(* *)
(**************************************************************************)
-include "Basic_2/grammar/tstc_vector.ma".
-include "Basic_2/substitution/lift_vector.ma".
-include "Basic_2/computation/cprs_tstc.ma".
+include "basic_2/grammar/tstc_vector.ma".
+include "basic_2/substitution/lift_vector.ma".
+include "basic_2/computation/cprs_tstc.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
(* Vector form of forward lemmas involving same top term constructor ********)
-(*
-lemma cpr_fwd_beta_vector: ∀L,V,W,T,U,Vs. L ⊢ ⒶVs. ⓐV. ⓛW. T ➡ U →
- ⒶVs. ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⒶVs. ⓓV. T ➡* U.
-#L #V #W #T #U * /2 width=1 by cpr_fwd_beta/
-#V0 #Vs #H
-elim (cpr_inv_appl1_simple … H ?) -H
-[ #V1 #T1 #_ #_ #H destruct /2 width=1/
-| elim Vs -Vs //
+
+(* Basic_1: was just: nf2_iso_appls_lref *)
+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/
+| #a #V0 #W0 #T0 #HV0 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+| #a #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: ∀a,L,Vs,V,W,T,U. L ⊢ ⒶVs. ⓐV. ⓛ{a}W. T ➡* U →
+ ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ L ⊢ ⒶVs. ⓓ{a}V. T ➡* U.
+#a #L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #U #H
+elim (cprs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
+| #b #V1 #W1 #T1 #HV01 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (tstc_inv_bind_appls_simple … HT1 ?) //
+ | @or_intror -W (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV1.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=1/
+ ]
+| #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (tstc_inv_bind_appls_simple … HT1 ?) //
+ | @or_intror -W (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/
+ ]
+]
+qed-.
+
+lemma cprs_fwd_delta_vector: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
+ ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
+ ∀Vs,U. L ⊢ ⒶVs.#i ➡* U →
+ ⒶVs.#i ≃ U ∨ L ⊢ ⒶVs.V2 ➡* U.
+#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=4 by cprs_fwd_delta/
+#V #Vs #IHVs #U #H -K -V1
+elim (cprs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
+| #b #V0 #W0 #T0 #HV0 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=1/
+ ]
+| #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=3/
+ ]
]
qed-.
-lemma cpr_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡ U →
- ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
+(* Basic_1: was: pr3_iso_appls_abbr *)
+lemma cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀a,V,T,U. L ⊢ ⒶV1s. ⓓ{a}V. T ➡* U →
+ ⒶV1s. ⓓ{a}V. T ≃ U ∨ L ⊢ ⓓ{a}V. ⒶV2s. T ➡* U.
#L #V1s #V2s * -V1s -V2s /3 width=1/
-#V1s #V2s #V1a #V2a #HV12a * -V1s -V2s /2 width=1 by cpr_fwd_theta/ -HV12a
-#V1s #V2s #V1b #V2b #_ #_ #V #U #T #H
-elim (cpr_inv_appl1_simple … H ?) -H //
-#V0 #T0 #_ #_ #H destruct /2 width=1/
+#V1s #V2s #V1a #V2a #HV12a #HV12s #a
+generalize in match HV12a; -HV12a
+generalize in match V2a; -V2a
+generalize in match V1a; -V1a
+elim HV12s -V1s -V2s /2 width=1 by cprs_fwd_theta/
+#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
+elim (cprs_inv_appl1 … H) -H *
+[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/
+| #b #V0a #W0 #T0 #HV10a #HT0 #HU
+ elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
+ [ -HV12a -HV12b -HV10a -HU
+ elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1s (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ elim (cprs_inv_abbr1 … HT0) -HT0 *
+ [ -HV12a -HV12b -HV10a #V1 #T1 #_ #_ #H destruct
+ | -V1b #X #HT1 #H #H0 destruct
+ elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+ @(cprs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s
+ @(cprs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [ /5 width=7/ ] -V -V2a -W1 -T1
+ @(cprs_strap2 … (ⓓ{b}V1a.T0)) [ /3 width=1/ ] -W0 /2 width=1/
+ ]
+ ]
+| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
+ elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
+ [ -HV12a -HV10a -HV0a -HU
+ elim (tstc_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1s -V1b (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ elim (cprs_inv_abbr1 … HT0) -HT0 *
+ [ #V1 #T1 #HV1 #HT1 #H destruct
+ lapply (cprs_lift (L.ⓓV) … HV12a … HV10a … HV0a) -V1a -V0a [ /2 width=1/ ] #HV2a
+ @(cprs_trans … (ⓓ{a}V.ⓐV2a.T1)) [ /3 width=1/ ] -T -V2b -V2s /3 width=1/
+ | #X #HT1 #H #H0 destruct
+ elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
+ lapply (cprs_lift (L.ⓓV0) … HV12a … HV10a … HV0a) -V0a [ /2 width=1/ ] #HV2a
+ @(cprs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s
+ @(cprs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /5 width=7/ ] -V -V1 -T1
+ @(cprs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/
+ ]
+ ]
+]
qed-.
-*)
(* Basic_1: was: pr3_iso_appls_cast *)
-lemma cprs_fwd_tau_vector: â\88\80L,Vs,W,T,U. L â\8a¢ â\92¶Vs. â\93£W. T ➡* U →
- â\92¶Vs. â\93£W. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U.
+lemma cprs_fwd_tau_vector: â\88\80L,Vs,W,T,U. L â\8a¢ â\92¶Vs. â\93\9dW. T ➡* U →
+ â\92¶Vs. â\93\9dW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U.
#L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/
#V #Vs #IHVs #W #T #U #H
elim (cprs_inv_appl1 … H) -H *
[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #V0 #W0 #T0 #HV0 #HT0 #HU
+| #b #V0 #W0 #T0 #HV0 #HT0 #HU
elim (IHVs … HT0) -IHVs -HT0 #HT0
[ elim (tstc_inv_bind_appls_simple … HT0 ?) //
- | @or_intror
- @(cprs_trans … HU) -HU
- @(cprs_strap1 … (ⓐV0.ⓛW0.T0)) /2 width=1/ -HV0 -HT0 /3 width=1/
+ | @or_intror -W (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV0.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=1/
]
-| #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+| #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
elim (IHVs … HT0) -IHVs -HT0 #HT0
[ elim (tstc_inv_bind_appls_simple … HT0 ?) //
- | @or_intror
- @(cprs_trans … HU) -HU
- @(cprs_strap1 … (ⓐV0.ⓓV2.T0)) /2 width=1/ -HV0 -HT0 /3 width=3/
+ | @or_intror -W (**) (* explicit constructor *)
+ @(cprs_trans … HU) -U
+ @(cprs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /3 width=3/
]
qed-.
-
-axiom cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡* U →
- ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
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