∃∃V2,T2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,K.ⓑ{I}V1⦄ ⊢ T1 ⬌η[h] T2 & ⓑ{p,I}V2.T2 = X2.
-#h #p0 #I0 #G #Y #V0 #T0 #X2
-@(insert_eq_0 … (ⓑ{p0,I0}V0.T0)) #X1 * -G -Y -X1 -X2
-[ #G #L #s #H destruct
-| #G #i #H destruct
-| #I #G #K #H destruct
-| #G #K #V #H destruct
-| #G #K #W #_ #H destruct
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H destruct
-| #G #L #l #H destruct
-| #p #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma cpce_inv_flat_sn (h) (I) (G) (L) (V1) (T1):
- ∀X2. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌η[h] X2 →
- ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,L⦄ ⊢ T1 ⬌η[h] T2 & ⓕ{I}V2.T2 = X2.
-#h #I0 #G #Y #V0 #T0 #X2
-@(insert_eq_0 … (ⓕ{I0}V0.T0)) #X1 * -G -Y -X1 -X2
-[ #G #L #s #H destruct
-| #G #i #H destruct
-| #I #G #K #H destruct
-| #G #K #V #H destruct
-| #G #K #W #_ #H destruct
-| #n #p #G #K #W #W1 #W2 #V #V1 #V2 #U #_ #_ #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H destruct
-| #G #L #l #H destruct
-| #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
-| #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/xoa/ex_7_8.ma".
-include "basic_2/rt_computation/cpms_drops.ma".
-include "basic_2/rt_conversion/cpce.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpce_ldef_drops (h) (G) (K) (V):
- ∀i,L. ⇩*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
-#h #G #K #V #i elim i -i
-[ #L #HLK
- lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
- /2 width=1 by cpce_ldef/
-| #i #IH #L #HLK
- elim (drops_inv_succ … HLK) -HLK #Z #Y #HYK #H destruct
- /3 width=3 by cpce_lref/
-]
-qed.
-
-lemma cpce_ldec_drops (h) (G) (K) (W):
- (∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
- ∀i,L. ⇩*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
-#h #G #K #W #HW #i elim i -i
-[ #L #HLK
- lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
- /3 width=5 by cpce_ldec/
-| #i #IH #L #HLK
- elim (drops_inv_succ … HLK) -HLK #Z #Y #HYK #H destruct
- /3 width=3 by cpce_lref/
-]
-qed.
-
-lemma cpce_eta_drops (h) (G) (K) (W):
- ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U →
- ∀W1. ⦃G,K⦄ ⊢ W ⬌η[h] W1 → ∀V1. ⦃G,K⦄ ⊢ V ⬌η[h] V1 →
- ∀i,L. ⇩*[i] L ≘ K.ⓛW → ∀W2. ⇧*[↑i] W1 ≘ W2 →
- ∀V2. ⇧*[↑i] V1 ≘ V2 → ⦃G,L⦄ ⊢ #i ⬌η[h] ⓝW2.+ⓛV2.ⓐ#0.#↑i.
-#h #G #K #W #n #p #V #U #HWU #W1 #HW1 #V1 #HV1 #i elim i -i
-[ #L #HLK #W2 #HW12 #V2 #HV12
- lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
- /2 width=8 by cpce_eta/
-| #i #IH #L #HLK #W2 #HW12 #V2 #HV12
- elim (drops_inv_succ … HLK) -HLK #I #Y #HYK #H destruct
- elim (lifts_split_trans … HW12 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #XW #HXW1 #HXW2
- elim (lifts_split_trans … HV12 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #XV #HXV1 #HXV2
- /6 width=9 by cpce_lref, lifts_push_lref, lifts_bind, lifts_flat/
-]
-qed.
-
-lemma cpce_lref_drops (h) (G) (K) (i):
- ∀T. ⦃G,K⦄ ⊢ #i ⬌η[h] T → ∀j,L. ⇩*[j] L ≘ K →
- ∀U. ⇧*[j] T ≘ U → ⦃G,L⦄ ⊢ #(j+i) ⬌η[h] U.
