+++ /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/cprs_cprs.ma".
-include "basic_2/computation/csn_lift.ma".
-include "basic_2/computation/csn_cpr.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_alt … H) -T #T #_ #IHT
-@csn_intro #T0 #HLT0 #HT0
-@IHT /2 width=2/ -IHT -HT0 /2 width=3/
-qed.
-
-lemma csn_abbr: ∀a,L,V. L ⊢ ⬊* V → ∀T. L. ⓓV ⊢ ⬊* T → L ⊢ ⬊* ⓓ{a}V. T.
-#a #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
- lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1
- lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ #HV1 @IHV // /2 width=1/ -HV1
- @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
- | -IHV -HLV1 * #H destruct /3 width=1/
- ]
-| -IHV -IHT -H2 #T0 #HLT0 #HT0
- lapply (csn_cpr_trans … HT … HLT0) -T #HLT0
- lapply (csn_inv_lift … HLT0 … HT0) -T0 /2 width=3/
-]
-qed.
-
-fact csn_appl_beta_aux: ∀a,L,W. L ⊢ ⬊* W → ∀U. L ⊢ ⬊* U →
- ∀V,T. U = ⓓ{a}V. T → L ⊢ ⬊* ⓐV. ⓛ{a}W. T.
-#a #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
-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 #b #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 #b #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: sn3_beta *)
-lemma csn_appl_beta: ∀a,L,W. L ⊢ ⬊* W → ∀V,T. L ⊢ ⬊* ⓓ{a}V. T →
- L ⊢ ⬊* ⓐV. ⓛ{a}W. T.
-/2 width=3/ qed.
-
-fact csn_appl_theta_aux: ∀a,L,U. L ⊢ ⬊* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀V,T. U = ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}V. T.
-#a #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 (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 (ⓓ{a}V.ⓐV2.T) (ⓓ{a}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 #HLT0 #HT0 #H0 destruct
- lapply (csn_cpr_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_cpr_trans … HVY) /2 width=1/
- ]
-| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #T0 #T1 #_ #_ #H destruct
-| -IHVT -H #b #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.
-
-lemma csn_appl_theta: ∀a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀L,V,T. L ⊢ ⬊* ⓓ{a}V. ⓐV2. T → L ⊢ ⬊* ⓐV1. ⓓ{a}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 → ⊥) → 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.
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