include "basic_2/notation/relations/peval_4.ma".
include "basic_2/computation/cprs.ma".
-include "basic_2/computation/csn.ma".
+include "basic_2/computation/csx.ma".
(* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************)
(* Basic_properties *********************************************************)
(* Basic_1: was just: nf2_sn3 *)
-lemma csn_cpre: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
-#h #g #G #L #T1 #H @(csn_ind … H) -T1
+lemma csx_cpre: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1
#T1 #_ #IHT1
elim (cnr_dec G L T1) /3 width=3/
* #T #H1T1 #H2T1
include "basic_2/notation/relations/peval_6.ma".
include "basic_2/computation/cpxs.ma".
-include "basic_2/computation/csn.ma".
+include "basic_2/computation/csx.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL EVALUATION ON TERMS ******************)
(* Basic_properties *********************************************************)
-lemma csn_cpxe: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] 𝐍⦃T2⦄.
-#h #g #G #L #T1 #H @(csn_ind … H) -T1
+lemma csx_cpxe: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] 𝐍⦃T2⦄.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1
#T1 #_ #IHT1
elim (cnx_dec h g G L T1) /3 width=3/
* #T #H1T1 #H2T1
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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/sn_5.ma".
-include "basic_2/reduction/cnx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-definition csn: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. SN … (cpx h g G L) (eq …).
-
-interpretation
- "context-sensitive extended strong normalization (term)"
- 'SN h g G L T = (csn h g G L T).
-
-(* Basic eliminators ********************************************************)
-
-lemma csn_ind: ∀h,g,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
- R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: sn3_pr2_intro *)
-lemma csn_intro: ∀h,g,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
- ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
-/4 width=1/ qed.
-
-lemma csn_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csn_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
-qed-.
-
-(* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csn: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-/2 width=1/ qed.
-
-lemma cnx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
-#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
-#Hkl @csn_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
-[ #H destruct elim HX //
-| -HX * #l0 #_ #H destruct -l0 /2 width=1/
-]
-qed.
-
-(* Basic_1: was just: sn3_cast *)
-lemma csn_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
-#h #g #G #L #W #HW @(csn_ind … HW) -W #W #HW #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpx_inv_cast1 … H1) -H1
-[ * #W0 #T0 #HLW0 #HLT0 #H destruct
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ /3 width=3 by csn_cpx_trans/
- | -HLW0 * #H destruct /3 width=1/
- ]
-|2,3: /3 width=3 by csn_cpx_trans/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-fact csn_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
-@csn_intro #V2 #HLV2 #HV2
-@(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
-qed-.
-
-(* Basic_1: was just: sn3_gen_head *)
-lemma csn_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
-/2 width=5 by csn_fwd_pair_sn_aux/ qed-.
-
-fact csn_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
-@csn_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{a,I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
-qed-.
-
-(* Basic_1: was just: sn3_gen_bind *)
-lemma csn_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
-/2 width=4 by csn_fwd_bind_dx_aux/ qed-.
-
-fact csn_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
-@csn_intro #T2 #HLT2 #HT2
-@(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
-qed-.
-
-(* Basic_1: was just: sn3_gen_flat *)
-lemma csn_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-/2 width=5 by csn_fwd_flat_dx_aux/ qed-.
-
-(* Basic_1: removed theorems 14:
- sn3_cdelta
- sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
- sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
- sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/acp_aaa.ma".
-include "basic_2/computation/csn_tstc_vector.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Main properties concerning atomic arity assignment ***********************)
-
-theorem aaa_csn: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #A #H
-@(acp_aaa … (csn_acp h g) (csn_acr h g) … H)
-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/snalt_5.ma".
-include "basic_2/computation/cpxs.ma".
-include "basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* alternative definition of csn *)
-definition csna: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. SN … (cpxs h g G L) (eq …).
-
-interpretation
- "context-sensitive extended strong normalization (term) alternative"
- 'SNAlt h g G L T = (csna h g G L T).
-
-(* Basic eliminators ********************************************************)
-
-lemma csna_ind: ∀h,g,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: sn3_intro *)
-lemma csna_intro: ∀h,g,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
- ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-/4 width=1/ qed.
-
-fact csna_intro_aux: ∀h,g,G,L,T1. (
- ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
- ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-/4 width=3/ qed-.
-
-(* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csna_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2.
-#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csna_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
-qed.
-
-(* Basic_1: was just: sn3_pr2_intro (old version) *)
-lemma csna_intro_cpx: ∀h,g,G,L,T1. (
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
- ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-#h #g #G #L #T1 #H
-@csna_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
-[ -H #H destruct #H
- elim H //
-| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
- elim (term_eq_dec T0 T) #HT0
- [ -HLT1 -HLT2 -H /3 width=1/
- | -IHT -HT12 /4 width=3/
- ]
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem csn_csna: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
-#h #g #G #L #T #H @(csn_ind … H) -T /4 width=1/
-qed.
-
-theorem csna_csn: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H @(csna_ind … H) -T /4 width=1/
-qed.
-
-(* Basic_1: was just: sn3_pr3_trans *)
-lemma csn_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csn_cpx_trans/
-qed-.
-
-(* Main eliminators *********************************************************)
-
-lemma csn_ind_alt: ∀h,g,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1
-@H0 -H0 /2 width=1/ -HT1 /3 width=1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cnx_lift.ma".
-include "basic_2/computation/acp.ma".
-include "basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was just: sn3_lift *)
-lemma csn_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
- ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
-@csn_intro #T #HLT2 #HT2
-elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1/
-qed.
-
-(* Basic_1: was just: sn3_gen_lift *)
-lemma csn_inv_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
- ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
-@csn_intro #T #HLT2 #HT2
-elim (lift_total T d e) #T0 #HT0
-lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1/
-qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was just: sn3_abbr *)
-lemma csn_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
-#h #g #I #G #L #K #V #i #HLK #HV
-@csn_intro #X #H #Hi
-elim (cpx_inv_lref1 … H) -H
-[ #H destruct elim Hi //
-| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
- lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
- @(csn_lift … HLK HV1) -HLK -HV1
- @(csn_cpx_trans … HV) -HV //
-]
-qed.
-
-lemma csn_appl_simple: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
-#h #g #G #L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
-@csn_intro #X #H1 #H2
-elim (cpx_inv_appl1_simple … H1) // -H1
-#V0 #T0 #HLV0 #HLT10 #H destruct
-elim (eq_false_inv_tpair_dx … H2) -H2
-[ -IHV -HT1 #HT10
- @(csn_cpx_trans … (ⓐV.T0)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // /2 width=1/
-| -HLT10 * #H #HV0 destruct
- @IHV -IHV // -HT1 /2 width=1/ -HV0
- #T2 #HLT02 #HT02
- @(csn_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // -HLT02 /2 width=1/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: sn3_gen_def *)
-lemma csn_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
-#h #g #I #G #L #K #V #i #HLK #Hi
-elim (lift_total V 0 (i+1)) #V0 #HV0
-lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
-@(csn_inv_lift … H0LK … HV0) -H0LK
-@(csn_cpx_trans … Hi) -Hi /2 width=7/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem csn_acp: ∀h,g. acp (cpx h g) (eq …) (csn h g).
-#h #g @mk_acp
-[ #G #L elim (deg_total h g 0)
- #l #Hl lapply (cnx_sort_iter … L … Hl) /2 width=2/
-| @cnx_lift
-| /2 width=3 by csn_fwd_flat_dx/
-| /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/grammar/tstc_tstc.ma".
-include "basic_2/computation/cpxs_cpxs.ma".
-include "basic_2/computation/csn_alt.ma".
-include "basic_2/computation/csn_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Advanced properties ******************************************************)
-
-lemma csn_lpx_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
- ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #L2 #HL12 #T #H @(csn_ind_alt … H) -T #T #_ #IHT
-@csn_intro #T0 #HLT0 #HT0
-@IHT /2 width=2/ -IHT -HT0 /2 width=3 by lpx_cpx_trans/
-qed.
-
-lemma csn_abst: ∀h,g,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
- ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
-#h #g #a #G #L #W #HW @(csn_ind … HW) -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpx_inv_abst1 … H1) -H1
-#W0 #T0 #HLW0 #HLT0 #H destruct
-elim (eq_false_inv_tpair_sn … H2) -H2
-[ -IHT #H lapply (csn_cpx_trans … HLT0) // -HT
- #HT0 lapply (csn_lpx_conf … (L.ⓛW0) … HT0) -HT0 /2 width=1/ /3 width=1/
-| -IHW -HLW0 -HT * #H destruct /3 width=1/
-]
-qed.
-
-lemma csn_abbr: ∀h,g,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
- ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
-#h #g #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpx_inv_abbr1 … H1) -H1 *
-[ #V1 #T1 #HLV1 #HLT1 #H destruct
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ #HV1 @IHV // /2 width=1/ -HV1
- @(csn_lpx_conf … (L.ⓓV)) /2 width=1/ -HLV1 /2 width=3 by csn_cpx_trans/
- | -IHV -HLV1 * #H destruct /3 width=1/
- ]
-| -IHV -IHT -H2 #T0 #HLT0 #HT0
- lapply (csn_cpx_trans … HT … HLT0) -T #HLT0
- lapply (csn_inv_lift … HLT0 … HT0) -T0 /2 width=3/
-]
-qed.
