(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predstar_5.ma".
+include "basic_2/notation/relations/predstar_6.ma".
include "basic_2/unfold/sstas.ma".
include "basic_2/reduction/cnx.ma".
include "basic_2/computation/cprs.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-definition cpxs: ∀h. sd h → lenv → relation term ≝
- λh,g. LTC … (cpx h g).
+definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
+ λh,g,G. LTC … (cpx h g G).
interpretation "extended context-sensitive parallel computation (term)"
- 'PRedStar h g L T1 T2 = (cpxs h g L T1 T2).
+ 'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
(* Basic eliminators ********************************************************)
-lemma cpxs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 →
+lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 →
(∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) →
∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2.
-#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12
+#h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
-lemma cpxs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 →
+lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 →
(∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) →
∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1.
-#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12
+#h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
-lemma cpxs_refl: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
+lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
/2 width=1/ qed.
-lemma cpx_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
/2 width=1/ qed.
-lemma cpxs_strap1: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
+lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
normalize /2 width=3/ qed.
-lemma cpxs_strap2: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
normalize /2 width=3/ qed.
-lemma lsubr_cpxs_trans: ∀h,g. lsub_trans … (cpxs h g) lsubr.
+lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
/3 width=5 by lsubr_cpx_trans, TC_lsub_trans/
qed-.
-lemma sstas_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •* [h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 //
+lemma sstas_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •* [h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(sstas_ind … H) -T2 //
/3 width=4 by cpxs_strap1, ssta_cpx/
qed.
-lemma cprs_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
+lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
qed.
-lemma cpxs_bind_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
- ∀I,T1,T2. ⦃h, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+ ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
+#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
/3 width=1/ /3 width=3/
qed.
-lemma cpxs_flat_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h, g] ⓕ{I} V2. T2.
-#h #g #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
qed.
-lemma cpxs_flat_sn: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h, g] ⓕ{I} V2. T2.
-#h #g #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
qed.
-lemma cpxs_zeta: ∀h,g,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
- ⦃h, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
-#h #g #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
+lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
+ ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
+#h #g #G #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
qed.
-lemma cpxs_tau: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
-#h #g #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
+lemma cpxs_tau: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+ ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
qed.
-lemma cpxs_ti: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
-#h #g #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
+lemma cpxs_ti: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+ ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
+#h #g #G #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
qed.
-lemma cpxs_beta_dx: ∀h,g,a,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
+lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
/4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *)
qed.
-lemma cpxs_theta_dx: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
/4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *)
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
+lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
∃∃n,l. deg h g k (n+l) & U2 = ⋆((next h)^n k).
-#h #g #L #U2 #k #H @(cpxs_ind … H) -U2
+#h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2
[ elim (deg_total h g k) #l #Hkl
@(ex2_2_intro … 0 … Hkl) -Hkl //
| #U #U2 #_ #HU2 * #n #l #Hknl #H destruct
]
qed-.
-lemma cpxs_inv_cast1: ∀h,g,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
+lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
| ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
| ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
-#h #g #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
+#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
#U2 #U #_ #HU2 * /3 width=3/ *
#W #T #HW1 #HT1 #H destruct
elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3/ *
lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/
qed-.
-lemma cpxs_inv_cnx1: ∀h,g,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U.
-#h #g #L #T #U #H @(cpxs_ind_dx … H) -T //
+lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U.
+#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
#T0 #T #H1T0 #_ #IHT #H2T0
lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
qed-.
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