sn3/props sn3_change
sn3/props sn3_gen_def
sn3/props sn3_cdelta
-sn3/props sn3_appl_appls
-sn3/props sn3_appls_lref
sn3/props sns3_lifts
sty0/fwd sty0_gen_sort
(* *)
(**************************************************************************)
-include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/cnf.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
qed-.
+lemma cprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍[T] → T = U.
+#L #T #U #H @(cprs_ind_dx … H) -T //
+#T0 #T #H1T0 #_ #IHT #H2T0
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+qed-.
+
(* Basic_1: removed theorems 6:
clear_pr3_trans pr3_cflat pr3_gen_bind
pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind
(* Forward lemmas involving same top term constructor ***********************)
+lemma cprs_fwd_cnf: ∀L,T. L ⊢ 𝐍[T] → ∀U. L ⊢ T ➡* U → T ≃ U.
+#L #T #HT #U #H
+>(cprs_inv_cnf1 … H HT) -L -T //
+qed-.
+
(* Basic_1: was: pr3_iso_beta *)
lemma cprs_fwd_beta: ∀L,V,W,T,U. L ⊢ ⓐV. ⓛW. T ➡* U →
ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⓓV. T ➡* U.
(* Vector form of forward lemmas involving same top term constructor ********)
+lemma cprs_fwd_cnf_vector: ∀L,T. 𝐒[T] → L ⊢ 𝐍[T] → ∀Vs,U. L ⊢ ⒶVs.T ➡* U → ⒶVs.T ≃ U.
+#L #T #H1T #H2T #Vs elim Vs -Vs [ @(cprs_fwd_cnf … H2T) ] (**) (* /2 width=3 by cprs_fwd_cnf/ does not work *)
+#V #Vs #IHVs #U #H
+elim (cprs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
+| #V0 #W0 #T0 #HV0 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+| #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+]
+qed-.
+
(* Basic_1: was: pr3_iso_appls_beta *)
lemma cprs_fwd_beta_vector: ∀L,Vs,V,W,T,U. L ⊢ ⒶVs. ⓐV. ⓛW. T ➡* U →
ⒶVs. ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⒶVs. ⓓV. T ➡* U.
lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
/2 width=5/ qed-.
-(* Basic_1: removed theorems 9:
+(* Basic_1: removed theorems 10:
sn3_gen_cflat sn3_cflat
sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
- sn3_bind sn3_appl_bind sn3_appls_bind
+ sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
*)
(**************************************************************************)
include "basic_2/computation/acp_aaa.ma".
-include "basic_2/computation/csn_lcpr_vector.ma".
+include "basic_2/computation/csn_tstc_vector.ma".
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/acp_cr.ma".
-include "basic_2/computation/cprs_tstc_vector.ma".
-include "basic_2/computation/csn_lcpr.ma".
-include "basic_2/computation/csn_vector.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
-
-(* Advanced properties ******************************************************)
-(*
-(* Basic_1: was only: sn3_appl_appls *)
-lemma csn_appl_appls_simple_tstc: ∀L,Vs,V,T1. L ⊢ ⬇* V → L ⊢ ⬇* T1 →
- (∀T2. L ⊢ ⒶVs.T1 ➡* T2 → (ⒶVs.T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
- 𝐒[T1] → L ⊢ ⬇* ⓐV. ⒶVs. T1.
-#L *
-[ #V #T1 #HV
- @csn_appl_simple_tstc //
-| #V0 #Vs #V #T1 #HV #H1T1 #H2T1 #H3T1
- @csn_appl_simple_tstc // -HV
- [ @H2T1
-]
-qed.
-*)
-(* Basic_1: was: sn3_appls_beta *)
-lemma csn_applv_beta: ∀L,W. L ⊢ ⬇* W →
- ∀Vs,V,T. L ⊢ ⬇* ⒶVs.ⓓV.T →
- L ⊢ ⬇* ⒶVs. ⓐV.ⓛW. T.
-#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
-#V0 #Vs #IHV #V #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 (cprs_fwd_beta_vector … H) -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
-| -H1T elim Vs -Vs //
-]
-qed.
-
-lemma csn_applv_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
- ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
- ∀Vs.L ⊢ ⬇* (ⒶVs. V2) → L ⊢ ⬇* (ⒶVs. #i).
-#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=4/
-| #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 (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
- | -L -K -V -V1 -V2 elim Vs -Vs //
- ]
-]
-qed.
-
-(* Basic_1: was: sn3_appls_abbr *)
-lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T. L ⊢ ⬇* ⓓV. ⒶV2s. T → L ⊢ ⬇* V →
- L ⊢ ⬇* ⒶV1s. ⓓV. T.