-
-fact csn_appl_beta_aux: ∀h,g,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
- ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
-#h #g #a #G #L #X #H @(csn_ind … H) -X
-#X #HT1 #IHT1 #V #W #T1 #H1 destruct
-@csn_intro #X #H1 #H2
-elim (cpx_inv_appl1 … H1) -H1 *
-[ -HT1 #V0 #Y #HLV0 #H #H0 destruct
- elim (cpx_inv_abst1 … H) -H #W0 #T0 #HLW0 #HLT0 #H destruct
- @IHT1 -IHT1 [4: // | skip |3: #H destruct /2 width=1/ ] -H2
- lapply (lsubr_cpx_trans … HLT0 (L.ⓓⓝW.V) ?) -HLT0 [ /2 width=1/ ] /3 width=1/
-| -IHT1 -H2 #b #V0 #W0 #W2 #T0 #T2 #HLV0 #HLW02 #HLT02 #H1 #H3 destruct
- lapply (lsubr_cpx_trans … HLT02 (L.ⓓⓝW0.V) ?) -HLT02 [ /2 width=1/ ] #HT02
- @(csn_cpx_trans … HT1) -HT1 /3 width=1/
-| -HT1 -IHT1 -H2 #b #V0 #V1 #W0 #W1 #T0 #T3 #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: sn3_beta *)
-lemma csn_appl_beta: ∀h,g,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
-/2 width=3 by csn_appl_beta_aux/ qed.
-
-fact csn_appl_theta_aux: ∀h,g,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
-#h #g #a #G #L #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
-lapply (csn_fwd_pair_sn … HVT) #HV
-lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
-@csn_intro #X #HL #H
-elim (cpx_inv_appl1 … HL) -HL *
-[ -HV #V0 #Y #HLV10 #HL #H0 destruct
- elim (cpx_inv_abbr1 … HL) -HL *
- [ #V3 #T3 #HV3 #HLT3 #H0 destruct
- elim (lift_total V0 0 1) #V4 #HV04
- elim (term_eq_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V3.ⓐV4.T3))
- [ -IHVT #H0 destruct
- elim (eq_false_inv_tpair_sn … H) -H
- [ -HLV10 -HV3 -HLT3 -HVT
- >(lift_inj … HV12 … HV04) -V4
- #H elim H //
- | * #_ #H elim H //
- ]
- | -H -HVT #H
- lapply (cpx_lift … HLV10 (L. ⓓV) … HV12 … HV04) -HLV10 -HV12 /2 width=1/ #HV24
- @(IHVT … H … HV04) -IHVT // -H -HV04 /4 width=1/
- ]
- | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
- lapply (csn_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1/ -T #HVT0
- lapply (csn_inv_lift … L … 1 HVT0 ? ? ?) -HVT0 [ /2 width=4/ |2,3: skip | /2 width=1/ ] -V2 -T0 #HVY
- @(csn_cpx_trans … HVY) /2 width=1/
- ]
-| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
-| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
- lapply (cpx_lift … HLV10 (L. ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
- @csn_abbr /2 width=3 by csn_cpx_trans/ -HV
- @(csn_lpx_conf … (L. ⓓW0)) /2 width=1/ -W1
- @(csn_cpxs_trans … HVT) -HVT /3 width=1/
-]
-qed-.
-
-lemma csn_appl_theta: ∀h,g,a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
-/2 width=5 by csn_appl_theta_aux/ qed.
-
-(* Basic_1: was just: sn3_appl_appl *)
-lemma csn_appl_simple_tstc: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≃ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
-#h #g #G #L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
-@csn_intro #X #HL #H
-elim (cpx_inv_appl1_simple … HL) -HL //
-#V0 #T0 #HLV0 #HLT10 #H0 destruct
-elim (eq_false_inv_tpair_sn … H) -H
-[ -IHT1 #HV0
- @(csn_cpx_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
- @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
- #T2 #HLT12 #HT12
- @(csn_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
- @H2T1 -H2T1 // -HLT12 /2 width=1/
-| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
- elim (tstc_dec T1 T0) #H2T10
- [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
- #T2 #HLT02 #HT02
- @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
- | -IHT1 -H3T1 -H1T10
- @H2T1 -H2T1 /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/computation/acp_cr.ma".
-include "basic_2/computation/cpxs_tstc_vector.ma".
-include "basic_2/computation/csn_lpx.ma".
-include "basic_2/computation/csn_vector.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was just: sn3_appls_lref *)
-lemma csn_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
- ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
-#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csn … H2T) ] (**) (* /2 width=1/ does not work *)
-#V #Vs #IHV #H
-elim (csnv_inv_cons … H) -H #HV #HVs
-@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs
-#X #H #H0
-lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
-elim (H0) -H0 //
-qed.
-
-lemma csn_applv_sort: ∀h,g,G,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus … l) -l [ /3 width=1/ ]
-#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
-#Hkl #Vs elim Vs -Vs /2 width=1/
-#V #Vs #IHVs #HVVs
-elim (csnv_inv_cons … HVVs) #HV #HVs
-@csn_appl_simple_tstc // -HV /2 width=1/ -IHVs -HVs
-#X #H #H0
-elim (cpxs_fwd_sort_vector … H) -H #H
-[ elim H0 -H0 //
-| -H0 @(csn_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1/
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_beta *)
-lemma csn_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
- ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
-#h #g #a #G #L #Vs elim Vs -Vs /2 width=1/
-#V0 #Vs #IHV #V #W #T #H1T
-lapply (csn_fwd_pair_sn … H1T) #HV0
-lapply (csn_fwd_flat_dx … H1T) #H2T
-@csn_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T
-#X #H #H0
-elim (cpxs_fwd_beta_vector … H) -H #H
-[ -H1T elim H0 -H0 //
-| -H0 @(csn_cpxs_trans … H1T) -H1T /2 width=1/
-]
-qed.
-
-lemma csn_applv_delta: ∀h,g,I,G,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
- ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
-#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
-[ #H
- lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
- lapply (csn_inv_lift … H … HLK0 HV12) -V2 -HLK0 /2 width=5/
-| #V #Vs #IHV #H1T
- lapply (csn_fwd_pair_sn … H1T) #HV
- lapply (csn_fwd_flat_dx … H1T) #H2T
- @csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
- #X #H #H0
- elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
- [ -H1T elim H0 -H0 //
- | -H0 @(csn_cpxs_trans … H1T) -H1T /2 width=1/
- ]
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_abbr *)
-lemma csn_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
- ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
-#h #g #a #G #L #V1s #V2s * -V1s -V2s /2 width=1/
-#V1s #V2s #V1 #V2 #HV12 #H
-generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
-elim H -V1s -V2s /2 width=3/
-#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H
-lapply (csn_appl_theta … HW12 … H) -H -HW12 #H
-lapply (csn_fwd_pair_sn … H) #HW1
-lapply (csn_fwd_flat_dx … H) #H1
-@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -H1 #X #H1 #H2
-elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
-[ -H #H elim H2 -H2 //
-| -H2 #H1 @(csn_cpxs_trans … H) -H /2 width=1/
-]
-qed.
-
-(* Basic_1: was just: sn3_appls_cast *)
-lemma csn_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
- ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
-#h #g #G #L #Vs elim Vs -Vs /2 width=1/
-#V #Vs #IHV #W #T #H1W #H1T
-lapply (csn_fwd_pair_sn … H1W) #HV
-lapply (csn_fwd_flat_dx … H1W) #H2W
-lapply (csn_fwd_flat_dx … H1T) #H2T
-@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2W -H2T
-#X #H #H0
-elim (cpxs_fwd_cast_vector … H) -H #H
-[ -H1W -H1T elim H0 -H0 //
-| -H1W -H0 @(csn_cpxs_trans … H1T) -H1T /2 width=1/
-| -H1T -H0 @(csn_cpxs_trans … H1W) -H1W /2 width=1/
-]
-qed.
-
-theorem csn_acr: ∀h,g. acr (cpx h g) (eq …) (csn h g) (λG,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T).
-#h #g @mk_acr //
-[ /3 width=1/
-|2,3,6: /2 width=1/
-| /2 width=7/
-| #G #L #V1s #V2s #HV12s #a #V #T #H #HV
- @(csn_applv_theta … HV12s) -HV12s
- @(csn_abbr) //
-| @csn_lift
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/term_vector.ma".
-include "basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
-
-definition csnv: ∀h. sd h → relation3 genv lenv (list term) ≝
- λh,g,G,L. all … (csn h g G L).