-#h #G #K #i #T #Hi #j elim j -j
-[ #L #HLK #U #HTU
- lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
- lapply (lifts_fwd_isid … HTU ?) -HTU [ // ] #H destruct //
-| #j #IH #Y #HYK #X #HTX -Hi
- elim (drops_inv_succ … HYK) -HYK #I #L #HLK #H destruct
- elim (lifts_split_trans … HTX (𝐔❴j❵) (𝐔❴1❵)) [| // ] #U #HTU #HUX
- /3 width=3 by cpce_lref/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-axiom cpce_inv_lref_sn_drops_pair (h) (G) (i) (L):
- ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
- ∀I,K,W. ⇩*[i] L ≘ K.ⓑ{I}W →
- ∨∨ ∧∧ Abbr = I & #i = X2
- | ∧∧ Abst = I & ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
- | ∃∃n,p,W1,W2,V,V1,V2,U. Abst = I & ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
- & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
- & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
- & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
-
-axiom cpce_inv_lref_sn_drops_ldef (h) (G) (i) (L):
- ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
- ∀K,V. ⇩*[i] L ≘ K.ⓓV → #i = X2.
-
-axiom cpce_inv_lref_sn_drops_ldec (h) (G) (i) (L):
- ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
- ∀K,W. ⇩*[i] L ≘ K.ⓛW →
- ∨∨ ∧∧ ∀n,p,V,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #i = X2
- | ∃∃n,p,W1,W2,V,V1,V2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U
- & ⦃G,K⦄ ⊢ W ⬌η[h] W1 & ⇧*[↑i] W1 ≘ W2
- & ⦃G,K⦄ ⊢ V ⬌η[h] V1 & ⇧*[↑i] V1 ≘ V2
- & ⓝW2.+ⓛV2.ⓐ#0.#(↑i) = X2.
-(*
-#h #G #i elim i -i
-[ #L #X2 #HX2 #I #K #HLK
- lapply (drops_fwd_isid … HLK ?) -HLK [ // ] #H destruct
- /2 width=1 by cpce_inv_zero_sn/
-| #i #IH #L0 #X0 #HX0 #J #K #H0
- elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
- elim (cpce_inv_lref_sn … HX0) -HX0 #X2 #HX2 #HX20
- elim (IH … HX2 … HLK) -IH -I -L *
- [ #HJ #H destruct
- lapply (lifts_inv_lref1_uni … HX20) -HX20 #H destruct
- /4 width=7 by or_introl, conj/
- | #n #p #W #V1 #V2 #W2 #U #HWU #HV12 #HVW2 #H1 #H2 destruct
- elim (lifts_inv_bind1 … HX20) -HX20 #X2 #X #HWX2 #HX #H destruct
- elim (lifts_inv_flat1 … HX) -HX #X0 #X1 #H0 #H1 #H destruct
- lapply (lifts_inv_push_zero_sn … H0) -H0 #H destruct
- elim (lifts_inv_push_succ_sn … H1) -H1 #j #Hj #H destruct
- lapply (lifts_inv_lref1_uni … Hj) -Hj #H destruct
- /4 width=12 by lifts_trans_uni, ex5_7_intro, or_intror/
- ]
-]
-qed-.
-
-lemma cpce_inv_zero_sn_drops (h) (G) (i) (L):
- ∀X2. ⦃G,L⦄ ⊢ #i ⬌η[h] X2 →
- ∀I,K. ⇩*[i] L ≘ K.ⓘ{I} →
- (∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
- #i = X2.
-#h #G #i #L #X2 #HX2 #I #K #HLK #HI
-elim (cpce_inv_lref_sn_drops_bind … HX2 … HLK) -L *
-[ #_ #H //
-| #n #p #W #V1 #V2 #W2 #U #HWU #_ #_ #H destruct
- elim (HI … HWU) -n -p -K -X2 -V1 -V2 -W2 -U -i //
-]
-qed-.
