include "basic_2/notation/relations/predtystar_5.ma".
include "basic_2/rt_transition/cpx.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
definition cpxs: sh → relation4 genv lenv term term ≝
- λh,G. LTC … (cpx h G).
+ λh,G. CTC … (cpx h G).
-interpretation "uncounted context-sensitive parallel rt-computation (term)"
+interpretation "unbound context-sensitive parallel rt-computation (term)"
'PRedTyStar h G L T1 T2 = (cpxs h G L T1 T2).
(* Basic eliminators ********************************************************)
-lemma cpxs_ind: ∀h,G,L,T1. ∀R:predicate term. R T1 →
- (â\88\80T,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h] T2 â\86\92 R T â\86\92 R T2) →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 R T2.
-#h #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
+lemma cpxs_ind: ∀h,G,L,T1. ∀Q:predicate term. Q T1 →
+ (â\88\80T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88*[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â¬\88[h] T2 â\86\92 Q T â\86\92 Q T2) →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88*[h] T2 â\86\92 Q T2.
+#h #L #G #T1 #Q #HT1 #IHT1 #T2 #HT12
@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
-lemma cpxs_ind_dx: ∀h,G,L,T2. ∀R:predicate term. R T2 →
- (â\88\80T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h] T2 â\86\92 R T â\86\92 R T1) →
- â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 R T1.
-#h #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
+lemma cpxs_ind_dx: ∀h,G,L,T2. ∀Q:predicate term. Q T2 →
+ (â\88\80T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â¬\88*[h] T2 â\86\92 Q T â\86\92 Q T1) →
+ â\88\80T1. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88*[h] T2 â\86\92 Q T1.
+#h #G #L #T2 #Q #HT2 #IHT2 #T1 #HT12
@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
-lemma cpxs_refl: â\88\80h,G,L,T. â¦\83G, Lâ¦\84 ⊢ T ⬈*[h] T.
+lemma cpxs_refl: â\88\80h,G,L,T. â\9dªG,Lâ\9d« ⊢ T ⬈*[h] T.
/2 width=1 by inj/ qed.
-lemma cpx_cpxs: â\88\80h,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2.
+lemma cpx_cpxs: â\88\80h,G,L,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2.
/2 width=1 by inj/ qed.
-lemma cpxs_strap1: â\88\80h,G,L,T1,T. â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap1: â\88\80h,G,L,T1,T. â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â¬\88[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by step/ qed-.
-lemma cpxs_strap2: â\88\80h,G,L,T1,T. â¦\83G, Lâ¦\84 ⊢ T1 ⬈[h] T →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap2: â\88\80h,G,L,T1,T. â\9dªG,Lâ\9d« ⊢ T1 ⬈[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â¬\88*[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by TC_strap/ qed-.
(* Basic_2A1: was just: cpxs_sort *)
-lemma cpxs_sort: â\88\80h,G,L,s,n. â¦\83G, Lâ¦\84 ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
+lemma cpxs_sort: â\88\80h,G,L,s,n. â\9dªG,Lâ\9d« ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
#h #G #L #s #n elim n -n /2 width=1 by cpx_cpxs/
#n >iter_S /2 width=3 by cpxs_strap1/
qed.
-lemma cpxs_bind_dx: â\88\80h,G,L,V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ⬈[h] V2 →
- â\88\80I,T1,T2. â¦\83G, L. â\93\91{I}V1â¦\84 ⊢ T1 ⬈*[h] T2 →
- â\88\80p. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\88*[h] â\93\91{p,I}V2.T2.
+lemma cpxs_bind_dx: â\88\80h,G,L,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] V2 →
+ â\88\80I,T1,T2. â\9dªG,L. â\93\91[I]V1â\9d« ⊢ T1 ⬈*[h] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â¬\88*[h] â\93\91[p,I]V2.T2.
#h #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
qed.
-lemma cpxs_flat_dx: â\88\80h,G,L,V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ⬈[h] V2 →
- â\88\80T1,T2. â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2 →
- â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88*[h] â\93\95{I}V2.T2.
+lemma cpxs_flat_dx: â\88\80h,G,L,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈[h] V2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2 →
+ â\88\80I. â\9dªG,Lâ\9d« â\8a¢ â\93\95[I]V1.T1 â¬\88*[h] â\93\95[I]V2.T2.
#h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_flat_sn: â\88\80h,G,L,T1,T2. â¦\83G, Lâ¦\84 ⊢ T1 ⬈[h] T2 →
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ⬈*[h] V2 →
- â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88*[h] â\93\95{I}V2.T2.
+lemma cpxs_flat_sn: â\88\80h,G,L,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈[h] T2 →
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈*[h] V2 →
+ â\88\80I. â\9dªG,Lâ\9d« â\8a¢ â\93\95[I]V1.T1 â¬\88*[h] â\93\95[I]V2.T2.