-
-interpretation
- "context-sensitive strong normalization (term vector)"
- 'SN h g G L Ts = (csnv h g G L Ts).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma csnv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
- ⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
-normalize // qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma csn_fwd_applv: ∀h,g,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Ⓐ Vs.T →
- ⦃G, L⦄ ⊢ ⬊*[h, g] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #Vs elim Vs -Vs /2 width=1/
-#V #Vs #IHVs #HVs
-lapply (csn_fwd_pair_sn … HVs) #HV
-lapply (csn_fwd_flat_dx … HVs) -HVs #HVs
-elim (IHVs HVs) -IHVs -HVs /3 width=1/
-qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/sn_5.ma".
+include "basic_2/reduction/cnx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+definition csx: ∀h. sd h → relation3 genv lenv term ≝
+ λh,g,G,L. SN … (cpx h g G L) (eq …).
+
+interpretation
+ "context-sensitive extended strong normalization (term)"
+ 'SN h g G L T = (csx h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
+ R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: sn3_pr2_intro *)
+lemma csx_intro: ∀h,g,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+/4 width=1/ qed.
+
+lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+@csx_intro #T #HLT2 #HT2
+elim (term_eq_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1/
+| -HT1 -HT2 /3 width=4/
+qed-.
+
+(* Basic_1: was just: sn3_nf2 *)
+lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/2 width=1/ qed.
+
+lemma cnx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
+#h #g #G #L #k elim (deg_total h g k)
+#l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
+#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
+#Hkl @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
+[ #H destruct elim HX //
+| -HX * #l0 #_ #H destruct -l0 /2 width=1/
+]
+qed.
+
+(* Basic_1: was just: sn3_cast *)
+lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
+#h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_cast1 … H1) -H1
+[ * #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2
+ [ /3 width=3 by csx_cpx_trans/
+ | -HLW0 * #H destruct /3 width=1/
+ ]
+|2,3: /3 width=3 by csx_cpx_trans/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+ ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #V2 #HLV2 #HV2
+@(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
+qed-.
+
+(* Basic_1: was just: sn3_gen_head *)
+lemma csx_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
+
+fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+ ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓑ{a,I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed-.
+
+(* Basic_1: was just: sn3_gen_bind *)
+lemma csx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
+
+fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed-.
+
+(* Basic_1: was just: sn3_gen_flat *)
+lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
+
+(* Basic_1: removed theorems 14:
+ sn3_cdelta
+ sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
+ sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
+ sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/acp_aaa.ma".
+include "basic_2/computation/csx_tstc_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Main properties concerning atomic arity assignment ***********************)
+
+theorem aaa_csx: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #A #H
+@(acp_aaa … (csx_acp h g) (csx_acr h g) … H)
+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/snalt_5.ma".
+include "basic_2/computation/cpxs.ma".
+include "basic_2/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* alternative definition of csx *)
+definition csxa: ∀h. sd h → relation3 genv lenv term ≝
+ λh,g,G,L. SN … (cpxs h g G L) (eq …).
+
+interpretation
+ "context-sensitive extended strong normalization (term) alternative"
+ 'SNAlt h g G L T = (csxa h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csxa_ind: ∀h,g,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: sn3_intro *)
+lemma csxa_intro: ∀h,g,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
+ ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+/4 width=1/ qed.
+
+fact csxa_intro_aux: ∀h,g,G,L,T1. (
+ ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+ ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+/4 width=3/ qed-.
+
+(* Basic_1: was just: sn3_pr3_trans (old version) *)
+lemma csxa_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2.
+#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+@csxa_intro #T #HLT2 #HT2
+elim (term_eq_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1/
+| -HT1 -HT2 /3 width=4/
+qed.
+
+(* Basic_1: was just: sn3_pr2_intro (old version) *)
+lemma csxa_intro_cpx: ∀h,g,G,L,T1. (
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+ ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+#h #g #G #L #T1 #H
+@csxa_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
+[ -H #H destruct #H
+ elim H //
+| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
+ elim (term_eq_dec T0 T) #HT0
+ [ -HLT1 -HLT2 -H /3 width=1/
+ | -IHT -HT12 /4 width=3/
+ ]
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem csx_csxa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
+#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1/
+qed.
+
+theorem csxa_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1/
+qed.
+
+(* Basic_1: was just: sn3_pr3_trans *)
+lemma csx_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csx_cpx_trans/
+qed-.
+
+(* Main eliminators *********************************************************)
+
+lemma csx_ind_alt: ∀h,g,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 #T1 #HT1 #IHT1
+@H0 -H0 /2 width=1/ -HT1 /3 width=1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cnx_lift.ma".
+include "basic_2/computation/acp.ma".
+include "basic_2/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was just: sn3_lift *)
+lemma csx_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+ ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+@csx_intro #T #HLT2 #HT2
+elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1/
+qed.
+
+(* Basic_1: was just: sn3_gen_lift *)
+lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,d,e. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+ ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+@csx_intro #T #HLT2 #HT2
+elim (lift_total T d e) #T0 #HT0
+lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1/
+qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_abbr *)
+lemma csx_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
+#h #g #I #G #L #K #V #i #HLK #HV
+@csx_intro #X #H #Hi
+elim (cpx_inv_lref1 … H) -H
+[ #H destruct elim Hi //
+| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
+ lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
+ @(csx_lift … HLK HV1) -HLK -HV1
+ @(csx_cpx_trans … HV) -HV //
+]
+qed.
+
+lemma csx_appl_simple: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
+ 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
+#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
+@csx_intro #X #H1 #H2
+elim (cpx_inv_appl1_simple … H1) // -H1
+#V0 #T0 #HLV0 #HLT10 #H destruct
+elim (eq_false_inv_tpair_dx … H2) -H2
+[ -IHV -HT1 #HT10
+ @(csx_cpx_trans … (ⓐV.T0)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // /2 width=1/
+| -HLT10 * #H #HV0 destruct
+ @IHV -IHV // -HT1 /2 width=1/ -HV0
+ #T2 #HLT02 #HT02
+ @(csx_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // -HLT02 /2 width=1/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: sn3_gen_def *)
+lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
+#h #g #I #G #L #K #V #i #HLK #Hi
+elim (lift_total V 0 (i+1)) #V0 #HV0
+lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
+@(csx_inv_lift … H0LK … HV0) -H0LK
+@(csx_cpx_trans … Hi) -Hi /2 width=7/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem csx_acp: ∀h,g. acp (cpx h g) (eq …) (csx h g).
+#h #g @mk_acp
+[ #G #L elim (deg_total h g 0)
+ #l #Hl lapply (cnx_sort_iter … L … Hl) /2 width=2/
+| @cnx_lift
+| /2 width=3 by csx_fwd_flat_dx/
+| /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/grammar/tstc_tstc.ma".
+include "basic_2/computation/cpxs_cpxs.ma".
+include "basic_2/computation/csx_alt.ma".
+include "basic_2/computation/csx_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Advanced properties ******************************************************)
+
+lemma csx_lpx_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T #T #_ #IHT
+@csx_intro #T0 #HLT0 #HT0
+@IHT /2 width=2/ -IHT -HT0 /2 width=3 by lpx_cpx_trans/
+qed.
+
+lemma csx_abst: ∀h,g,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+ ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
+#h #g #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_abst1 … H1) -H1
+#W0 #T0 #HLW0 #HLT0 #H destruct
+elim (eq_false_inv_tpair_sn … H2) -H2
+[ -IHT #H lapply (csx_cpx_trans … HLT0) // -HT
+ #HT0 lapply (csx_lpx_conf … (L.ⓛW0) … HT0) -HT0 /2 width=1/ /3 width=1/
+| -IHW -HLW0 -HT * #H destruct /3 width=1/
+]
+qed.
+
+lemma csx_abbr: ∀h,g,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
+ ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
+#h #g #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_abbr1 … H1) -H1 *
+[ #V1 #T1 #HLV1 #HLT1 #H destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2
+ [ #HV1 @IHV // /2 width=1/ -HV1
+ @(csx_lpx_conf … (L.ⓓV)) /2 width=1/ -HLV1 /2 width=3 by csx_cpx_trans/
+ | -IHV -HLV1 * #H destruct /3 width=1/
+ ]
+| -IHV -IHT -H2 #T0 #HLT0 #HT0
+ lapply (csx_cpx_trans … HT … HLT0) -T #HLT0
+ lapply (csx_inv_lift … HLT0 … HT0) -T0 /2 width=3/
+]
+qed.
+
+fact csx_appl_beta_aux: ∀h,g,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
+ ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
+#h #g #a #G #L #X #H @(csx_ind … H) -X
+#X #HT1 #IHT1 #V #W #T1 #H1 destruct
+@csx_intro #X #H1 #H2
+elim (cpx_inv_appl1 … H1) -H1 *
+[ -HT1 #V0 #Y #HLV0 #H #H0 destruct
+ elim (cpx_inv_abst1 … H) -H #W0 #T0 #HLW0 #HLT0 #H destruct
+ @IHT1 -IHT1 [4: // | skip |3: #H destruct /2 width=1/ ] -H2
+ lapply (lsubr_cpx_trans … HLT0 (L.ⓓⓝW.V) ?) -HLT0 [ /2 width=1/ ] /3 width=1/
+| -IHT1 -H2 #b #V0 #W0 #W2 #T0 #T2 #HLV0 #HLW02 #HLT02 #H1 #H3 destruct
+ lapply (lsubr_cpx_trans … HLT02 (L.ⓓⓝW0.V) ?) -HLT02 [ /2 width=1/ ] #HT02
+ @(csx_cpx_trans … HT1) -HT1 /3 width=1/
+| -HT1 -IHT1 -H2 #b #V0 #V1 #W0 #W1 #T0 #T3 #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: sn3_beta *)
+lemma csx_appl_beta: ∀h,g,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
+/2 width=3 by csx_appl_beta_aux/ qed.
