include "basic_2/computation/cprs_cprs.ma".
include "basic_2/computation/csn_lift.ma".
include "basic_2/computation/csn_cpr.ma".
-include "basic_2/computation/csn_cprs.ma".
+include "basic_2/computation/csn_alt.ma".
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
(* Advanced properties ******************************************************)
lemma csn_lcpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
-#L1 #L2 #HL12 #T #H @(csn_ind_cprs … H) -T #T #_ #IHT
+#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/
qed.
lemma csn_abbr: ∀L,V. L ⊢ ⬇* V → ∀T. L. ⓓV ⊢ ⬇* T → L ⊢ ⬇* ⓓV. T.
-#L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_cprs … HT) -T #T #HT #IHT
+#L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT
@csn_intro #X #H1 #H2
elim (cpr_inv_abbr1 … H1) -H1 *
[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
fact csn_appl_beta_aux: ∀L,W. L ⊢ ⬇* W → ∀U. L ⊢ ⬇* U →
∀V,T. U = ⓓV. T → L ⊢ ⬇* ⓐV. ⓛW. T.
-#L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V #T #H destruct
+#L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V #T #H destruct
lapply (csn_fwd_pair_sn … HVT) #HV
lapply (csn_fwd_bind_dx … HVT) #HT -HVT
@csn_intro #X #H #H2
qed.
(* Basic_1: was: sn3_beta *)
-lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* (ⓓV. T) → (**)
+lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* ⓓV. T →
L ⊢ ⬇* ⓐV. ⓛW. T.
/2 width=3/ qed.
fact csn_appl_theta_aux: ∀L,U. L ⊢ ⬇* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
∀V,T. U = ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
-#L #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+#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
]
qed.
-(* Basic_1: was: sn3_appl_bind *)
lemma csn_appl_theta: ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
∀L,V,T. L ⊢ ⬇* ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
/2 width=5/ qed.
(* Basic_1: was only: sn3_appl_appl *)
lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬇* V → ∀T1.
L ⊢ ⬇* T1 →
- (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
- 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
+ (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → ⊥) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒⦃T1⦄ → L ⊢ ⬇* ⓐV. T1.
#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 (cpr_inv_appl1_simple … HL ?) -HL //