#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_pair_sn: â\88\80h,I,G,L,V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ⬈*[h] V2 →
- â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â¬\88*[h] â\91¡{I}V2.T.
+lemma cpxs_pair_sn: â\88\80h,I,G,L,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈*[h] V2 →
+ â\88\80T. â\9dªG,Lâ\9d« â\8a¢ â\91¡[I]V1.T â¬\88*[h] â\91¡[I]V2.T.
#h #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_pair_sn/
qed.
-lemma cpxs_zeta: ∀h,G,L,V,T1,T,T2. ⬆*[1] T2 ≡ T →
- ⦃G, L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
-#h #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_zeta (h) (G) (L) (V):
+ ∀T1,T. ⇧*[1] T ≘ T1 →
+ ∀T2. ❪G,L❫ ⊢ T ⬈*[h] T2 → ❪G,L❫ ⊢ +ⓓV.T1 ⬈*[h] T2.
+#h #G #L #V #T1 #T #HT1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_zeta/
+qed.
+
+(* Basic_2A1: was: cpxs_zeta *)
+lemma cpxs_zeta_dx (h) (G) (L) (V):
+ ∀T2,T. ⇧*[1] T2 ≘ T →
+ ∀T1. ❪G,L.ⓓV❫ ⊢ T1 ⬈*[h] T → ❪G,L❫ ⊢ +ⓓV.T1 ⬈*[h] T2.
+#h #G #L #V #T2 #T #HT2 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
qed.
-lemma cpxs_eps: â\88\80h,G,L,T1,T2. â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2 →
- â\88\80V. â¦\83G, Lâ¦\84 ⊢ ⓝV.T1 ⬈*[h] T2.
+lemma cpxs_eps: â\88\80h,G,L,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2 →
+ â\88\80V. â\9dªG,Lâ\9d« ⊢ ⓝV.T1 ⬈*[h] T2.
#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
qed.
(* Basic_2A1: was: cpxs_ct *)
-lemma cpxs_ee: â\88\80h,G,L,V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ⬈*[h] V2 →
- â\88\80T. â¦\83G, Lâ¦\84 ⊢ ⓝV1.T ⬈*[h] V2.
+lemma cpxs_ee: â\88\80h,G,L,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈*[h] V2 →
+ â\88\80T. â\9dªG,Lâ\9d« ⊢ ⓝV1.T ⬈*[h] V2.
#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ee/
qed.
lemma cpxs_beta_dx: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ W1 ⬈[h] W2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â¬\88*[h] â\93\93{p}ⓝW2.V2.T2.
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â¬\88*[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ W1 ⬈[h] W2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â¬\88*[h] â\93\93[p]ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
qed.
lemma cpxs_theta_dx: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â¬\86*[1] V â\89¡ V2 â\86\92 â¦\83G, L.â\93\93W1â¦\84 ⊢ T1 ⬈*[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h] W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â¬\88*[h] â\93\93{p}W2.ⓐV2.T2.
-#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
+ â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[h] V â\86\92 â\87§*[1] V â\89\98 V2 â\86\92 â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈*[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[h] W2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â¬\88*[h] â\93\93[p]W2.ⓐV2.T2.
+#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: wa just: cpxs_inv_sort1 *)
-lemma cpxs_inv_sort1: â\88\80h,G,L,X2,s. â¦\83G, Lâ¦\84 ⊢ ⋆s ⬈*[h] X2 →
+lemma cpxs_inv_sort1: â\88\80h,G,L,X2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ⬈*[h] X2 →
∃n. X2 = ⋆((next h)^n s).
#h #G #L #X2 #s #H @(cpxs_ind … H) -X2 /2 width=2 by ex_intro/
#X #X2 #_ #HX2 * #n #H destruct
elim (cpx_inv_sort1 … HX2) -HX2 #H destruct /2 width=2 by ex_intro/
-@(ex_intro â\80¦ (⫯n)) >iter_S //
+@(ex_intro â\80¦ (â\86\91n)) >iter_S //
qed-.
-lemma cpxs_inv_cast1: â\88\80h,G,L,W1,T1,U2. â¦\83G, Lâ¦\84 ⊢ ⓝW1.T1 ⬈*[h] U2 →
- â\88¨â\88¨ â\88\83â\88\83W2,T2. â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h] W2 & â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
- | â¦\83G, Lâ¦\84 ⊢ T1 ⬈*[h] U2
- | â¦\83G, Lâ¦\84 ⊢ W1 ⬈*[h] U2.
+lemma cpxs_inv_cast1: â\88\80h,G,L,W1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓝW1.T1 ⬈*[h] U2 →
+ â\88¨â\88¨ â\88\83â\88\83W2,T2. â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88*[h] W2 & â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
+ | â\9dªG,Lâ\9d« ⊢ T1 ⬈*[h] U2
+ | â\9dªG,Lâ\9d« ⊢ W1 ⬈*[h] U2.
#h #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
#W #T #HW1 #HT1 #H destruct
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