-#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 #HV
-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 -HV -H1 #X #H1 #H2
-elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
-[ -H #H elim (H2 ?) -H2 //
-| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
-]
-qed.
-
-(* Basic_1: was: sn3_appls_cast *)
-lemma csn_applv_tau: ∀L,W. L ⊢ ⬇* W →
- ∀Vs,T. L ⊢ ⬇* ⒶVs. T →
- L ⊢ ⬇* ⒶVs. ⓣW. T.
-#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
-#V #Vs #IHV #T #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 (cprs_fwd_tau_vector … H) -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
-| -H1T elim Vs -Vs //
-]
-qed.
-(*
-theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
-@mk_acr //
-[
-| /2 width=1/
-| /2 width=6/
-| #L #V1 #V2 #HV12 #V #T #H #HVT
- @(csn_applv_theta … HV12) -HV12 //
- @(csn_abbr) //
-| /2 width=1/
-| @csn_lift
-]
-qed.
-*)
-axiom csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/acp_cr.ma".
+include "basic_2/computation/cprs_tstc_vector.ma".
+include "basic_2/computation/csn_lcpr.ma".
+include "basic_2/computation/csn_vector.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: sn3_appls_lref *)
+lemma csn_applv_cnf: ∀L,T. 𝐒[T] → L ⊢ 𝐍[T] →
+ ∀Vs. L ⊢ ⬇* Vs → L ⊢ ⬇* ⒶVs.T.
+#L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(csn_cnf … 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 (cprs_fwd_cnf_vector … H) -H // -H1T -H2T #H
+ elim (H0 ?) -H0 //
+| -L -V elim Vs -Vs //
+]
+qed.
+
+(* Basic_1: was: sn3_appls_beta *)
+lemma csn_applv_beta: ∀L,W. L ⊢ ⬇* W →
+ ∀Vs,V,T. L ⊢ ⬇* ⒶVs.ⓓV.T →
+ L ⊢ ⬇* ⒶVs. ⓐV.ⓛW. T.
+#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
+#V0 #Vs #IHV #V #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 (cprs_fwd_beta_vector … H) -H #H
+ [ -H1T elim (H0 ?) -H0 //
+ | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
+ ]
+| -H1T elim Vs -Vs //
+]
+qed.
+
+lemma csn_applv_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
+ ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
+ ∀Vs.L ⊢ ⬇* (ⒶVs. V2) → L ⊢ ⬇* (ⒶVs. #i).
+#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=4/
+| #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 (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
+ [ -H1T elim (H0 ?) -H0 //
+ | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
+ ]
+ | -L -K -V -V1 -V2 elim Vs -Vs //
+ ]
+]
+qed.
+
+(* Basic_1: was: sn3_appls_abbr *)
+lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T. L ⊢ ⬇* ⓓV. ⒶV2s. T → L ⊢ ⬇* V →
+ L ⊢ ⬇* ⒶV1s. ⓓV. T.
+#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 #HV
+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 -HV -H1 #X #H1 #H2
+elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
+[ -H #H elim (H2 ?) -H2 //
+| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
+]
+qed.
+
+(* Basic_1: was: sn3_appls_cast *)
+lemma csn_applv_tau: ∀L,W. L ⊢ ⬇* W →
+ ∀Vs,T. L ⊢ ⬇* ⒶVs. T →
+ L ⊢ ⬇* ⒶVs. ⓣW. T.
+#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
+#V #Vs #IHV #T #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 (cprs_fwd_tau_vector … H) -H #H
+ [ -H1T elim (H0 ?) -H0 //
+ | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
+ ]
+| -H1T elim Vs -Vs //
+]
+qed.
+
+theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
+@mk_acr //
+[ /3 width=1/
+| /2 width=1/
+| /2 width=6/
+| #L #V1 #V2 #HV12 #V #T #H #HVT
+ @(csn_applv_theta … HV12) -HV12 //
+ @(csn_abbr) //
+| /2 width=1/
+| @csn_lift
+]
+qed.
"context-sensitive strong normalization (term vector)"
'SN L Ts = (csnv L Ts).
+(* Basic properties *********************************************************)
+
+lemma all_csnv: ∀L,Vs. all … (csn L) Vs → L ⊢ ⬇* Vs.
+#L #Vs elim Vs -Vs //
+#V #Vs #IHVs * /3 width=1/
+qed.
+
(* Basic inversion lemmas ***************************************************)
fact csnv_inv_cons_aux: ∀L,Ts. L ⊢ ⬇* Ts → ∀U,Us. Ts = U :: Us →
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