-*)
-(* Properties with uniform slicing for local environments *******************)
-
-axiom cpce_lifts_sn (h) (G):
- d_liftable2_sn … lifts (cpce h G).
-(*
-#h #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #G #K #s #b #f #L #HLK #X #HX
- lapply (lifts_inv_sort1 … HX) -HX #H destruct
- /2 width=3 by cpce_sort, lifts_sort, ex2_intro/
-| #G #i #b #f #L #HLK #X #HX
- elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
- @(ex2_intro … (#j))
- [ /2 width=1 by lifts_lref/
- | @cpce_zero_drops #n #p #Y #W #V #U #HY #_
- elim (drops_inv_atom2 … HLK) -HLK #j1 #g #HLK #Hg
- elim (after_at_fwd … Hf … Hg) -f #j2 #_ #Hj -g -i
- lapply (at_inv_uni … Hj) -Hj #H destruct
- lapply (drops_conf … HLK … HY ??) -L [3:|*: // ] #H
- elim (drops_inv_atom1 … H) -H #H #_ destruct
- ]
-| #G #K #I #HI #b #f #L #HLK #X #HX
- elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
- @(ex2_intro … (#j))
- [ /2 width=1 by lifts_lref/
- | elim (drops_split_trans_bind2 … HLK … Hf) -HLK -Hf #J #Y1 #HY1 #HK #HIJ
- @cpce_zero_drops #n #p #Y2 #W #V #U #HY2 #HWU
- lapply (drops_mono … HY2 … HY1) -L #H destruct
- elim (liftsb_inv_pair_dx … HIJ) -HIJ #X #HXW #H destruct
- elim (cpms_inv_lifts_sn … HWU … HK … HXW) -b -Y1 -W #X0 #H #HXU
- elim (lifts_inv_bind2 … H) -H #V0 #U0 #_ #_ #H destruct -f -j -V -U
- /2 width=7 by/
- ]
-| #n #p #G #K #W #V1 #V2 #W2 #U #HWU #_ #HVW2 #IH #b #f #L #HLK #X #HX
- elim (lifts_inv_lref1 … HX) -HX #j #Hf #H destruct
- elim (drops_split_trans_bind2 … HLK … Hf) -HLK #J #Y #HY #HK #HIJ
- elim (liftsb_inv_pair_sn … HIJ) -HIJ #W0 #HW0 #H destruct
- elim (cpms_lifts_sn … HWU … HK … HW0) -HWU -HW0 #X #H #HWU0
- elim (lifts_inv_bind1 … H) -H #V0 #U0 #HV10 #HU0 #H destruct
- elim (IH … HK … HV10) -IH -HK -HV10 #VX #HV2X #HV0X
- elim (lifts_total W2 f) #WX2 #HWX2
- lapply (lifts_trans … HVW2 … HWX2 ??) [3:|*: // ] -HVW2 #HVX2
- @(ex2_intro … (+ⓛWX2.ⓐ#O.#(↑j)))
- [ /5 width=1 by lifts_lref, lifts_bind, lifts_flat, at_S1/
- | /4 width=18 by cpce_eta_drops, lifts_conf, after_uni_succ_dx/
- ]
-| #I #G #K #T #U #i #_ #HTU #IH #b #f #L #HLK #X #HX
- elim (lifts_inv_lref1 … HX) -HX #x #Hf #H destruct
- elim (at_inv_nxx … Hf) -Hf [|*: // ] #j #Hf #H destruct
- elim (drops_split_trans_bind2 … HLK) -HLK [|*: // ] #Z #Y #HLY #HYK #_ -I
- lapply (drops_isuni_fwd_drop2 … HLY) -HLY [ // ] #HLY
- elim (IH … HYK) -IH -HYK [|*: /2 width=2 by lifts_lref/ ] -i #T0 #HT0 #Hj
- elim (lifts_total U f) #U0 #HU0
- lapply (lifts_trans … HTU … HU0 ??) [3:|*: // ] -HTU #HTU0
- lapply (lifts_conf … HT0 … HTU0 ??) -T
- [3:|*: /2 width=3 by after_uni_succ_dx/ ] #HTU0 >plus_S1
- /3 width=7 by cpce_lref_drops, ex2_intro/