+
+fact csx_appl_theta_aux: ∀h,g,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+ ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
+#h #g #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+lapply (csx_fwd_pair_sn … HVT) #HV
+lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1 … HL) -HL *
+[ -HV #V0 #Y #HLV10 #HL #H0 destruct
+ elim (cpx_inv_abbr1 … HL) -HL *
+ [ #V3 #T3 #HV3 #HLT3 #H0 destruct
+ elim (lift_total V0 0 1) #V4 #HV04
+ elim (term_eq_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V3.ⓐV4.T3))
+ [ -IHVT #H0 destruct
+ elim (eq_false_inv_tpair_sn … H) -H
+ [ -HLV10 -HV3 -HLT3 -HVT
+ >(lift_inj … HV12 … HV04) -V4
+ #H elim H //
+ | * #_ #H elim H //
+ ]
+ | -H -HVT #H
+ lapply (cpx_lift … HLV10 (L. ⓓV) … HV12 … HV04) -HLV10 -HV12 /2 width=1/ #HV24
+ @(IHVT … H … HV04) -IHVT // -H -HV04 /4 width=1/
+ ]
+ | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
+ lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1/ -T #HVT0
+ lapply (csx_inv_lift … L … 1 HVT0 ? ? ?) -HVT0 [ /2 width=4/ |2,3: skip | /2 width=1/ ] -V2 -T0 #HVY
+ @(csx_cpx_trans … HVY) /2 width=1/
+ ]
+| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
+| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
+ lapply (cpx_lift … HLV10 (L. ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
+ @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
+ @(csx_lpx_conf … (L. ⓓW0)) /2 width=1/ -W1
+ @(csx_cpxs_trans … HVT) -HVT /3 width=1/
+]
+qed-.
+
+lemma csx_appl_theta: ∀h,g,a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
+ ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
+/2 width=5 by csx_appl_theta_aux/ qed.
+
+(* Basic_1: was just: sn3_appl_appl *)
+lemma csx_appl_simple_tstc: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≃ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
+ 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
+#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1_simple … HL) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (eq_false_inv_tpair_sn … H) -H
+[ -IHT1 #HV0
+ @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
+ @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
+ #T2 #HLT12 #HT12
+ @(csx_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+ @H2T1 -H2T1 // -HLT12 /2 width=1/
+| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
+ elim (tstc_dec T1 T0) #H2T10
+ [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
+ #T2 #HLT02 #HT02
+ @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
+ | -IHT1 -H3T1 -H1T10
+ @H2T1 -H2T1 /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/computation/acp_cr.ma".
+include "basic_2/computation/cpxs_tstc_vector.ma".
+include "basic_2/computation/csx_lpx.ma".
+include "basic_2/computation/csx_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_appls_lref *)
+lemma csx_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
+ ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
+#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
+#V #Vs #IHV #H
+elim (csxv_inv_cons … H) -H #HV #HVs
+@csx_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs
+#X #H #H0
+lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
+elim (H0) -H0 //
+qed.
+
+lemma csx_applv_sort: ∀h,g,G,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
+#h #g #G #L #k elim (deg_total h g k)
+#l generalize in match k; -k @(nat_ind_plus … l) -l [ /3 width=1/ ]
+#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
+#Hkl #Vs elim Vs -Vs /2 width=1/
+#V #Vs #IHVs #HVVs
+elim (csxv_inv_cons … HVVs) #HV #HVs
+@csx_appl_simple_tstc // -HV /2 width=1/ -IHVs -HVs
+#X #H #H0
+elim (cpxs_fwd_sort_vector … H) -H #H
+[ elim H0 -H0 //
+| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_beta *)
+lemma csx_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
+#h #g #a #G #L #Vs elim Vs -Vs /2 width=1/
+#V0 #Vs #IHV #V #W #T #H1T
+lapply (csx_fwd_pair_sn … H1T) #HV0
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T
+#X #H #H0
+elim (cpxs_fwd_beta_vector … H) -H #H
+[ -H1T elim H0 -H0 //
+| -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
+]
+qed.
+
+lemma csx_applv_delta: ∀h,g,I,G,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
+ ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
+#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ #H
+ lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
+ lapply (csx_inv_lift … H … HLK0 HV12) -V2 -HLK0 /2 width=5/
+| #V #Vs #IHV #H1T
+ lapply (csx_fwd_pair_sn … H1T) #HV
+ lapply (csx_fwd_flat_dx … H1T) #H2T
+ @csx_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
+ #X #H #H0
+ elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
+ [ -H1T elim H0 -H0 //
+ | -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
+ ]
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_abbr *)
+lemma csx_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
+#h #g #a #G #L #V1s #V2s * -V1s -V2s /2 width=1/
+#V1s #V2s #V1 #V2 #HV12 #H
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1s -V2s /2 width=3/
+#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H
+lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
+lapply (csx_fwd_pair_sn … H) #HW1
+lapply (csx_fwd_flat_dx … H) #H1
+@csx_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
+[ -H #H elim H2 -H2 //
+| -H2 #H1 @(csx_cpxs_trans … H) -H /2 width=1/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_cast *)
+lemma csx_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
+#h #g #G #L #Vs elim Vs -Vs /2 width=1/
+#V #Vs #IHV #W #T #H1W #H1T
+lapply (csx_fwd_pair_sn … H1W) #HV
+lapply (csx_fwd_flat_dx … H1W) #H2W
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_tstc // -HV /2 width=1/ -IHV -H2W -H2T
+#X #H #H0
+elim (cpxs_fwd_cast_vector … H) -H #H
+[ -H1W -H1T elim H0 -H0 //
+| -H1W -H0 @(csx_cpxs_trans … H1T) -H1T /2 width=1/
+| -H1T -H0 @(csx_cpxs_trans … H1W) -H1W /2 width=1/
+]
+qed.
+
+theorem csx_acr: ∀h,g. acr (cpx h g) (eq …) (csx h g) (λG,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T).
+#h #g @mk_acr //
+[ /3 width=1/
+|2,3,6: /2 width=1/
+| /2 width=7/
+| #G #L #V1s #V2s #HV12s #a #V #T #H #HV
+ @(csx_applv_theta … HV12s) -HV12s
+ @(csx_abbr) //
+| @csx_lift
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/term_vector.ma".
+include "basic_2/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
+ λh,g,G,L. all … (csx h g G L).
+
+interpretation
+ "context-sensitive strong normalization (term vector)"
+ 'SN h g G L Ts = (csxv h g G L Ts).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma csxv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
+normalize // qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma csx_fwd_applv: ∀h,g,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Ⓐ Vs.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #Vs elim Vs -Vs /2 width=1/
+#V #Vs #IHVs #HVs
+lapply (csx_fwd_pair_sn … HVs) #HV
+lapply (csx_fwd_flat_dx … HVs) -HVs #HVs
+elim (IHVs HVs) -IHVs -HVs /3 width=1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/btpredstarproper_8.ma".
+include "basic_2/reduction/fpbc.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
+
+inductive fpbg (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbg_inj : ∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ fpbg h g G1 L1 T1 G2 L2 T2
+| fpbg_step: ∀G,L,L2,T. fpbg h g G1 L1 T1 G L T → ⦃G, L⦄ ⊢ ➡[h, g] L2 → fpbg h g G1 L1 T1 G L2 T
+.
+
+interpretation "'big tree' proper parallel computation (closure)"
+ 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
+
+(* Basic forvard lemmas *****************************************************)
+
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+/3 width=5 by fpbs_strap1, fpbc_fpb, fpb_lpx/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_inj, fpbg_step/ qed.
+
+lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lapply (fpbg_fwd_fpbs … H1) #H0
+elim (fpb_inv_fpbc … H2) -H2 [| * #HG2 #HL2 #HT2 destruct ]
+/2 width=5 by fpbg_inj, fpbg_step/
+qed-.
+
+lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -G2 -L2 -T2
+/3 width=5 by fpbg_step, fpbg_inj, fpbs_strap2/
+qed-.
+
+lemma fpbg_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(fpbs_ind … HT2) -G2 -L2 -T2
+/2 width=5 by fpbg_strap1/
+qed-.
+
+lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #HT1 @(fpbs_ind … HT1) -G -L -T
+/3 width=5 by fpbg_strap2/
+qed-.
