(* *)
(**************************************************************************)
+include "Basic_2/grammar/tstc_tstc.ma".
include "Basic_2/reducibility/lcpr_cpr.ma".
-include "Basic_2/computation/cprs_lcpr.ma".
-include "Basic_2/computation/csn_cprs.ma".
+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".
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
(* Advanced properties ******************************************************)
-lemma csn_lcpr_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
+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
@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 @(csn_ind … 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_cprs … HT) -T #T #HT #IHT
@csn_intro #X #H1 #H2
elim (cpr_inv_abbr1 … H1) -H1 *
[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1
lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
- elim (eq_false_inv_tpair … H2) -H2
+ elim (eq_false_inv_tpair_sn … H2) -H2
[ #HV1 @IHV // /2 width=1/ -HV1
- @(csn_lcpr_trans (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
| -IHV -HLV1 * #H destruct /3 width=1/
]
| -IHV -IHT -H2 #T0 #HT0 #HLT0
lapply (csn_inv_lift … HT … HT0) -HT /2 width=3/
]
qed.
+
+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
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) #HT -HVT
+@csn_intro #X #H #H2
+elim (cpr_inv_appl1 … H) -H *
+[ #V0 #Y #HLV0 #H #H0 destruct
+ elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_beta … H2) -H2
+ [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+ | -IHW -HLW0 -HV -HT * #H #HVT0 destruct
+ @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/
+ ]
+| -IHW -IHVT -H2 #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct
+ lapply (lcpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+| -IHW -IHVT -HV -HT -H2 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: sn3_beta *)
+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
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
+@csn_intro #X #HL #H
+elim (cpr_inv_appl1 … HL) -HL *
+[ -HV #V0 #Y #HLV10 #HL #H0 destruct
+ elim (cpr_inv_abbr1 … HL) -HL *
+ [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct
+ lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34
+ elim (lift_total V0 0 1) #V5 #HV05
+ elim (term_eq_dec (ⓓV.ⓐV2.T) (ⓓV4.ⓐV5.T3))
+ [ -IHVT #H0 destruct
+ elim (eq_false_inv_tpair_sn … H) -H
+ [ -HLV10 -HLV34 -HV3 -HLT3 -HVT
+ >(lift_inj … HV12 … HV05) -V5
+ #H elim (H ?) //
+ | * #_ #H elim (H ?) //
+ ]
+ | -H -HVT #H
+ lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25
+ lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3
+ @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/
+ ]
+ | -H -IHVT #T0 #HT0 #HLT0
+ @(csn_cpr_trans … (ⓐV1.T0)) /2 width=1/ -V0 -Y
+ @(csn_inv_lift … 0 1 HVT) /2 width=1/
+ ]
+| -HV -HV12 -HVT -IHVT -H #V0 #W0 #T0 #T1 #_ #_ #H destruct
+| -IHVT -H #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct
+ lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
+ lapply (lcpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓW0)) /2 width=1/ -W1
+ @(csn_cprs_trans … HVT) -HVT /2 width=1/
+]
+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.
+#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 //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (eq_false_inv_tpair_sn … H) -H
+[ -IHT1 #HV0
+ @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
+ @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
+ #T2 #HLT12 #HT12
+ @(csn_cpr_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.