-| #G #K #l #b #f #L #HLK #X #HX
- lapply (lifts_inv_gref1 … HX) -HX #H destruct
- /2 width=3 by cpce_gref, lifts_gref, ex2_intro/
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
- elim (lifts_inv_bind1 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HLK … HVW1) -IHV #W2 #HVW2 #HW12
- elim (IHT … HTU1) -IHT -HTU1 [|*: /3 width=3 by drops_skip, ext2_pair/ ] -HVW1 #U2 #HTU2 #HU12
- /3 width=5 by cpce_bind, lifts_bind, ex2_intro/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
- elim (lifts_inv_flat1 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HLK … HVW1) -IHV -HVW1 #W2 #HVW2 #HW12
- elim (IHT … HLK … HTU1) -IHT -HTU1 -HLK #U2 #HTU2 #HU12
- /3 width=5 by cpce_flat, lifts_flat, ex2_intro/
-]
-qed-.
-*)
-lemma cpce_lifts_bi (h) (G):
- d_liftable2_bi … lifts (cpce h G).
-/3 width=12 by cpce_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
-
-(* Inversion lemmas with uniform slicing for local environments *************)
-
-axiom cpce_inv_lifts_sn (h) (G):
- d_deliftable2_sn … lifts (cpce h G).
-(*
-#h #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #G #K #s #b #f #L #HLK #X #HX
- lapply (lifts_inv_sort2 … HX) -HX #H destruct
- /2 width=3 by cpce_sort, lifts_sort, ex2_intro/
-| #G #i #b #f #L #HLK #X #HX
- elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
- @(ex2_intro … (#j))
- [ /2 width=1 by lifts_lref/
- | @cpce_zero_drops #n #p #Y #W #V #U #HY #_ -n -p -G -V -U -i
- elim (drops_inv_atom1 … HLK) -HLK #H #_ destruct -b -f
- elim (drops_inv_atom1 … HY) -HY #H #_ destruct
- ]
-| #G #K #I #HI #b #f #L #HLK #X #HX
- elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
- @(ex2_intro … (#j))
- [ /2 width=1 by lifts_lref/
- | elim (at_inv_xxp … Hf) -Hf [| // ] #g #H1 #H2 destruct
- elim (drops_inv_skip1 … HLK) -HLK #J #Y #HKY #HIJ #H destruct
- @cpce_zero #n #p #W #V #U #H #HWU destruct
- elim (liftsb_inv_pair_sn … HIJ) -HIJ #X #HXW #H destruct
- elim (cpms_lifts_sn … HWU … HKY … HXW) -b -Y -W #X0 #H #HXU
- elim (lifts_inv_bind1 … H) -H #V0 #U0 #_ #_ #H destruct -V -U
- /2 width=7 by/
- ]
-| #n #p #G #K #W #V1 #V2 #W2 #U #HWU #_ #HVW2 #IH #b #f #L #HLK #X #HX
- elim (lifts_inv_lref2 … HX) -HX #j #Hf #H destruct
- elim (at_inv_xxp … Hf) -Hf [| // ] #g #H1 #H2 destruct
- elim (drops_inv_skip1 … HLK) -HLK #J #Y #HKY #HIJ #H destruct
- elim (liftsb_inv_pair_dx … HIJ) -HIJ #W0 #HW0 #H destruct
- elim (cpms_inv_lifts_sn … HWU … HKY … HW0) -HWU -HW0 #X #H #HWU0
- elim (lifts_inv_bind2 … H) -H #V0 #U0 #HV10 #HU0 #H destruct
- elim (IH … HKY … HV10) -IH -HKY -HV10 #VX #HV2X #HV0X
- lapply (lifts_trans … HV2X … HVW2 (↑g) ?)
