(**************************************************************************)
include "basic_2/grammar/tstc_vector.ma".
-include "basic_2/relocation/lift_vector.ma".
+include "basic_2/substitution/lift_vector.ma".
include "basic_2/computation/cpxs_tstc.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
(* Vector form of forward lemmas involving same top term constructor ********)
-(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
- â\88\80Vs,U. â¦\83G, Lâ¦\84 â\8a¢ â\92¶Vs.T â\9e¡*[h, g] U â\86\92 â\92¶Vs.T â\89\83 U.
+(* Basic_1: was just: nf2_iso_applv_lref *)
+lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
+ â\88\80Vs,U. â¦\83G, Lâ¦\84 â\8a¢ â\92¶Vs.T â\9e¡*[h, g] U â\86\92 â\92¶Vs.T â\89\82 U.
#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
#V #Vs #IHVs #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair/
| #a #W0 #T0 #HT0 #HU
lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (tstc_inv_bind_appls_simple … HT0) //
+ elim (tstc_inv_bind_applv_simple … HT0) //
| #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (tstc_inv_bind_appls_simple … HT0) //
+ elim (tstc_inv_bind_applv_simple … HT0) //
]
qed-.
lemma cpxs_fwd_sort_vector: ∀h,g,G,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
- â\92¶Vs.â\8b\86k â\89\83 U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
+ â\92¶Vs.â\8b\86k â\89\82 U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
#h #g #G #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
#V #Vs #IHVs #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
| #a #W1 #T1 #HT1 #HU
elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tstc_inv_bind_appls_simple … HT1) //
+ [ elim (tstc_inv_bind_applv_simple … HT1) //
| @or_intror (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) [ /2 width=1/ ] -Vs /3 width=1/
+ @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
]
| #a #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tstc_inv_bind_appls_simple … HT1) //
+ [ elim (tstc_inv_bind_applv_simple … HT1) //
| @or_intror (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) [ /2 width=1/ ] -Vs /3 width=3/
+ @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
]
]
qed-.
-(* Basic_1: was just: pr3_iso_appls_beta *)
+(* Basic_1: was just: pr3_iso_applv_beta *)
lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
- â\92¶Vs. â\93\90V. â\93\9b{a}W. T â\89\83 U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
+ â\92¶Vs. â\93\90V. â\93\9b{a}W. T â\89\82 U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
#V0 #Vs #IHVs #V #W #T #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
| #b #W1 #T1 #HT1 #HU
elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tstc_inv_bind_appls_simple … HT1) //
+ [ elim (tstc_inv_bind_applv_simple … HT1) //
| @or_intror (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓛ{b}W1.T1)) [ /2 width=1/ ] -V -Vs -T /3 width=1/
+ @(cpxs_strap1 … (ⓐV0.ⓛ{b}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
]
| #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (tstc_inv_bind_appls_simple … HT1) //
+ [ elim (tstc_inv_bind_applv_simple … HT1) //
| @or_intror (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) [ /2 width=1/ ] -V -V0 -Vs -T /3 width=3/
+ @(cpxs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
]
]
qed-.
-lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⇩[i] L ≡ K.ⓑ{I}V1 →
∀V2. ⇧[0, i + 1] V1 ≡ V2 →
∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
- â\92¶Vs.#i â\89\83 U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
+ â\92¶Vs.#i â\89\82 U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
#V #Vs #IHVs #U #H -K -V1
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0) //
+ [ elim (tstc_inv_bind_applv_simple … HT0) //
| @or_intror -i (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) [ /2 width=1/ ] -V2 -Vs /3 width=1/
+ @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
]
| #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0) //
+ [ elim (tstc_inv_bind_applv_simple … HT0) //
| @or_intror -i (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) [ /2 width=1/ ] -V -V2 -Vs /3 width=3/
+ @(cpxs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
]
]
qed-.
-(* Basic_1: was just: pr3_iso_appls_abbr *)
+(* Basic_1: was just: pr3_iso_applv_abbr *)
lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
- â\92¶V1s. â\93\93{a}V. T â\89\83 U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
-#h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1/
+ â\92¶V1s. â\93\93{a}V. T â\89\82 U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
+#h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1 by or_intror/
#V1s #V2s #V1a #V2a #HV12a #HV12s #a
generalize in match HV12a; -HV12a
generalize in match V2a; -V2a
elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1/
+[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
[ -HV12a -HV12b -HU
[ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
| -V1b #X #HT1 #H #H0 destruct
elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1/ ] -T -V2b -V2s
- @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0)) [2: /2 width=1/ ]
- /4 width=7 by cpx_zeta, lift_bind, lift_flat/ (* auto too slow without trace *)
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+ @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0))
+ /4 width=7 by cpxs_beta_dx, cpx_zeta, lift_bind, lift_flat/
]
]
| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
@(cpxs_trans … HU) -U
elim (cpxs_inv_abbr1 … HT0) -HT0 *
[ #V1 #T1 #HV1 #HT1 #H destruct
- lapply (cpxs_lift … HV10a (L.ⓓV) … HV12a … HV0a) -V1a -V0a [ /2 width=1/ ] #HV2a
- @(cpxs_trans … (ⓓ{a}V.ⓐV2a.T1)) [ /3 width=1/ ] -T -V2b -V2s /3 width=1/
+ lapply (cpxs_lift … HV10a (L.ⓓV) (Ⓕ) … HV12a … HV0a) -V1a -V0a [ /2 width=1 by drop_drop/ ] #HV2a
+ @(cpxs_trans … (ⓓ{a}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
| #X #HT1 #H #H0 destruct
elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
- lapply (cpxs_lift … HV10a (L.ⓓV0) … HV12a … HV0a) -V0a [ /2 width=1/ ] #HV2a
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1/ ] -T -V2b -V2s
- @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1 (* auto too slow without trace *)
- @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) [ /3 width=3/ ] -V1a /3 width=1/
+ lapply (cpxs_lift … HV10a (L.ⓓV0) (Ⓕ) … HV12a … HV0a) -V0a [ /2 width=1 by drop_drop/ ] #HV2a
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+ @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1
+ @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
]
]
]
qed-.
-(* Basic_1: was just: pr3_iso_appls_cast *)
+(* Basic_1: was just: pr3_iso_applv_cast *)
lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
- â\88¨â\88¨ â\92¶Vs. â\93\9dW. T â\89\83 U
+ â\88¨â\88¨ â\92¶Vs. â\93\9dW. T â\89\82 U
| ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
| ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
#h #g #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
#V #Vs #IHVs #W #T #U #H
elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #b #W0 #T0 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0) //
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or3_intro0/
+| #b #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (tstc_inv_bind_applv_simple … HT0) //
| @or3_intro1 -W (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1/
+ @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
| @or3_intro2 -T (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1/
+ @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
]
| #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0) //
+ [ elim (tstc_inv_bind_applv_simple … HT0) //
| @or3_intro1 -W (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -T /2 width=3/
+ @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
| @or3_intro2 -T (**) (* explicit constructor *)
@(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) [ /2 width=1/ ] -V -Vs -W /2 width=3/
+ @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
]
]
qed-.
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