+
+lemma fsupp_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2
+/3 width=5 by fsupp_fpbs, fpbc_fsup, fpbc_fpbg, fpbg_inj/
+qed.
+
+lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+[ #H elim H //
+| #T #T2 #_ #HT2 #IHT1 #HT12
+ elim (term_eq_dec T1 T) #H destruct
+ [ -IHT1 /4 width=1/
+ | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
+ @(fpbg_strap1 … HT1) -HT1 /2 width=1 by fpb_cpx/
+ ]
+]
+qed.
+
+lemma cprs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+/3 width=1 by cprs_cpxs, cpxs_fpbg/ 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/computation/fpbg.ma".
+
+(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+
+(* Main properties **********************************************************)
+
+theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
+/3 width=5 by fpbg_fwd_fpbs, fpbg_fpbs_trans/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/lsstas_lift.ma".
+include "basic_2/reduction/fpb_lift.ma".
+include "basic_2/reduction/fpbc_lift.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lsstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 → (T1 = T2 → ⊥) →
+ ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #l2 #H @(lsstas_ind_dx … H) -l2 -T2
+[ #H elim H //
+| #l2 #T #T2 #HT1 #HT2 #IHT1 #HT12 #l1 #Hl21
+ elim (term_eq_dec T1 T) #H destruct [ -IHT1 |]
+ [ elim (le_inv_plus_l … Hl21) -Hl21 #_ #Hl1
+ >(plus_minus_m_m … Hl1) -Hl1 /3 width=5 by ssta_fpbc, fpbg_inj/
+ | #Hl1 >commutative_plus in Hl21; #Hl21
+ elim (le_inv_plus_l … Hl21) -Hl21 #Hl12 #Hl21
+ lapply (lsstas_da_conf … HT1 … Hl1) -HT1
+ >(plus_minus_m_m … Hl12) -Hl12
+ /4 width=5 by ssta_fpb, fpbg_strap1/
+ ]
+]
+qed.
include "basic_2/notation/relations/btpredstar_8.ma".
include "basic_2/substitution/fsupp.ma".
include "basic_2/reduction/fpb.ma".
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/lprs.ma".
+include "basic_2/computation/cpxs.ma".
+include "basic_2/computation/lpxs.ma".
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
/3 width=5 by fpb_fsup, tri_step, fpb_fpbs/
qed.
-lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2
-/3 width=5 by fpb_cpr, fpbs_strap1/
+lemma cpxs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpb_cpx, fpbs_strap1/
qed.
-lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
-#h #g #G #L1 #L2 #T #H @(lprs_ind … H) -L2
-/3 width=5 by fpb_lpr, fpbs_strap1/
+lemma lpxs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
+/3 width=5 by fpb_lpx, fpbs_strap1/
qed.
+lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
+
+lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
+
lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, fpb_lpr, fpb_cpr/ qed.
-
-lemma ssta_fpbs: ∀h,g,G,L,T,U,l.
- ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h, g] U →
- ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
-/3 width=2 by fpb_fpbs, fpb_ssta/ qed.
+/4 width=5 by fpbs_strap1, lpr_fpb, cpr_fpb/ qed.
(* *)
(**************************************************************************)
-include "basic_2/computation/fpbs.ma".
+include "basic_2/computation/fpbs_lift.ma".
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
(**************************************************************************)
include "basic_2/unfold/lsstas_lift.ma".
+include "basic_2/reduction/fpb_lift.ma".
include "basic_2/computation/fpbs.ma".
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
(* Advanced properties ******************************************************)
+lemma ssta_fpbs: ∀h,g,G,L,T,U,l.
+ ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h, g] U →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
+/3 width=2 by fpb_fpbs, ssta_fpb/ qed.
+
lemma lsstas_fpbs: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 →
∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
#h #g #G #L #T1 #T2 #l2 #H @(lsstas_ind_dx … H) -l2 -T2 //
elim (le_inv_plus_l … Hl21) -Hl21 #Hl12 #Hl21
lapply (lsstas_da_conf … HT1 … Hl1) -HT1
>(plus_minus_m_m … Hl12) -Hl12
-/3 width=5 by fpb_ssta, fpbs_strap1/
+/3 width=5 by ssta_fpb, fpbs_strap1/
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/btpredstarproper_8.ma".
-include "basic_2/reduction/ysc.ma".
-include "basic_2/computation/fpbs.ma".
-
-(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
-
-inductive ygt (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| ygt_inj : ∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ygt h g G1 L1 T1 G2 L2 T2
-| ygt_step: ∀G,L,L2,T. ygt h g G1 L1 T1 G L T → ⦃G, L⦄ ⊢ ➡ L2 → ygt h g G1 L1 T1 G L2 T
-.
-
-interpretation "'big tree' proper parallel computation (closure)"
- 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (ygt h g G1 L1 T1 G2 L2 T2).
-
-(* Basic forvard lemmas *****************************************************)
-
-lemma ygt_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
-/3 width=5 by fpbs_strap1, ysc_fpb, fpb_lpr/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ysc_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by ygt_inj, ygt_step/ qed.
-
-lemma ygt_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
-lapply (ygt_fwd_fpbs … H1) #H0
-elim (fpb_inv_ysc … H2) -H2 [| * #HG2 #HL2 #HT2 destruct ]
-/2 width=5 by ygt_inj, ygt_step/
-qed-.
-
-lemma ygt_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -G2 -L2 -T2
-/3 width=5 by ygt_step, ygt_inj, fpbs_strap2/
-qed-.
-
-lemma ygt_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(fpbs_ind … HT2) -G2 -L2 -T2
-/2 width=5 by ygt_strap1/
-qed-.
-
-lemma fpbs_ygt_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #HT1 @(fpbs_ind … HT1) -G -L -T
-/3 width=5 by ygt_strap2/
-qed-.
-
-lemma fsupp_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2
-/3 width=5 by fsupp_fpbs, ysc_fsup, ysc_ygt, ygt_inj/
-qed.
-
-lemma cprs_ygt: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
- ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2
-[ #H elim H //
-| #T #T2 #_ #HT2 #IHT1 #HT12
- elim (term_eq_dec T1 T) #H destruct
- [ -IHT1 /4 width=1/
- | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
- @(ygt_strap1 … HT1) -HT1 /2 width=1 by fpb_cpr/
- ]
-]
-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/lsstas_lift.ma".
-include "basic_2/computation/ygt.ma".
-
-(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lsstas_ygt: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 → (T1 = T2 → ⊥) →
- ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #l2 #H @(lsstas_ind_dx … H) -l2 -T2
-[ #H elim H //
-| #l2 #T #T2 #HT1 #HT2 #IHT1 #HT12 #l1 #Hl21
- elim (term_eq_dec T1 T) #H destruct [ -IHT1 |]
- [ elim (le_inv_plus_l … Hl21) -Hl21 #_ #Hl1
- >(plus_minus_m_m … Hl1) -Hl1 /3 width=5 by ysc_ssta, ygt_inj/
- | #Hl1 >commutative_plus in Hl21; #Hl21
- elim (le_inv_plus_l … Hl21) -Hl21 #Hl12 #Hl21
- lapply (lsstas_da_conf … HT1 … Hl1) -HT1
- >(plus_minus_m_m … Hl12) -Hl12
- /4 width=5 by fpb_ssta, ygt_strap1/
- ]
-]
-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/computation/ygt.ma".
-
-(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
-
-(* Main properties **********************************************************)
-
-theorem ygt_trans: ∀h,g. tri_transitive … (ygt h g).
-/3 width=5 by ygt_fwd_fpbs, ygt_fpbs_trans/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/computation/csn_aaa.ma".
+include "basic_2/computation/csx_aaa.ma".
include "basic_2/computation/cpds_aaa.ma".
include "basic_2/equivalence/cpcs_aaa.ma".
include "basic_2/dynamic/snv.ma".
]
qed-.
-lemma snv_fwd_csn: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma snv_fwd_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
#h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
qed-.
include "basic_2/unfold/lsstas_lsstas.ma".
include "basic_2/computation/fpbs_lift.ma".
-include "basic_2/computation/ygt.ma".
+include "basic_2/computation/fpbg.ma".
include "basic_2/equivalence/cpes_cpds.ma".
include "basic_2/dynamic/snv.ma".