- [ /3 width=5 by after_isid_sn, after_next/ ] -V2 #H
- elim (lifts_split_trans … H 𝐔❴1❵ (⫯g) ?)
- [| /3 width=7 by after_isid_dx, after_push/ ] #VX2 #HVX2 #HVW2
- /5 width=10 by cpce_eta, lifts_flat, lifts_bind, lifts_lref, ex2_intro/
-| #I #G #K #T #U #i #_ #HTU #IH #b #f #L #HLK #X #HX
- elim (lifts_inv_lref2 … HX) -HX #x #Hf #H destruct
-(**) (* this part should be a lemma *)
- elim (at_inv_xxn … Hf) -Hf [2,4: // ] *
- [ #g #j #Hij #H1 #H2 destruct
- elim (drops_inv_skip1 … HLK) -HLK #J #Y #HLK #_ #H destruct -I
- | #g #Hij #H destruct
- lapply (drops_inv_drop1 … HLK) -HLK #HLK
- ]
-(**)
- elim (IH … HLK) -IH -HLK [1,4:|*: /2 width=2 by lifts_lref/ ] -i #T0 #HT0 #Hj
- lapply (lifts_trans … HT0 … HTU (↑g) ?)
- [1,3: /3 width=5 by after_isid_sn, after_next/ ] -T #H
- elim (lifts_split_trans … H 𝐔❴1❵ (⫯g) ?)
- [2,4: /3 width=7 by after_isid_dx, after_push/ ] #U0 #HTU0 #HU0
- /3 width=5 by cpce_lref, ex2_intro/
-| #G #K #l #b #f #L #HLK #X #HX
- lapply (lifts_inv_gref2 … HX) -HX #H destruct
- /2 width=3 by cpce_gref, lifts_gref, ex2_intro/
-| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
- elim (lifts_inv_bind2 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HLK … HVW1) -IHV #W2 #HVW2 #HW12
- elim (IHT … HTU1) -IHT -HTU1 [|*: /3 width=3 by drops_skip, ext2_pair/ ] -HVW1 #U2 #HTU2 #HU12
- /3 width=5 by cpce_bind, lifts_bind, ex2_intro/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #b #f #L #HLK #X #HX
- elim (lifts_inv_flat2 … HX) -HX #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HLK … HVW1) -IHV -HVW1 #W2 #HVW2 #HW12
- elim (IHT … HLK … HTU1) -IHT -HTU1 -HLK #U2 #HTU2 #HU12
- /3 width=5 by cpce_flat, lifts_flat, ex2_intro/
-]
-qed-.
-*)
-lemma cpce_inv_lifts_bi (h) (G):
- d_deliftable2_bi … lifts (cpce h G).
-/3 width=12 by cpce_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/cext2.ma".
-include "basic_2/rt_conversion/cpce.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR BINDERS ********************)
-
-definition cpce_ext (h) (G): relation3 lenv bind bind ≝ cext2 (cpce h G).
-
-interpretation
- "context-sensitive parallel eta-conversion (binder)"
- 'PConvEta h G L I1 I2 = (cpce_ext h G L I1 I2).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 "static_2/relocation/lex.ma".
-include "basic_2/notation/relations/pconveta_4.ma".
-include "basic_2/rt_conversion/cpce_ext.ma".
-
-(* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
-
-definition lpce (h) (G): relation lenv ≝ lex (cpce h G).
-
-interpretation
- "parallel eta-conversion on all entries (local environment)"
- 'PConvEta h G L1 L2 = (lpce h G L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lpce_bind (h) (G):
- ∀K1,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 →
- ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] K2.ⓘ{I2}.
-/2 width=1 by lex_bind/ qed.
