(* *)
(**************************************************************************)
-include "basic_2/syntax/tsts.ma".
+include "basic_2/syntax/tsts_tdeq.ma".
+include "basic_2/rt_computation/cpxs_lsubr.ma".
+include "basic_2/rt_computation/cpxs_cnx.ma".
include "basic_2/rt_computation/lfpxs_cpxs.ma".
(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
(* Forward lemmas with same top term structure ******************************)
-(*
-lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ⬈*[h, o] U → T ≂ U.
-#h #o #G #L #T #HT #U #H
->(cpxs_inv_cnx1 … H HT) -G -L -T //
-qed-.
-*)
-lemma cpxs_fwd_sort: ∀h,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
- ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
-#h #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
+
+lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
+ ⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
+#h #o #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
[ #H destruct /2 width=1 by or_introl/
| #n #H destruct
@or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
qed-.
(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
- â\93\90V.â\93\9b{p}W.T â\89\82 U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
-#h #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
+lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
+ â\93\90V.â\93\9b{p}W.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
+#h #o #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
qed-.
(* Note: probably this is an inversion lemma *)
-lemma cpxs_fwd_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
- ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
- ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h, o] U →
- #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h, o] U.
+(* Basic_2A1: was: cpxs_fwd_delta *)
+lemma cpxs_fwd_delta_drops: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[⫯i] V1 ≡ V2 →
+ ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h] U →
+ #i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h] U.
#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
-elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
+elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
-lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
-/4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
+lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
+elim (cpxs_lifts … HVU0 (Ⓣ) … L … HV12) -HVU0 -HV12 /2 width=3 by drops_isuni_fwd_drop2/ #X #H
+<(lifts_mono … HU0 … H) -U0 -X /2 width=1 by or_intror/
qed-.
-lemma cpxs_fwd_theta: ∀h,o,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ⬈*[h, o] U →
- ∀V2. ⬆[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
- ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ⬈*[h, o] U.
-#h #o #a #G #L #V1 #V #T #U #H #V2 #HV12
+lemma cpxs_fwd_theta: ∀h,o,p,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] U →
+ ∀V2. ⬆*[1] V1 ≡ V2 → ⓐV1.ⓓ{p}V.T ⩳[h, o] U ∨
+ ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U.
+#h #o #p #G #L #V1 #V #T #U #H #V2 #HV12
elim (cpxs_inv_appl1 … H) -H *
[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
-| #b #W #T0 #HT0 #HU
+| #q #W #T0 #HT0 #HU
elim (cpxs_inv_abbr1 … HT0) -HT0 *
[ #V3 #T3 #_ #_ #H destruct
| #X #HT2 #H #H0 destruct
- elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
+ elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
@or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
- /4 width=7 by cpx_zeta, lift_bind, lift_flat/
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
+ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
]
-| #b #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
+| #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
@or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
elim (cpxs_inv_abbr1 … HT0) -HT0 *
[ #V5 #T5 #HV5 #HT5 #H destruct
- lapply (cpxs_lift … HV13 (L.ⓓV) … HV12 … HV34) -V1 -V3
- /3 width=2 by cpxs_flat, cpxs_bind, drop_drop/
+ elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV) … HV12) -V1 /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV34 … H) -V3 -X /3 width=1 by cpxs_flat, cpxs_bind/
| #X #HT1 #H #H0 destruct
- elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
- lapply (cpxs_lift … HV13 (L.ⓓV0) … HV12 … HV34) -V3 /2 width=2 by drop_drop/ #HV24
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V5 -T5
- @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
+ elim (lifts_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
+ elim (cpxs_lifts … HV13 (Ⓣ) … (L.ⓓV0) … HV12) -HV13 /3 width=1 by drops_refl, drops_drop/ #X #H
+ <(lifts_mono … HV34 … H) -V3 -X #HV24
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{q}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V5 -T5
+ @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
]
]
qed-.
-lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h, o] U →
- â\88¨â\88¨ â\93\9dW. T â\89\82 U | â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] U | â¦\83G, Lâ¦\84 â\8a¢ W â¬\88*[h, o] U.
+lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] U →
+ â\88¨â\88¨ â\93\9dW. T ⩳[h, o] U | â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h] U | â¦\83G, Lâ¦\84 â\8a¢ W â¬\88*[h] U.
#h #o #G #L #W #T #U #H
elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
#W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
qed-.
+
+lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
+ ∀U. ⦃G, L⦄ ⊢ T ⬈*[h] U → T ⩳[h, o] U.
+/3 width=4 by cpxs_inv_cnx1, tdeq_tsts/ qed-.
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