∀G,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
-@(cprs_ind … H) -T2 /4 width=6 by ygt_fpbs_trans, cprs_fpbs/
+@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
qed-.
fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
-@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, ygt_fpbs_trans, cprs_fpbs/
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
qed-.
fact da_cpcs_aux: ∀h,g,G0,L0,T0.
elim (IH1 … Hl21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
[2: /3 width=12 by da_cprs_lpr_aux/
|3: /3 width=10 by snv_cprs_lpr_aux/
-|4: /3 width=5 by ygt_fpbs_trans, cprs_fpbs/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
] -G0 -L0 -T0 -T1 -T -l1 #U2 #HTU2 #HU2
/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
qed-.
| lapply (lsstas_da_conf … HT1T … Hl1) #Hl1l
lapply (lsstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
elim (lsstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L)
- /3 width=8 by ygt_fpbs_trans, lsstas_fpbs, monotonic_le_minus_l/ -T #U2 #HTU2 #HU12
+ /3 width=8 by fpbg_fpbs_trans, lsstas_fpbs, monotonic_le_minus_l/ -T #U2 #HTU2 #HU12
/3 width=5 by cpcs_cpes, ex3_2_intro/
]
qed-.
lapply (fsupp_lref … G1 … HLK1)
elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- /4 width=10 by da_ldef, da_ldec, fsupp_ygt/
+ /4 width=10 by da_ldef, da_ldec, fsupp_fpbg/
|2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
lapply (ldrop_mono … H … HLK1) -H #H destruct
lapply (fsupp_lref … G1 … HLK1)
elim (lpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
lapply (ldrop_fwd_ldrop2 … HLK2) -V2
- /4 width=7 by da_lift, fsupp_ygt/
+ /4 width=7 by da_lift, fsupp_fpbg/
]
| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
lapply (da_inv_bind … H2) -H2
elim (cpr_inv_bind1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=9 by da_bind, fsupp_ygt, lpr_pair/
+ /4 width=9 by da_bind, fsupp_fpbg, lpr_pair/
| #T2 #HT12 #HT2 #H1 #H2 destruct
- /4 width=11 by da_inv_lift, fsupp_ygt, lpr_pair, ldrop_ldrop/
+ /4 width=11 by da_inv_lift, fsupp_fpbg, lpr_pair, ldrop_ldrop/
]
| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct
elim (snv_inv_appl … H1) -H1 #b0 #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
lapply (da_inv_flat … H2) -H2 #Hl
elim (cpr_inv_appl1 … H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fsupp_ygt/
+ [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fsupp_fpbg/
| #b #V2 #W #W2 #U1 #U2 #HV12 #HW2 #HU12 #H1 #H2 destruct
elim (snv_inv_bind … HT1) -HT1 #HW #HU1
lapply (da_inv_bind … Hl) -Hl #Hl
lapply (cprs_div … HW3 … HW10) -W3 #HWW1
lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #H
elim (snv_fwd_da … HW) #l1 #Hl1
- lapply (IH3 … HV1 … 1 … Hl0 W1 ?) /2 width=2 by fsupp_ygt, ssta_lsstas/ #HW1
+ lapply (IH3 … HV1 … 1 … Hl0 W1 ?) /2 width=2 by fsupp_fpbg, ssta_lsstas/ #HW1
lapply (da_cpcs_aux … IH2 IH1 … Hl1 … H … HWW1) -H
- /3 width=5 by ygt_fpbs_trans, fsupp_ygt, ssta_fpbs/ #H destruct
- lapply (IH1 … HV1 … Hl0 … HV12 … HL12) -HV1 -Hl0 -HV12 [ /2 by fsupp_ygt/ ] #Hl0
- lapply (IH1 … Hl1 … HW2 … HL12) -Hl1 // /2 width=1 by fsupp_ygt/ -HW
- lapply (IH1 … HU1 … Hl … HU12 (L2.ⓛW2) ?) -IH1 -HU1 -Hl -HU12 [1,2: /2 by fsupp_ygt, lpr_pair/ ] -HL12 -HW2
+ /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, ssta_fpbs/ #H destruct
+ lapply (IH1 … HV1 … Hl0 … HV12 … HL12) -HV1 -Hl0 -HV12 [ /2 by fsupp_fpbg/ ] #Hl0
+ lapply (IH1 … Hl1 … HW2 … HL12) -Hl1 // /2 width=1 by fsupp_fpbg/ -HW
+ lapply (IH1 … HU1 … Hl … HU12 (L2.ⓛW2) ?) -IH1 -HU1 -Hl -HU12 [1,2: /2 by fsupp_fpbg, lpr_pair/ ] -HL12 -HW2
/4 width=6 by da_bind, lsubd_da_trans, lsubd_abbr/
| #b #V #V2 #W #W2 #U1 #U2 #HV1 #HV2 #HW2 #HU12 #H1 #H2 destruct -IH3 -IH2 -V -W0 -T0 -l0 -HV1 -HVW1
elim (snv_inv_bind … HT1) -HT1 #_
lapply (da_inv_bind … Hl) -Hl
- /5 width=9 by da_bind, da_flat, fsupp_ygt, lpr_pair/
+ /5 width=9 by da_bind, da_flat, fsupp_fpbg, lpr_pair/
]
| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
lapply (da_inv_flat … H2) -H2 #Hl
elim (cpr_inv_cast1 … H3) -H3
- [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fsupp_ygt/
- | /3 width=7 by fsupp_ygt/
+ [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fsupp_fpbg/
+ | /3 width=7 by fsupp_fpbg/
]
]
qed-.
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
lapply (fsupp_lref … G1 … HLK1) #HKL
elim (cpr_inv_lref1 … H2) -H2
- [ #H destruct -HLK1 /4 width=10 by fsupp_ygt, snv_lref/
+ [ #H destruct -HLK1 /4 width=10 by fsupp_fpbg, snv_lref/
| * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 /4 width=7 by fsupp_ygt, snv_lift/
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 /4 width=7 by fsupp_fpbg, snv_lift/
]
| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
elim (cpr_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fsupp_ygt, snv_bind, lpr_pair/
+ [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fsupp_fpbg, snv_bind, lpr_pair/
| #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
- /4 width=10 by fsupp_ygt, snv_inv_lift, lpr_pair, ldrop_ldrop/
+ /4 width=10 by fsupp_fpbg, snv_inv_lift, lpr_pair, ldrop_ldrop/
]
| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1
elim (cpr_inv_appl1 … H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct -IH4
- lapply (IH1 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HV2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fsupp_ygt/ #HT2
- lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #H2l0
- elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fsupp_ygt/ -HV1 #W2 #HVW2 #HW12
- elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fsupp_ygt/ -HT12 -HTU1 #X #HTU2 #H
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #HV2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fsupp_fpbg/ #HT2
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #H2l0
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fsupp_fpbg/ -HV1 #W2 #HVW2 #HW12
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fsupp_fpbg/ -HT12 -HTU1 #X #HTU2 #H
elim (cprs_inv_abst1 … H) -H #W20 #U2 #HW120 #_ #H destruct
lapply (lpr_cprs_conf … HL12 … HW10) -L1 #HW10
lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
elim (cpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
lapply (cprs_div … HW10 … HW230) -W30 #HW120
- lapply (snv_ssta_aux … IH4 … Hl0 … HVW1) /2 width=1 by fsupp_ygt/ #HW10
+ lapply (snv_ssta_aux … IH4 … Hl0 … HVW1) /2 width=1 by fsupp_fpbg/ #HW10
lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #HlW10
elim (snv_fwd_da … HW20) #l #Hl
lapply (da_cpcs_aux … IH1 IH2 … HlW10 … Hl … HW120) // -HlW10
- /3 width=5 by ygt_fpbs_trans, fsupp_ygt, ssta_fpbs/ #H destruct
- lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HlV2
- lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fsupp_ygt/ #HlW2
- elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #W3 #HV2W3 #HW103
+ /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, ssta_fpbs/ #H destruct
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #HlV2
+ lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fsupp_fpbg/ #HlW2
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #W3 #HV2W3 #HW103
lapply (ssta_da_conf … HV2W3 … HlV2) <minus_plus_m_m #HlW3
lapply (cpcs_cpr_strap1 … HW120 … HW202) -HW120 #HW102
lapply (lpr_cpcs_conf … HL12 … HW102) -HW102 #HW102
lapply (cpcs_canc_sn … HW103 … HW102) -W10 #HW32
- lapply (IH1 … HV12 … HL12) /2 width=1 by fsupp_ygt/ -HV1 #HV2
- lapply (IH1 … HW202 … HL12) /2 width=1 by fsupp_ygt/ -HW20 #HW2
- lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fsupp_ygt, lpr_pair/ -HT20 #HT2
+ lapply (IH1 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ -HV1 #HV2
+ lapply (IH1 … HW202 … HL12) /2 width=1 by fsupp_fpbg/ -HW20 #HW2
+ lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fsupp_fpbg, lpr_pair/ -HT20 #HT2
lapply (snv_ssta_aux … IH4 … HlV2 … HV2W3)
- /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/ #HW3
+ /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_fpbs/ #HW3
lapply (lsubsv_snv_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /3 width=3 by snv_bind, snv_cast/
- @(lsubsv_abbr … l) /3 width=7 by fsupp_ygt/ #W #W0 #l0 #Hl0 #HV2W #HW20
+ @(lsubsv_abbr … l) /3 width=7 by fsupp_fpbg/ #W #W0 #l0 #Hl0 #HV2W #HW20
lapply (lsstas_ssta_conf_pos … HV2W3 … HV2W) -HV2W #HW3W
@(lsstas_cpcs_lpr_aux … IH1 IH2 IH3 … HlW3 … HW3W … HlW2 … HW20 … HW32) //
- [ /3 width=9 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_ssta_fpbs/