-
-(* Advanced properties ******************************************************)
-
-lemma lpce_pair (h) (G):
- ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 →
- ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2.
-/2 width=1 by lex_pair/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpce_inv_atom_sn (h) (G):
- ∀L2. ⦃G,⋆⦄ ⊢ ⬌η[h] L2 → L2 = ⋆.
-/2 width=2 by lex_inv_atom_sn/ qed-.
-
-lemma lpce_inv_bind_sn (h) (G):
- ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬌η[h] L2 →
- ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L2 = K2.ⓘ{I2}.
-/2 width=1 by lex_inv_bind_sn/ qed-.
-
-lemma lpce_inv_atom_dx (h) (G):
- ∀L1. ⦃G,L1⦄ ⊢ ⬌η[h] ⋆ → L1 = ⋆.
-/2 width=2 by lex_inv_atom_dx/ qed-.
-
-lemma lpce_inv_bind_dx (h) (G):
- ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬌η[h] I2 & L1 = K1.ⓘ{I1}.
-/2 width=1 by lex_inv_bind_dx/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lpce_inv_unit_sn (h) (G):
- ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
- ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
-/2 width=1 by lex_inv_unit_sn/ qed-.
-
-lemma lpce_inv_pair_sn (h) (G):
- ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬌η[h] L2 →
- ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L2 = K2.ⓑ{I}V2.
-/2 width=1 by lex_inv_pair_sn/ qed-.
-
-lemma lpce_inv_unit_dx (h) (G):
- ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓤ{I} →
- ∃∃K1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L1 = K1.ⓤ{I}.
-/2 width=1 by lex_inv_unit_dx/ qed-.
-
-lemma lpce_inv_pair_dx (h) (G):
- ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬌η[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬌η[h] V2 & L1 = K1.ⓑ{I}V1.
-/2 width=1 by lex_inv_pair_dx/ qed-.
-
-lemma lpce_inv_pair (h) (G):
- ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬌η[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G,L1⦄ ⊢ ⬌η[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬌η[h] V2 & I1 = I2.
-/2 width=1 by lex_inv_pair/ 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 "static_2/relocation/drops_lex.ma".
-include "basic_2/rt_conversion/lpce.ma".
-
-(* PARALLEL ETA-CONVERSION FOR FULL LOCAL ENVIRONMENTS **********************)
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-lemma lpce_drops_conf (h) (G): dropable_sn (cpce h G).
-/2 width=3 by lex_dropable_sn/ qed-.
-
-lemma lpce_drops_trans (h) (G): dropable_dx (cpce h G).
-/2 width=3 by lex_dropable_dx/ qed-.
]
[ { "context-sensitive native validity" * } {
[ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ]
- [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_acle" + "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpmuwe" + "cnv_cpmuwe_cpme" + "cnv_eval" + "cnv_cpce" + "cnv_lpce" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" + "cnv_preserve_cpcs" * ]
+ [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_acle" + "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpmuwe" + "cnv_cpmuwe_cpme" + "cnv_eval" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" + "cnv_preserve_cpcs" * ]
}
]
}
]
class "blue"
[ { "rt-conversion" * } {
+(*
[ { "context-sensitive parallel eta-conversion" * } {
[ [ "for lenvs on all entries" ] "lpce ( ⦃?,?⦄ ⊢ ⬌η[?] ? )" "lpce_drops" * ]
[ [ "for binders" ] "cpce_ext" + "( ⦃?,?⦄ ⊢ ? ⬌η[?] ? )" * ]
[ [ "for terms" ] "cpce" + "( ⦃?,?⦄ ⊢ ? ⬌η[?] ? )" "cpce_drops" * ]
}
]
+*)
[ { "context-sensitive parallel r-conversion" * } {
[ [ "for terms" ] "cpc" + "( ⦃?,?⦄ ⊢ ? ⬌[?] ? )" "cpc_cpc" * ]
}
projects / helm.git / commitdiff