- | /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/
+ [ /3 width=9 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_ssta_fpbs/
+ | /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_fpbs/
]
| #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
elim (cpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
lapply (lpr_cprs_conf … HL12 … HW10) -HW10 #HW10
- elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fsupp_ygt, lpr_pair/ -HTU0 #X #HTU2 #H
+ elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fsupp_fpbg, lpr_pair/ -HTU0 #X #HTU2 #H
elim (cprs_inv_abst1 … H) -H #W #U2 #HW1 #_ #H destruct -U3
- elim (ssta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fsupp_ygt/ -IH3 -HVW1 #X #H1 #H2
+ elim (ssta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fsupp_fpbg/ -IH3 -HVW1 #X #H1 #H2
lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
elim (lift_total X 0 1) #W20 #H3
lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1 by ldrop_ldrop/ -H1 #HVW20
lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) /2 width=1 by cprs_bind/ -HW3 -HTU2 #HTU2
- lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fsupp_ygt/ -IH2 -Hl0 #Hl0
+ lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fsupp_fpbg/ -IH2 -Hl0 #Hl0
lapply (da_lift … Hl0 (L2.ⓓW2) … HV02) /2 width=1 by ldrop_ldrop/ -Hl0 #Hl0
- lapply (IH1 … HW02 … HL12) /2 width=1 by fsupp_ygt/ -HW0 #HW2
- lapply (IH1 … HV10 … HL12) /2 width=1 by fsupp_ygt/ -HV1 -HV10 #HV0
- lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fsupp_ygt, lpr_pair/ -L1 #HT2
+ lapply (IH1 … HW02 … HL12) /2 width=1 by fsupp_fpbg/ -HW0 #HW2
+ lapply (IH1 … HV10 … HL12) /2 width=1 by fsupp_fpbg/ -HV1 -HV10 #HV0
+ lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fsupp_fpbg, lpr_pair/ -L1 #HT2
lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /3 width=7 by snv_bind, snv_appl, ldrop_ldrop/
]
| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
elim (cpr_inv_cast1 … H2) -H2
[ * #W2 #T2 #HW12 #HT12 #H destruct
lapply (cpcs_cprs_strap1 … HUW1 W2 ?) /2 width=1 by cpr_cprs/ -HUW1 #H1
- lapply (IH1 … HW12 … HL12) /2 width=1 by fsupp_ygt/ -HW1 -HW12 #HW2
- lapply (IH1 … HT12 … HL12) /2 width=1 by fsupp_ygt/ -IH1 #HT2
- elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fsupp_ygt/ -IH3 -HTU1 #U2 #HTU2 #HU12
- lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fsupp_ygt/ -IH2 -HT1 -HT12 -Hl0 #Hl0
+ lapply (IH1 … HW12 … HL12) /2 width=1 by fsupp_fpbg/ -HW1 -HW12 #HW2
+ lapply (IH1 … HT12 … HL12) /2 width=1 by fsupp_fpbg/ -IH1 #HT2
+ elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fsupp_fpbg/ -IH3 -HTU1 #U2 #HTU2 #HU12
+ lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fsupp_fpbg/ -IH2 -HT1 -HT12 -Hl0 #Hl0
/4 width=7 by snv_cast, lpr_cpcs_conf, cpcs_canc_sn/
| #H -IH3 -IH2 -HW1 -HTU1 -HUW1
- lapply (IH1 … H … HL12) /2 width=1 by fsupp_ygt/
+ lapply (IH1 … H … HL12) /2 width=1 by fsupp_fpbg/
]
]
qed-.
lapply (ldrop_mono … HLK0 … HLK1) -HLK0 #H destruct
[ lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21 ]
lapply (fsupp_lref … G1 … HLK1) #H
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /4 width=8 by fsupp_ygt, snv_lift/
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /4 width=8 by fsupp_fpbg, snv_lift/
| #p #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
- elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fsupp_ygt, snv_bind/
+ elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fsupp_fpbg, snv_bind/
| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct
elim (snv_inv_appl … H1) -H1 #a #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
lapply (da_inv_flat … Hl1) -Hl1 #Hl1
elim (lsstas_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
- lapply (IH1 … HT1 … Hl1 … HTU1) /2 width=1 by fsupp_ygt/ #HU1
- elim (lsstas_cpds_aux … IH1 IH4 IH3 IH2 … Hl1 … HTU1 … HT10) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fsupp_ygt/ -T1 -l1 #X #l #_ #H #HU10 -l2
+ lapply (IH1 … HT1 … Hl1 … HTU1) /2 width=1 by fsupp_fpbg/ #HU1
+ elim (lsstas_cpds_aux … IH1 IH4 IH3 IH2 … Hl1 … HTU1 … HT10) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fsupp_fpbg/ -T1 -l1 #X #l #_ #H #HU10 -l2
elim (lsstas_inv_bind1 … H) -H #U0 #_ #H destruct -T0 -l
elim (cpes_inv_abst2 … HU10) -HU10 #W2 #U2 #HU12 #HU02
elim (cprs_inv_abst … HU02) -HU02 #HW02 #_
[ lapply (lsstas_inv_O … H2) -H2 #H destruct // ]
elim (snv_inv_cast … H1) -H1
lapply (da_inv_flat … Hl1) -Hl1
- lapply (lsstas_inv_cast1 … H2) -H2 /3 width=8 by fsupp_ygt/
+ lapply (lsstas_inv_cast1 … H2) -H2 /3 width=8 by fsupp_fpbg/
]
qed-.
|2,4: * #K0 #V0 #X0 #H #HVX0 #HX0
lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
]
- [ elim (IH1 … HWl1 … HW0 … HW12 … HK12) -IH1 -HW0 /2 width=1 by fsupp_ygt/ #V2 #HWV2 #HV2
+ [ elim (IH1 … HWl1 … HW0 … HW12 … HK12) -IH1 -HW0 /2 width=1 by fsupp_fpbg/ #V2 #HWV2 #HV2
elim (lift_total V2 0 (i+1))
- /6 width=11 by fsupp_ygt, cpcs_lift, lsstas_ldec, ex2_intro/
- | elim (IH1 … HVl1 … HW0 … HV12 … HK12) -IH1 -HVl1 -HW0 -HV12 -HK12 -IH2 /2 width=1 by fsupp_ygt/ #W2 #HVW2 #HW02
+ /6 width=11 by fsupp_fpbg, cpcs_lift, lsstas_ldec, ex2_intro/
+ | elim (IH1 … HVl1 … HW0 … HV12 … HK12) -IH1 -HVl1 -HW0 -HV12 -HK12 -IH2 /2 width=1 by fsupp_fpbg/ #W2 #HVW2 #HW02
elim (lift_total W2 0 (i+1))
/4 width=11 by cpcs_lift, lsstas_ldef, ex2_intro/
- | elim (IH1 … HVl1 … HW0 … HVX0 … HK12) -IH1 -HVl1 -HW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fsupp_ygt/ -l1 #W2 #HXW2 #HW02
+ | elim (IH1 … HVl1 … HW0 … HVX0 … HK12) -IH1 -HVl1 -HW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fsupp_fpbg/ -l1 #W2 #HXW2 #HW02
elim (lift_total W2 0 (i+1))
/3 width=11 by cpcs_lift, lsstas_lift, ex2_intro/
]
elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
elim (cpr_inv_bind1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fsupp_ygt, lpr_pair/ -T1
+ elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fsupp_fpbg, lpr_pair/ -T1
/4 width=5 by cpcs_bind2, lpr_cpr_conf, lsstas_bind, ex2_intro/
| #T3 #HT13 #HXT3 #H1 #H2 destruct
- elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fsupp_ygt, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
+ elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fsupp_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
elim (lsstas_inv_lift1 … HTU3 L2 … HXT3) -T3
/5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, ldrop_ldrop, ex2_intro/
]
elim (cpr_inv_appl1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH4 -IH3 -IH2
elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1
- /4 width=5 by fsupp_ygt, cpcs_flat, lpr_cpr_conf, lsstas_appl, ex2_intro/
+ /4 width=5 by fsupp_fpbg, cpcs_flat, lpr_cpr_conf, lsstas_appl, ex2_intro/
| #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
lapply (cprs_div … HW10 … HW20) -W0 #HW12
lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #H
elim (snv_fwd_da … HW2) #l #Hl
- lapply (IH4 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fsupp_ygt, ssta_lsstas/ #HW1
+ lapply (IH4 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fsupp_fpbg, ssta_lsstas/ #HW1
lapply (da_cpcs_aux … IH3 IH2 … H … Hl … HW12) // -H
- /3 width=5 by ygt_fpbs_trans, fsupp_ygt, ssta_fpbs/ #H destruct
- lapply (IH3 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HV2
- lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HV2l
- elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fsupp_ygt, ssta_lsstas/ -HVW1 #W4 #H #HW14
+ /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, ssta_fpbs/ #H destruct
+ lapply (IH3 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #HV2
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_fpbg/ #HV2l
+ elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fsupp_fpbg, ssta_lsstas/ -HVW1 #W4 #H #HW14
lapply (lsstas_inv_SO … H) #HV2W4
lapply (ssta_da_conf … HV2W4 … HV2l) <minus_plus_m_m #HW4l
- lapply (IH4 … HV2 … HV2l … H) -H /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/ #HW4
- lapply (IH3 … HW23 … HL12) /2 width=1 by fsupp_ygt/ #HW3
- lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fsupp_ygt/ #HW3l
- elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fsupp_ygt, lpr_pair/ #U3 #HTU3 #HU23
+ lapply (IH4 … HV2 … HV2l … H) -H /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_fpbs/ #HW4
+ lapply (IH3 … HW23 … HL12) /2 width=1 by fsupp_fpbg/ #HW3
+ lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fsupp_fpbg/ #HW3l
+ elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fsupp_fpbg, lpr_pair/ #U3 #HTU3 #HU23
lapply (cpcs_cpr_strap1 … HW12 … HW23) #H
lapply (lpr_cpcs_conf … HL12 … H) -H #H
lapply (cpcs_canc_sn … HW14 H) -H #HW43
@(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
/4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
| -U3
- @(lsubsv_abbr … l) /3 width=7 by fsupp_ygt/
+ @(lsubsv_abbr … l) /3 width=7 by fsupp_fpbg/
#W #W0 #l0 #Hl0 #HV2W #HW30
lapply (lsstas_ssta_conf_pos … HV2W4 … HV2W) -HV2W #HW4W
@(lsstas_cpcs_lpr_aux … IH3 IH2 IH1 … Hl0 … HW4W … Hl0 … HW30 … HW43) //
- [ /3 width=9 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_ssta_fpbs/
- | /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/
+ [ /3 width=9 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_ssta_fpbs/
+ | /3 width=5 by fpbg_fpbs_trans, fsupp_fpbg, cpr_lpr_fpbs/
]
- | -IH1 -IH3 -IH4 /3 width=9 by fsupp_ygt, lpr_pair/
+ | -IH1 -IH3 -IH4 /3 width=9 by fsupp_fpbg, lpr_pair/
]
| #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH4 -IH3 -IH2
elim (snv_inv_bind … HT1) -HT1 #_ #HT0
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
elim (lsstas_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fsupp_ygt, lpr_pair/ -T0 #U2 #HTU2 #HU02
+ elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fsupp_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
elim (cpr_inv_cast1 … H3) -H3
[ * #U2 #T2 #_ #HT12 #H destruct
elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /3 width=3 by fsupp_ygt, lsstas_cast, ex2_intro/
+ /3 width=3 by fsupp_fpbg, lsstas_cast, ex2_intro/
| #HT1X3
elim (IH1 … Hl1 … HTU1 … HT1X3 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /2 width=3 by fsupp_ygt, ex2_intro/
+ /2 width=3 by fsupp_fpbg, ex2_intro/
]
]
qed-.
i: irreducible form
n: normal form
-p: parallel transformation
+p: reflexive parallel transformation
+q: sequential transformation
r: reducible form
-s: sequential transformation
+s: strongly normalizing form
- third letter
- forth letter (if present)
+c: proper single step (successor)
e: reflexive transitive closure to normal form (evaluation)
+g: proper multiple step (greater)
p: non-reflexive transitive closure (plus)
q: reflexive closure (question)
s: reflexive transitive closure (star)
include "basic_2/notation/relations/btpred_8.ma".
include "basic_2/relocation/fsup.ma".
-include "basic_2/static/ssta.ma".
-include "basic_2/reduction/lpr.ma".
+include "basic_2/reduction/lpx.ma".
(* "BIG TREE" PARALLEL REDUCTION FOR CLOSURES *******************************)
inductive fpb (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fsup : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
-| fpb_lpr : ∀L2. ⦃G1, L1⦄ ⊢ ➡ L2 → fpb h g G1 L1 T1 G1 L2 T1
-| fpb_cpr : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡ T2 → fpb h g G1 L1 T1 G1 L1 T2
-| fpb_ssta : ∀T2,l. ⦃G1, L1⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G1, L1⦄ ⊢ T1 •[h, g] T2 → fpb h g G1 L1 T1 G1 L1 T2
+| fpb_fsup: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
+| fpb_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → fpb h g G1 L1 T1 G1 L1 T2
+| fpb_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → fpb h g G1 L1 T1 G1 L2 T1
.
interpretation
(* Basic properties *********************************************************)
lemma fpb_refl: ∀h,g. tri_reflexive … (fpb h g).
-/2 width=1 by fpb_cpr/ qed.
+/2 width=1 by fpb_cpx/ qed.
+
+lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+/3 width=1 by fpb_cpx, cpr_cpx/ qed.
+
+lemma lpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpb_lpx, lpr_lpx/ 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/reduction/cpx_lift.ma".
+include "basic_2/reduction/fpb.ma".
+
+(* "BIG TREE" PARALLEL REDUCTION FOR CLOSURES *******************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ssta_fpb: ∀h,g,G,L,T1,T2,l.
+ ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h, g] T2 →
+ ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+/3 width=5 by fpb_cpx, ssta_cpx/ 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/btpredproper_8.ma".
+include "basic_2/reduction/fpb.ma".
+
+(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
+
+inductive fpbc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbc_fsup : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpbc h g G1 L1 T1 G2 L2 T2
+| fpbc_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpbc h g G1 L1 T1 G1 L1 T2
+.
+
+interpretation
+ "'big tree' proper parallel reduction (closure)"
+ 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpbc_fpb: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/2 width=1 by fpb_fsup, fpb_cpx/
+qed.
+
+lemma cpr_fpbc: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+/3 width=1 by fpbc_cpx, cpr_cpx/ qed.
+
+(* Inversion lemmas on "big tree" parallel reduction for closures ***********)
+
+lemma fpb_inv_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
+ ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡[h, g] L2 & T1 = T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by and3_intro, or_introl, or_intror, fpbc_fsup/
+#T2 #HT12 elim (term_eq_dec T1 T2) #H destruct
+/4 width=1 by and3_intro, or_introl, or_intror, fpbc_cpx/
+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_ssta.ma".
+include "basic_2/reduction/cpx_lift.ma".
+include "basic_2/reduction/fpbc.ma".
+
+(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
+
+(* Advanced properties ******************************************************)
+
+lemma ssta_fpbc: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •[h, g] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (term_eq_dec T1 T2)
+/3 width=5 by fpbc_cpx, ssta_cpx/ #H destruct
+elim (ssta_inv_refl_pos … HT1 … HT12)
+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/btpredproper_8.ma".
-include "basic_2/reduction/fpb.ma".
-
-(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
-
-inductive ysc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| ysc_fsup : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ysc h g G1 L1 T1 G2 L2 T2
-| ysc_cpr : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → ysc h g G1 L1 T1 G1 L1 T2
-| ysc_ssta : ∀T2,l. ⦃G1, L1⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G1, L1⦄ ⊢ T1 •[h, g] T2 → ysc h g G1 L1 T1 G1 L1 T2
-.
-
-interpretation
- "'big tree' proper parallel reduction (closure)"
- 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (ysc h g G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma ysc_fpb: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/2 width=2 by fpb_fsup, fpb_cpr, fpb_ssta/
-qed.
-
-(* Inversion lemmas on "big tree" parallel reduction for closures ***********)
-
-lemma fpb_inv_ysc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
- ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡ L2 & T1 = T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=2 by and3_intro, or_introl, or_intror, ysc_fsup, ysc_ssta/
-#T2 #HT12 elim (term_eq_dec T1 T2) #H destruct
-/4 width=1 by and3_intro, or_introl, or_intror, ysc_cpr/
-qed-.
}
]
[ { "strongly normalizing computation" * } {
- [ "csn_vector ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csn_tstc_vector" + "csn_aaa" * ]
- [ "csn ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csn_alt ( ⦃?,?⦄ ⊢ ⬊⬊*[?,?] ? )" "csn_lift" + "csn_lpx" * ]
+ [ "csx_vector ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_tstc_vector" + "csx_aaa" * ]
+ [ "csx ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_alt ( ⦃?,?⦄ ⊢ ⬊⬊*[?,?] ? )" "csx_lift" + "csx_lpx" * ]
}
]
[ { "\"big tree\" parallel computation" * } {
- [ "ygt ( ⦃?,?,?⦄ >[?,?] ⦃?,?,?⦄ )" "ygt_lift" + "ygt_ygt" * ]
- [ "fpbs ( ? ⊢ ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_fpbs" * ]
+ [ "fpbg ( ⦃?,?,?⦄ >[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fpbg" * ]
+ [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_fpbs" * ]
}
]
[ { "decomposed extended computation" * } {
class "water"
[ { "reduction" * } {
[ { "\"big tree\" parallel reduction" * } {
- [ "fpb ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "ysc ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" * ]
+ [ "fpbc ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbc_lift" * ]
+ [ "fpb ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpb_lift" * ]
}
]
[ { "context-sensitive extended normal forms" * } {
projects / helm.git / commitdiff